Notes for Solid State Theory FFF051/FYST25. Andreas Wacker Matematisk Fysik Lunds Universitet

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1 Notes for Solid State Theory FFF05/FYST5 Andreas Wacker Matematisk Fysik Lunds Universitet Vårtermin 05

2 ii A. Wacker, Lund University: Solid State Theory, VT 05 These notes give a summary of the lecture and present additional material, which may be less accessible by standard text books. They should be studied together with standard text books of solid state physics, such as Snoke (008, Hofmann (008, Ibach and Lüth (003 or Kittel (996, to which is frequently referred. Solid state theory is a large field and thus a 7.5 point course must restrict the material. E.g., important issues such as calculation schemes for the electronic structure or a detailed account of crystal symmetries is not contained in this course. Sections marked with a present additional material on an advanced level, which may be treated very briefly or even skipped. They will not be relevant for the exam. The same holds for footnotes which shall point towards more sophisticated problems. Note that there are two different usages for the symbol e: In these note e > 0 denotes the elementary charge, which consitent with most textbooks (including Snoke (008,Ibach and Lüth (003, and Kittel (996. In contrast sometimes e < 0 denotes the charge of the electron, which I also used in previous versions of these notes. Thus, there may still be some places, where I forgot to change. Please report these together with other misprints and any other suggestion for improvement. I want the thank all former students for helping in improving the text. Any further suggestions as well as reports of misprints are welcome! Special thanks to Rikard Nelander for critical reading and preparing several figures.

3 Bibliography D. W. Snoke, Solid State Physics: Essential Concepts (Addison-Wesley, 008. P. Hofmann, Solid State Physics (Viley-VCH, Weinheim, 008. H. Ibach and H. Lüth, Solid-state physics (Springer, Berlin, 003. C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, New York, 996. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson Learning, 979. G. Czycholl, Festkörperphysik (Springer, Berlin, 004. D. Ferry, Semiconductors (Macmillan Publishing Company, New York, 99. E. Kaxiras, Atomic and Electronic Structure of Solids (Cambridge University Press, Cambridge, 003. C. Kittel, Quantum Theory of Solids (John Wiley & Sons, New York, 987. M. P. Marder, Condensed Matter Physics (John Wiley & Sons, New York, 000. J. R. Schrieffer, Theory of Superconductivity (Perseus, 983. K. Seeger, Semiconductor Physics (Springer, Berlin, 989. P. Y. Yu and M. Cardona, Fundamentals of Semiconductors (Springer, Berlin, 999. C. Kittel and H. Krömer, Thermal Physics (Freeman and Company, San Francisco, 980. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 998, 3rd ed. W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals (Springer, Berlin, 999. iii

4 iv A. Wacker, Lund University: Solid State Theory, VT 05 List of symbols symbol meaning page A(r, t magnetic vector potential a i primitive lattice vector B(r, t magnetic field D(E density of states 7 E F Fermi energy 7 E n (k Energy of Bloch state with band index n and Bloch vector k e elementary charge (positive! F(r, t electric field f(k occupation probability 6 g e Landé factor of the electron 30 g i primitive vector of reciprocal lattice G reciprocal lattice vector H magnetizing field 9 I radiation intensity Eq. (4.0 M Magnetization 9 m e electron mass m n effective mass m eff of band n 0 N Number of unit cells in normalization volume 3 n electron density (or spin density with unit /Volume 7 n refractive index 4 P m,n (k momentum matrix element 0 R lattice vector u nk (r lattice periodic function of Bloch state (n, k V Normalization volume 3 V c volume of unit cell v n (k velocity of Bloch state with band index n and Bloch vector k 0 α absorption coefficient 4 φ(r, t electrical potential µ magnetic dipole moment 9 µ chemical potential 6 µ 0 vacuum permeability 9 µ B Bohr magneton 30 µ k electric dipole moment 44 µ mobility 7 ν number of nearest neighbor sites in the lattice 36 χ magnetic/electric susceptibility 9/39

5 Contents Band structure. Bloch s theorem Derivation of Bloch s theorem by lattice symmetry Born-von Kármán boundary conditions Examples of band structures Plane wave expansion for a weak potential Superposition of localized orbits for bound electrons Density of states and Fermi level Parabolic and isotropic bands General scheme to determine the Fermi level Properties of the band structure and Bloch functions Kramers degeneracy Normalization Velocity and effective mass Envelope functions The effective Hamiltonian Motivation of Eq. ( Heterostructures Transport 5. Semiclassical equation of motion General aspects of electron transport Phonon scattering Scattering Probability Thermalization Boltzmann Equation Electrical conductivity Transport in inhomogeneous systems Diffusion and chemical potential v

6 vi A. Wacker, Lund University: Solid State Theory, VT Thermoelectric effects Details for Phonon quantization and scattering Quantized phonon spectrum Deformation potential interaction with longitudinal acoustic phonons Polar interaction with longitudinal optical phonons Magnetism 9 3. Classical magnetic moments Magnetic susceptibilities from independent electrons Larmor Diamagnetism Paramagnetism by thermal orientation of spins Pauli paramagnetism Ferromagnetism by interaction Many-Particle Schrödinger equation The band model for ferromagnetism Singlet and Triplet states Heisenberg model Spin waves Introduction to dielectric function and semiconductor lasers The dielectric function Kramers-Kronig relation Connection to oscillating fields Interaction with lattice vibrations Interaction with free carriers Optical transitions The semiconductor laser Phenomenological description of gain Threshold current Quantum kinetics of many-particle systems Occupation number formalism Definitions Anti-commutation rules Field operators Operators Temporal evolution of expectation values

7 vii 5.3 Density operator Semiconductor Bloch equation Free carrier gain spectrum Quasi-equilibrium gain spectrum Spectral hole burning Electron-Electron interaction Coulomb effects for interband transitions The Hamiltonian Semiconductor Bloch equations in HF approximation Excitons The Hartree-Fock approximation Proof Application to the Coulomb interaction The free electron gas A brief glimpse of density functional theory The Lindhard-Formula for the dielectric function Derivation Plasmons Static screening Superconductivity 7 7. Phenomenology BCS Theory The Cooper pair The BCS ground state Excitations from the BCS state Electron transport in the BCS state Justification of attractive interaction

8 viii A. Wacker, Lund University: Solid State Theory, VT 05

9 Chapter Band structure. Bloch s theorem Most solid materials (a famous exception is glass show a crystalline structure which exhibits a translation symmetry. The crystal is invariant under translations by all lattice vectors R l = l a + l a + l 3 a 3 (. where l i Z. The set of points associated with the end points of these vectors is called the Bravais lattice. The primitive vectors a i span the Bravais lattice and can be determined by X-ray spectroscopy for each material. The volume of the unit cell is V c = a (a a 3. In order to characterize the energy eigenstates of such a crystal, the following theorem is of utmost importance: Bloch s Theorem: The eigenstates of a lattice-periodic Hamiltonian satisfying Ĥ(r = Ĥ(r + R l for all l i Z can be written as Bloch functions in the form Ψ n,k (r = e ik r u n,k (r (. where k is the Bloch vector and u n,k (r is a lattice-periodic function. An equivalent defining relation for the Bloch functions is Ψ n,k (r + R l = e ik R l Ψn,k (r for all lattice vectors R l (sometimes called Bloch condition. For each Bravais lattice one can construct the corresponding primitive vectors of the reciprocal lattice g i by the relations g i a j = πδ ij. (.3 In analogy to the real lattice, they span the reciprocal lattice with vectors G m = m g +m g + m 3 g 3. More details on the real and reciprocal lattice are found in your textbook. We define the first Brillouin zone by the set of vectors k, satisfying k k G m for all G n, i.e. they are closer to the origin than to any other vector of the reciprocal lattice. Thus the first Brillouin zone is confined by the planes k G m = G m /. Then we can write each vector k as k = k + G m, where k is within the first Brillouin zone and G m is a vector of the reciprocal lattice. (This decomposition is unique unless k is on the boundary of the first Brillouin zone. Then we have Ψ n,k (r = e i k r e ign r u n,k (r = e i k r uñ, k(r = Ψñ, k(r Another rare sort of solid materials with high symmetry are quasi-crystals, which do not have an underlying Bravais lattice. Their discovery in 984 was awarded with the Nobel price in Chemistry 0 nobelprize.org/nobel_prizes/chemistry/laureates/0/sciback_0.pdf

10 A. Wacker, Lund University: Solid State Theory, VT 05 as uñ, k(r is also a lattice-periodic function. Therefore we can restrict our Bloch vectors to the first Brillouin zone without loss of generality. Band structure: For each k belonging to the first Brillouin zone, we have set of eigenstates of the Hamiltonian ĤΨ n,k (r = E n (kψ n,k (r (.4 where E n (k is a continuous function in k for each band index n. Bloch s theorem can be derived by examining the plane wave expansion of arbitrary wave functions and using u n,k (r = a (n,k m e igm r, {m i } see, e.g. chapter 7. of Ibach and Lüth (003 or chapter 7 of Kittel (996. In the subsequent section an alternate proof is given on the basis of the crystal symmetry. The treatment follows essentially chapter 8 of Ashcroft and Mermin (979 and chapter.3 of Snoke ( Derivation of Bloch s theorem by lattice symmetry We define the translation operator ˆT R by its action on arbitrary wave functions Ψ(r by ˆT R Ψ(r = Ψ(r + R where R is an arbitrary lattice vector. We find for arbitrary wave functions: ˆT R ˆTR Ψ(r = ˆT R Ψ(r + R = Ψ(r + R + R = ˆT R+R Ψ(r (.5 As ˆT R+R = ˆT R +R we find the commutation relation for all pairs of lattice vectors R, R. [ ˆT R, ˆT R ] = 0 (.6 Now we investigate the eigenfunctions Ψ α (r of the translation operator, satisfying ˆT R Ψ α (r = c α (RΨ α (r Let us write without loss of generality c α (a i = e πix i for the primitive lattice vectors a i with x i C (it will be shown below that only x i R is of relevance for bulk crystals. From Eqs. (.,.6 we find ˆT Rn Ψ α (r = ˆT n a ˆT n a ˆT n 3 a 3 Ψ α (r = e πi(n x +n x +n 3 x 3 Ψ α (r = e ikα Rn Ψ α (r where k α = x g + x g + x 3 g 3 and Eq. (.3 is used. Now we define u α (r = e ikα r Ψ α (r and find Thus we find: u α (r R n = e ikα (r Rn Ψ α (r R n = e ikα r e ikα Rn Ψ α (r R n = u } {{ } α (r =Ψ α(r If Ψ α (r is eigenfunction to all translation-operators ˆT R of the lattice, it has the form Ψ α (r = e ikα r u α (r (.7

11 Chapter : Band structure 3 where u α (r is a lattice periodic function. The vector k α is called Bloch vector. For infinite crystals we have k α R. This is proven by contradiction. Let, e.g., Im{k α a } = λ > 0. Then we find Ψ α ( na = T na Ψ α (0 = e πnikα a Ψ α (0 = e 4πnλ Ψ α (0 and the wave function diverges for n, i.e. in the direction opposite to a. Thus Ψ α (r has its weight at the boundaries of the crystal but does not contribute in the bulk of the crystal. As the crystal lattice is invariant to translations by lattice vectors R, the Hamiltonian Ĥ(r for the electrons in the crystal satisfies for all lattice vectors R. Thus we find Ĥ(r = Ĥ(r + R ˆT R Ĥ(rΨ(r = Ĥ(r + RΨ(r + R = Ĥ(r ˆT R Ψ(r or ( ˆTR Ĥ(r Ĥ(r ˆT R Ψ(r = 0 } {{ } =[ ˆT R,Ĥ(r] As this holds for arbitrary wave functions we find the commutation relation [ ˆT R, Ĥ(r] = 0. Thus {Ĥ, ˆT R, ˆT R,...} are a set of pairwise commuting operators and quantum mechanics tells us, that there is a complete set of functions Ψ α (r, which are eigenfunctions to each of these operators, i.e. ĤΨ α (r = E α Ψ α (r and ˆTR Ψ α (r = c α (RΨ α (r As Eq. (.7 holds, we may replace the index α by n, k, where k = k α is the Bloch vector and n describes different energy states for a fixed k. This provides us with Bloch s theorem... Born-von Kármán boundary conditions In order to count the Bloch states and obtain normalizable wave functions one can use the following trick. We assume a finite crystal in the shape of a parallelepiped with N N N 3 unit cells. Thus the entire volume is V = N N N 3 V c. For the wave functions we assume periodic boundary conditions Ψ(r+N i a i = Ψ(r for simplicity. For the Bloch functions this requires k N i a i = πn i with n i Z or k = n g + n g + n 3 g 3 N N N 3 Restricting to the first Brillouin zone 3 gives N N N 3 different values for k. Thus, Each band n has within the Brillouin zone as many states as there are unit cells in the crystal (twice as many for spin degeneracy. If all N i become large, the values k become close to each other and we can replace a sum over k by an integral. As the volume of the Brillouin zone is V ol(g, g, g 3 = (π 3 /V c we find that This does not hold close to a surface perpendicular to a. Therefor surface states can be described by a complex Bloch vector k α. 3 If the Brillouin zone is a parallelepiped, we can write N i / < n i < N i /. But typically the Brillouin zone is more complicated.

12 4 A. Wacker, Lund University: Solid State Theory, VT 05 Γ X W L Γ K X 3 E(k [h /(m e a ] 0 [000] [00] [½0] [½½½] [000] [¾¾0] [00] k [π/a] Figure.: Free electron band structure for an fcc crystal together with a sketch of the Brillouin zone (modified file from Wikipedia. The energy scale is /m (π/a. For the lattice constant a = 4.05Å of Al, we obtain 9.7 ev or 0.67 Rydberg. the continuum limit is given by k f(k = V cn N N 3 (π 3 V ol( k, k, k 3 f(k k V (π 3 d 3 kf(k (.8 for arbitrary functions f(k.. Examples of band structures The calculation of the band structure for a crystal is an intricate task and many approximation schemes have been developed, see, e.g., chapter 0 of Marder (000. Here we discuss two simple approximations in order to provide insight into the main features... Plane wave expansion for a weak potential In case of a constant potential U 0 the eigenstates of the Hamiltonian are free particle states, i.e. plan waves e ik r with energy k /m + U 0. These can be written as Bloch states by decomposing k = k + G n, where k is within the first Brillouin zone. Then we find Ψ n k(r = e i k r u n k(r and E n ( k = ( k + G n m + U 0 The corresponding band structure is shown in Fig.. for a fcc lattice. A weak periodic potential will split the degeneracies at crossings (in particular at zone boundaries and at k = 0, thus providing gaps (see exercise. In this way the band structure of many metals such as aluminum can be well understood, see Fig...

13 Chapter : Band structure E f E f Energy [Ry] V c Z Q Λ Σ Γ X W L Γ K V c Figure.: Calculated band structure of aluminum. [After E.C. Snow, Phys. Rev. 58, 683 (967].. Superposition of localized orbits for bound electrons Alternatively, one may start from a set of localized atomic wave functions φ j (r satisfying ] [ m + V A(r φ j (r = E j φ j (r for the atomic potential V A (r of a single unit cell. l V A(r R l and we construct Bloch states as Defining v(r = h 0 V A(r R h we find lj Ψ nk (r = e ik R l c (n,k j φ j (r R l N l,j The total crystal potential is given by ĤΨ nk (r = e ik R l c (n,k [ j Ej φ j (r R l + v(r R l φ j (r R l ]! = E n (kψ nk (r N Taking the scalar product by the operation N d 3 rφ i (r we find E i c (n,k i + d 3 rφ i (rv(rφ j (rc (n,k j + e ik R l d 3 rφ i (r [ E j + v(r R l ] φ j (r R l c (n,k j j l 0,j ( = E n (k c (n,k i + e ik R l d 3 rφ i (rφ j (r R l c (n,k j l 0,j which provides a matrix equation for the coefficients c (n,k i. For a given energy E n (k, the atomic levels E i E n (k dominate, and thus one can restrict oneself to a finite set of levels in the energy region of interest (e.g., the 3s and 3p levels for the conduction and valence band of Si. Restricting to a single atomic S-level and next-neighbor interactions in a simple cubic crystal with lattice constant a we find with A = d 3 rφ S(rv(rφ S (r, B = E(k E S + A + B(cos k xa + cos k y a + cos k z a + C(cos k x a + cos k y a + cos k z a d 3 rφ S(rv(r ae x φ S (r ae x, C = d 3 rφ S(rφ S (r ae x

14 6 A. Wacker, Lund University: Solid State Theory, VT 05 0 Cl - 3p Figure.3: Upper filled bands of KCl as a function of the lattice constant in atomic unit (equal to the Bohr radius a B = 0.59Å [After L.P. Howard, Phys. Rev. 09, 97 (958]. One can clearly see, how the atomic orbitals of the ions broaden to bands with decreasing distance between the ions. Energy [Rydbergs] - Cl - 3s - K + 3p -3 K + 3s -4 a 0 = au Lattice constant [a u] E F = 0 - Energy below E F [ev] Figure.4: Band structure of copper with experimental data. [After -8 R. Courths and S. Hüfner, Physics Reports, 53 (984] Calculation with parameter fit Experimental data L Λ Γ X K Σ Γ Thus the band is essentially of cosine shape, where the band width depends on the overlap B between next-neighbor wave functions. An example is shown in Fig..3 for Potassium chloride (KCl, where the narrow band can be well described by this approach. The outer shell of transition metals exhibits both s and d electrons. Here the wave function of the 3d-electrons does not reach out as far as the 4selectrons (with approximately equal total energy, as a part of the total energy is contained in the angular momentum. Thus bands resulting from the d electrons have a much smaller band width compared to bands resulting from the s electrons, which have essentially the character of free electrons. Taking into account avoided crossings, this results in the band structure for copper (Cu shown in Fig..4. The band structure of Si and GaAs as well as similar materials is dominated by the outer s and p shells of the constituents. This results in four occupied bands (valence bands and four empty bands (conduction bands with a gap of the order of ev between. See Fig..5.

15 Chapter : Band structure 7 Energy [ev] L 4,5 L 6 GaAs Γ 8 Γ 7 L 6 L 4,5 Γ 6 Γ8 L 6 Γ 7 X 7 X 6 X 7 X 6 Γ 8 Γ 7 Γ 6 Γ 8 Γ 7-8 L 6 X 6-0 L 6 X 6 - Γ 6 Γ 6 L Λ Γ X U,K Σ Γ Wave vector k Figure.5: Band structure of GaAs [After J.R. Chelikowsky and M.L. Cohen, Phys. Rev B 4,556 (976] and Si [from Wikipedia Commons, after J.R. Chelikowsky and M.L. Cohen, Phys. Rev B 0,5095 (974].3 Density of states and Fermi level The density of states gives the number of states per volume and energy interval. From Eq. (.8 we obtain For a single band n the density of states is defined by D n (E = d 3 k δ(e E (π 3 n (k.bz The total density of states is then the sum over all bands. The density of states is obviously zero in a band gap, where there are no states. On the other hand, it is particularly large, if the bands are flat as there are plenty of k-states within a small energy interval. Thus, copper has a large density of states in the energy range of 4eV < E < ev, see Fig..4. Bulk crystals cannot exhibit macroscopic space charges. Thus, the electron density n must equal the positive charge density of the ions. As double occupancy of levels is forbidden by the Pauli principle, the low lying energy levels with energies up to the Fermi energy E F are occupied at zero temperature. If the Fermi energy E F is within a band the crystal is a metal exhibiting a high electrical conductivity (see the next chapter. In contrast, if the Fermi energy is located in a band gap, we have a semiconductor (with moderate conductivity which is strongly increasing with temperature, or an insulator (with vanishingly small conductivity. This distinction is not well-defined; semiconductors have typically band gaps of the order of ev, while the band gap is much larger for insulators.

16 8 A. Wacker, Lund University: Solid State Theory, VT Parabolic and isotropic bands For a parabolic isotropic band (e.g. close to the Γ point, or in good approximation for metals we have E n (k = E n + k m eff. Setting E k = k /(m eff, we find the density of states n (E =(for spin (π 3 = π π dϕ 4π 3 parabolic 3D D.Bz dϑ sin θ 0 } 0 {{ } 4π d 3 k δ (E E n E k kmax = m eff meff (E E n Θ(E E π 3 n dk k 0 } {{ } Ek max m 0 de eff meff E k k 3 δ (E E n E k Here Θ(x is the Heaviside function with Θ(x = for x > 0 and 0 for x < 0. In the same spirit we obtain parabolic D Dn (E = (for spin m eff π Θ(E E n (.0 in two dimensions [see Sec..7 of Ibach and Lüth (003 or Sec..7. of Snoke (008]. If the states up to the Fermi energy are occupied, we find in the conventional three dimensional case the electron density n c = EF E c (.9 parabolic 3D dedn (E = [m eff(e F E c ] 3/. (. 3π 3 This an be used to estimate the Fermi energy of metals. E.g., Aluminium has a nuclear charge of 5 protons and two electrons are tightly bound to the nucleus within the s shell. Thus charge neutrality requires 3 free electron per unit cell of volume 6.6 Å 3 (a forth of the cubic cell a 3 for the fcc lattice. Using the free electron mass, this provides E F E c = m e ( /3 3π 3 6.6Å 3 =.7eV in good agreement with more detailed calculations displayed in Fig General scheme to determine the Fermi level For ionic crystals like KCl, it is good to start with the atomic orbitals of the isolated atoms/ions. Here the 3s and 3p states are entirely occupied both for the Cl and the K + ion. Combining these states to bands does not change the occupation. Thus, the resulting bands should all be occupied and the Fermi level is in the gap above the band dominated by the 3p states Cl, see Fig..3. A more general argument is the counting rule derived in Sec.... Here one first determines the number of electrons per unit cell required for the upper bands to achieve charge neutrality. Assuming spin degeneracy, this is twice the number of bands which need to be occupied. E.g., Aluminium requires 3 outer electrons per unit cell and thus.5 bands should be occupied in average. Indeed for any k-point, one or two bands lie below the Fermi energy in Fig... The same argument applies for Cu, where outer electrons per atom (in the 4s and 3d shell require the occupation of 5.5 bands in average, see Fig..4.

17 Chapter : Band structure 9 If there is more than one atom per unit cell, all charges have to be considered together. E.g. silicon crystallizes in the diamond lattice with two atoms per unit cell. As each Si atom has four electrons in the outer 3s/p shell, we need to populate 8 states per unit cell, i.e. 4 bands. Fig..5 shows that these are just the four bands below the gap (all eight bands displayed result from the 3s/p levels, and the Fermi level is in the gap. The same argument holds for GaAs..4 Properties of the band structure and Bloch functions Most of the following properties are given without proof Kramers degeneracy As Ĥ is a hermitian operator and the eigenenergies are real, we find from ĤΨ n,k(r = E n (kψ n,k (r the relation ĤΨ n,k(r = E n (kψ n,k(r Thus Ψ n,k (r = e ik r u n,k (r Ψ n, k(r is an eigenfunction of the Hamilton-operator with Bloch vector k and E n ( k = E n (k. The band structure satisfies the symmetry E n ( k = E n (k. If the band-structure depends on spin, the spin must be flipped as well..4. Normalization The lattice-periodic functions can be chosen such that V c d 3 r u m,k(ru n,k (r = V c δ m,n (. Furthermore they form a complete set of lattice periodic functions. Then the Bloch functions can be normalized in two different ways: For infinite systems we have a continuous spectrum of k values and set ϕ n,k (r = eik r u (π 3/ n,k (r d 3 r ϕ m,k (rϕ n,k(r = δ m,n δ(k k For finite systems of volume V and Born-von Kármán boundary conditions we have a discrete set of k values and set ϕ n,k (r = e ik r u n,k (r d 3 r ϕ m,k (rϕ n,k(r = δ m,n δ k,k V.4.3 Velocity and effective mass The stationary Schrödinger equation for the electron in a crystal reads in spatial representation ( + V (r Ψ n,k (r = E n (kψ n,k (r m e 4 Details can be found in textbooks, such as Snoke (008, Marder (000, Czycholl (004, or Kittel (987. V

18 0 A. Wacker, Lund University: Solid State Theory, VT 05 Inserting the the Bloch functions Ψ n,k (r = e ik r u nk (r can be expressed in terms of the lattice periodic functions u nk (r as E n (ku nk (r = ( k + k m e m e i + V (r u nk (r m e Now we start with a solution u nk0 (r with energy E n (k 0 and investigate small changes δk: E n (k 0 + δku nk0 +δk(r = (Ĥ 0 + k 0 δk + δk m e m e i + δk u nk0 +δk(r m e with Ĥ0 = k 0 m e + m e k 0 i m e + V (r Using the Taylor expansion of E n (k, we can write E n (k 0 + δk = E n (k 0 + E n(k 0 k δk + i,j=x,y,z (.3 m n (k 0 i,j δk i δk j + O(δk 3 (.4 where we defined the effective mass tensor ( = E n (k (.5 m n (k i,j k i k j Now we want to relate the expansion coeficients in Eq. (.4 to physical terms by considering Eq. (.3 in the spirit of perturbation theory. In first order perturbation theory, the change in energy is given by the expectation value of the perturbation δk with the unperturbed state: E n (k 0 + δk =E n (k 0 + u nk0 k 0 + V c m e m e i u nk 0 δk + O(δk =E n (k 0 + m e P n,n (k 0 δk + O(δk with the momentum matrix element P m,n (k = V c d 3 r Ψ m,k (r i Ψ n,k(r V c d 3 r Ψ n,k (r. (.6 Comparing with Eq. (.4 we can identify E n (k k = m e P n,n On the other hand, the quantum-mechanical current density of a Bloch electron is J(r = e { Re Ψ m nk(r } e i Ψ nk(r and P n,n (k/m e = J /e n is just the average velocity in a unit cell. Thus we identify the velocity of the Bloch state v n (k = E n (k k (.7

19 Chapter : Band structure The δk in Eq. (.3 provides together with the second order perturbation theory for the terms δk ( = δ i,j + P n,m;i (kp m,n;j (k. (.8 m n (k i,j m e m e E n (k E m (k m(m n This shows that the effective mass deviates from the bare electron mass m e due to the presence of neighboring bands. 5 We further see, that a small band gap of a semiconductor (e.g. InSb is related to a small effective mass..5 Envelope functions.5. The effective Hamiltonian Now we want to investigate crystals with the lattice potential V (r and additional inhomogeneities, e.g. additional electro-magnetic fields with scalar potential φ(r, t and vector potential A(r, t, which relate to the electric field F and the magnetic induction B via F(r, t = φ(r, t A(r, t t B(r, t = A(r, t (.9 Then the single particle Schrödinger equation reads, see e.g. staff/andreas.wacker/scripts/quantmagnetfield.pdf i [ ] (ˆp + ea(r, t t Ψ(r, t = + V (r eφ(r, t Ψ(r, t (.0 m e where e is the negative charge of the electron. For vanishing fields (i.e. A = 0 and φ = 0 Bloch s theorem provides the band structure E n (k and the eigenstates Ψ nk (r. As these eigenstates fopr a complete set of states, any wave functions Ψ(r, t can be expanded in terms of the Bloch functions. Now we assume that only the components of a single band with index n are of relevance, which is a good approximation if the energetical seperation between the bands is much larger than the terms in the Hamiltonian corresponding to the fields. Thus we can write Ψ(r, t = d 3 k c(k, tψ nk (r (. With the expansion coefficients c(k, t we can construct an envelope function 6 f(r, t = d 3 k c(k, t eik r (. (π 3/ which does not contain the (strongly oscillating lattice periodic functions u nk (r. If A(r and φ(r are constant on the lattice scale (e.g. their Fourier components A(q, φ(q are small unless q g i the envelope functions f(r, t satisfies the equation (to be motivated below i [ ( t f(r, t = E n i + e ] A(r, t eφ(r, t f(r, t (.3 where [E n ( i e A + eφ] is the effective Hamiltonian. Here one replaces the wavevector k in the dispersion relation E n (k by an operator. 5 This is used as a starting point for k p theory, see, e.g., Chow and Koch (999; Yu and Cardona ( This is the Wannier-Slater envelope function, see M.G. Burt, J. Phys.: Cond. Matter, R53 (999 for a wider class of envelope functions.

20 A. Wacker, Lund University: Solid State Theory, VT 05 Close to an extremum in the band structure at k 0 we find with Eq. (.5 i t f(r, t = [ E n (k 0 + ij ( ( + ea i i x i m n (k 0 i,j ( + ea j i x j eφ ] f(r, t (.4 which is called effective mass approximation 7. For crystals with high symmetry the mass tensor is diagonal for k 0, and Eq. (.4 has the form of a Schrödinger equation (.0 with the electron mass replaced by the effective mass. It is interesting to note, that Eq. (.4 can also be derived for a slightly different envelope function f n (r, t, which is defined via the wave function as Ψ(r, t = f n (r, tu nk0 (r. Both definitions are equivalent close to the extremum of the band. The definition of Eqs. (.,. has the advantage, that it holds in the entire band. On the other hand f n (r, t allows for a multiband description, which is used in k p theory..5. Motivation of Eq. (.3 Eq. (.3 is difficult to proof. 8 Here we restrict us to A(r, t = 0, i.e. without a magnetic field. For the electric potential we use the Fourier decomposition φ(r = d 3 iq r q φ(qe and insert Eq. (. into Eq. (.0. Multiplying by Ψ kn (r and performing the integration d 3 r provides us with the terms (omitting the band index: d 3 r Ψ k(r d 3 r Ψ k(ri d 3 k ċ(k, tψ k (r =i ċ(k, t ( p d 3 k c(k, t m + V (r Ψ k (r =E n (kc(k, t Substituting k = k + q we the potential part reads d 3 r Ψ k(r( e d 3 q φ(qe iq r d 3 k c(k eik r, t (π u 3/ k (r = d 3 r Ψ k(r d 3 k e ik r (π u 3/ k q(r( e d 3 q φ(qc(k q, t (u k q u k for small q ( e d 3 q φ(qc(k q, t providing us with i ċ(k, t E n (kc(k, t e d 3 q φ(qc(k q, t (.5 7 section 4.. of Yu and Cardona (999 8 See, e.g., the original article by Luttinger, Physical Review 84, 84, (95 using Wannier functions. A rigorous justification, as well as the range of validity is a subtle issue, see G. Nenciu, Reviews of Modern Physics 63, 9 (99.

21 Chapter : Band structure 3 Thus we find the following dynamics of the envelope function (.: i t f(r, t = (.5 d 3 k i ċ(k, t eik r (π 3/ d 3 e ik r k E n (k c(k, t e } {{ } (π 3/ =E n( i d 3 k d 3 iq r e q φ(qe ik r (π } {{ } 3/ c(k, t =φ(r (.6 where we replaced k = k q in the last term. Thus Eq. (.6 becomes Eq. ( Heterostructures For semiconductor heterostructures, the band structure varies in space (more details can be found in section.7 of Ibach and Lüth (003, e.g.. For several semiconductor materials like GaAs, InAs, or Al x Ga x As (for x 0.45 the minimum of the conduction band is at the Γ-point and due to symmetry the effective mass equation (.4 becomes i ] t f c(r, t = [E c (r m c (r f c (r, t (.7 where we neglected external potentials here. 9 The derivatives are written in this peculiar way to guarantee the hermiticity of the effective Hamiltonian when the conduction band edge E c (r = E conductionband (k = 0 and the effective mass m c (r become spatially dependent. 0 At the interface located at z = 0 with material A at z < 0 and material B at z > 0 this implies the boundary connection rules m A c f c ( r z 0 = f c ( r z 0 + (.8 f c ( r = f c ( r (.9 z z 0 m B c z z 0 + which allow for the study of the electron dynamics in the conduction band in semiconductor heterostructures. The valence band is more complicated as there are degenerate bands at k = 0. As an example we consider a quantum well, i.e. a slap of material (e.g. GaAs with thickness w embedded by regions of a different material with a higher energy E c (e.g. Al 0.3 Ga 0.7 As, see Fig..6. For zero temperature and without impurities, charge neutrality implies that the valence bands of both materials are fully occupied and the conduction bands are empty. If additional electrons are provided (e.g. by doping they will assemble in the region of lowest energy, i.e., in the range 0 < z < w. If w is small with respect to the electron wavelength (typically several tens of nanometers at room temperature, quantization effects are important. They can be taken into account using Eq. (.7. Here the conduction band edge E c (z is 9 A motivation is given in M.G. Burt, Phys. Rev. B 50, 758 (994 0 This approach is due to D. J. Ben Daniel and C. B. Duke, Physical Review 5, 683 (966. While it is used by default, it is still under debate, see e.g. B. A. Foreman, Physical Review Letters 80, 383 (998. Here k p theory is used, see, e.g., chapters 5+6 of Chow and Koch (999. Strictly speaking Al 0.3 Ga 0.7 As is not a crystal but an alloy. However most physical properties, such as the bandstructure, can be well described by a weighted average of the GaAs and AlAs properties. A collection of relevant parameters can be found in I. Vurgaftman, J.R. Meyer, L.R. Ram-Mohan, J. Appl. Phys., 89, 585 (00.

22 4 A. Wacker, Lund University: Solid State Theory, VT 05 Energy Energy Bound state E c (z E v (z Valence band GaAs Valence band Al 0.3 Ga 0.7 As Conduction band GaAs Conduction band Al 0.3 Ga 0.7 As 0 Bloch vector k 0 w z-direction Figure.6: Band structure for two similar semiconductors and energy spectrum of a heterostructure together with a bound state in conduction band of the quantum well. Here the zero line for the wave function is put to the energy of the state, a common practice. given by the upper black line in the right hand side of Fig..6. As the system is translational invariant in x, y direction, the ansatz f c (r = ϕ ν (ze i(kxx+kyy (.30 is appropriate. For the z component we obtain the stationary Schrödinger equation for k = 0 [ E ν ϕ ν (z = ] z m c (z z + E c(z ϕ ν (z For E c (GaAs < E ν < E c (Al 0.3 Ga 0.7 As we obtain the solutions Ae λz for z < 0 ϕ ν (z = Be iqz + Ce iqz for 0 < z < w De λz for z > w with λ = m A (E c (Al 0.3 Ga 0.7 As E ν / and q = m G (E ν E c (GaAs/, where m A and m G are the effective masses at the Γ minimum of the conduction band of Al 0.3 Ga 0.7 As and GaAs, respectively. Now the boundary conditions (.8,.9 provide two equations at each interface, which can be subsumed in the matrix equation: A 0 M(E ν B C = 0 with M(E ν = λ/m A iq/m G iq/m G 0 0 e iqw e iqw e λw D 0 iqe iqw /m G iqe iqw /m G λe λw /m A where the E ν -dependence arises by q and λ. The existence of nontrivial solutions requires detm(e ν = 0, which provides an equation to determine one or several discrete values E ν. From the corresponding eigenstates (A ν, B ν, C ν, D ν tr the eigenfunctions ϕ ν (z can be directly constructed. An example is shown in red in Fig. (.6. For finite k, we may use first order perturbation theory 3 and find the total energy of the state (.30 E ν (k x, k y = E ν + (kx + ky with = dz ϕ ν(z m ν m ν m c (z This implies a parabolic dispersion in k in addition to the quantized energy E ν, which is referred to as a subband. 3 A correct solution would require to solve a different z-equation for each k x, k y, which seems however rarely be done in practice.

23 Chapter Transport. Semiclassical equation of motion Ehrenfest s theorem tells us that the classical limit of the quantum evolution is given by the equations of motion H(r, p H(r, p ṙ = and ṗ = p r where the classic canonical momentum p replaces ( /i in the quantum mechanical Hamiltonian. From the effective Hamiltonian (.3 we find with k = (p+ea(r, t/ : (In the following indicates the quantity on which = / r operates in a product ṙ = E n (k = v n (k k ṗ = e [ ] En (k A(r, t +e φ(r, t k } {{ } v n(k [ A]+ A(v n(k [ ] = ev n (k B(r, t ef(r, t e (ṙ A(r, A(r, t t + t } {{ } = da(r,t dt where the definitions of the electromagnetic potentials (.9 are used. This provides the semiclassical equation of motion for electrons in a band n. ṙ = v n (k (. k = ev n (k B(r, t ef(r, t (. Thus k follows the same acceleration law as the kinetic momentum mv of a classical particle. Accordingly, k is frequently referred to as crystal momentum. Such a classical description in terms of position and momentum is only valid on length scales larger than the wavelength of the electronic states. In a semiconductor with a parabolic band, we find k = m eff E(k/ π/5 nm for an energy E(k 5 mev at room temperature and an effective mass of m eff 0.m e. Thus the semiclassical model makes sense on the micron scale, but fails on the nanometer We follow Luttinger, Physical Review 84, 84, (95 5

24 6 A. Wacker, Lund University: Solid State Theory, VT 05 scale. Even if the treatment looks classical, the velocity v n (k contains the information of the energy bands, which is reflected by the term semiclassical. One has to be aware, that the single-particle Bloch states considered here are an approximation where the interactions between the electrons are treated by an effective potential. This approximation has a certain justification for single-particle excitations from the ground sate of the interacting system. The properties of these excitations may however differ quite significantly from individual particles (e.g., their magnetic g-factor can differ essentially from g for individual electrons and thus one refers to them as quasi-particles. An important property is the acceleration in electric fields. Here the inertia of the quasiparticle is given by the direction-dependent effective mass tensor (.5, as shown in the exercises.. General aspects of electron transport Summing over all quasi-particles and performing the continuum limit, the current density is given by (see e.g., 9. of Ibach and Lüth (003 (for spin( e J = v n (kf n (k = e d 3 k v V (π 3 n (kf n (k (.3 n,k where 0 f n (k is the occupation probability of the Bloch state nk. In thermal equilibrium the occupation probability is given by the Fermi distribution n.bz f n (k = exp ( En(k µ int k B T + = f Fermi (E n (k where µ int is the internal chemical potential 3. Thus the current (.3 vanishes due to Kramer s degeneracy. Therefore we only find an electric current, if the carriers change their crystal momentum due to acceleration by an electric field. For small fields the current is linear in the field strength F, which is described by the conductivity tensor σ ij J = σf As the electric field (. does not change the band index for weak fields, bands do not contribute to the current if they are either entirely occupied or entirely empty. Thus, we find: Only partially occupied bands close to the Fermi energy contribute to the transport. For metals, (e.g., Al in Fig.. the Fermi energy cuts through the energy bands, which are partially occupied and this provides a high conductivity. In contrast, semiconductors and insulators (e.g., Si and GaAs in Fig..5 have an energy gap between the valence band and the conduction band. For an ideal crystal at zero temperature, the valence band is entirely occupied and the conduction band is entirely empty. Thus the conductivity is zero in this case. For semiconductors it is relatively easy to add electrons into the conduction band (by doping, heating, irradiation, or electrostatic induction, which allows to modify the conductivity in a wide range. E.g., g 0.4 for conduction band electrons in GaAs, M. Oestreich and W.W. Rühle, Phys. Rev. Lett. 74, 35 ( I follow the notation of Kittel and Krömer (980. µ int can be understood as the chemical potential relative to a fixed point in the band structure. For given band structure µ int is a function of temperature and electron density, but does not depend on the absolute electric potential.

25 Chapter : Transport 7 In the same way one may remove electrons from the valence band (index v of a semiconductor, which increases the conductivity as well. Now the excitation is a missing electron, and the corresponding quasi-particle is called a hole. The properties of holes (index h can be extracted from the valence band E v (k as follows [see Sec. 8 of Kittel (996]: q h = e k h = k E h (k h = E v (k µ (h int = µ int v h (k h = v v (k m h (k h = m v (k f h (k h = f(k (.4 and inserting into Eq. (.3 provides the current J = (for spinq h (π 3.Bz d 3 k h v h (k h f h (k while the acceleration law (. becomes k h = q h v h (k h B(r, t + q h F(r, t. Note, that the hole mass m h is positive at the maximum of the valence band, where the curvature is negative. Taking a closer look at the acceleration law (., we find that the Bloch vector k is continuously increasing in a constant electric field. If it reaches the boundary of the Brillouin zone (i.e. gets closer to a certain reciprocal lattice vector G n than to the origin, it continues at the equivalent point k G n. In this way k performs a periodic motion through the Brillouin zone with zero average velocity. 4 This is, however, not the generic situation. Typically, scattering processes restore the equilibrium situation on a picosecond time scale. In a simplified description the average velocity v of the quasi-particle satisfies dv dt = qf v m eff τ m where τ m is an average scattering time, m eff the average effective mass, and q = ±e the charge of the relevant quasiparticles. The stationary solution for a constant electric field provides v = sign{q} µf with the mobility µ = eτ m m eff (.5 and the conductivity becomes σ = en QP µ e, where n QP Sec..4. a more detailed derivation will be given. is the density of quasi-particles. In The key point is that electric conductivity is an interplay between acceleration of quasi-particles by an electric field and scattering events. While we defined the acceleration in Sec.., on the basis of the Bloch states of a perfect crystal, scattering is related to the deviations from ideality. In practice, no material is perfectly periodic due to the presence of impurities, lattice vacancies, lattice vibrations, etc. Their contribution can be considered as a perturbation potential to the crystal Hamiltonian studied so far. In the spirit of Fermi s golden rule, this provides transition rates between the Bloch states with different k, as the Bloch vector is not a good quantum number of the total Hamiltonian any longer. Such transitions are generally referred to as scattering processes..3 Phonon scattering Similar to the electron Bloch states, the lattice vibrations of a crystal can be characterized by a wave vector q and the mode (e.g. acoustic/optical or longitudinal/transverse, described by 4 This is called Bloch oscillation and can actually be observed in superlattices, C. Waschke et al., Phys. Rev. Lett. 70, 339 (993

26 8 A. Wacker, Lund University: Solid State Theory, VT 05 an index j. The corresponding angular frequency is ω j (q. Like any harmonic oscillation the energy of the lattice vibrations is quantized in portions ω j (q, which are called phonons. Using the standard raising (b j and lowering (b j operators (see, e.g., Andreas.Wacker/Scripts/oscilquant.pdf, the Hamiltonian for the lattice vibrations reads Ĥ phon = [ ω j (q b j (qb j(q + ] (.6 j,q The eigenvalues of the number operator b j (qb j(q are the integer phonon occupation numbers n j (q 0. In thermal equilibrium the expectation value of the phonon occupations is given by the Bose distribution. n j (q = e ω j(q/k BT = f Bose( ω j (q (.7 The elongation of the ions follows ( s(r, t e iq R b j (qe iωj(qt + b j ( qeiω j(qt where the use of ( q in the raising operator guarantees that the operator is hermitian. See subsection.5. for details. These lattice oscillations couple to the electronic states due to different mechanisms, such as the deformation potential (subsection.5. or the lattice polarization for optical phonons in polar crystals (subsection.5.3. The essence is that lattice vibrations distort the crystal periodicity and constitute the perturbation potential V (r, t = [ ] U (q,j (re iq r b j (qe iωj(qt + b j ( qeiω j(qt (.8 qj Here U (qj (r + R = U (qj (r is a lattice periodic function, describing the local details of the microscopic interaction mechanism 5 In order to provide a Hermitian operator, [ U (q,j (r ] = U ( q,j (r holds..3. Scattering Probability Fermi s golden rule (see, e.g., pdf gives us the transition probability per time 6 between Bloch states Ψ n,k and Ψ n,k. W nk n k = π qj [ Ψn,k, n j(q U qj (re iq rˆbj (q Ψ n,k, n j (q δ (E n (k E n (k ω j (q } {{ } Phonon absorption + Ψn,k, n j( q + U qj (re iq rˆb j ( q Ψ n,k, n j ( q ] δ (E n (k E n (k + ω j (q } {{ } Phonon emission (.9 5 U qj (r is constant for the deformation potential of acoustic phonons. It has a spatial dependence for scattering at polar phonons, which is however neglected in the averaging procedure applied in Sec sometimes called scattering probability, which is unfortunate as this suggests a dimensionless quantity.

27 Chapter : Transport 9 Here the presence of the phonon lowering and raising operators requires that the occupation n j (q of the phonon mode involved in the transition must change together with the electron state. This is indicated by combining the phonon occupation with the electron state. We find, that the first term relates to absorption of a phonon by the electron from the phonon mode (q, j, while the second relates to phonon emission into the mode ( q, j. Concomitantly, the δ-function tells us, that the energy of the final state E n (k is enlarged/decreased by ω j (q compared to E n (k. Let us now analyze the matrix elements: From quantum mechanics we know ˆb j (q n j (q = n j (q n j (q and ˆb j (q n j(q = nj (q + n j (q +. Therefore the phonon part gives us a pre-factor n j (q in the absorption [n j ( q+] in the emission term. This shows that phonon absorption vanishes for T 0 when the phonons are not excited, while there is always a finite probability for phonon emission. The spatial part can be treated as follows for a general function F (ˆp, ˆr, which is lattice-periodic in r. Ψ n,k F (ˆp, ˆreiq r Ψ n,k = d 3 r e ik r u n V k (rf (ˆp, ˆreiq r e ik r u nk (r V = e i(q+k k R d 3 r e ik r u n N c V k ( rf (ˆp, ˆru nk( re i(q+k r (.0 c R V c } {{ } } {{ } =F n k,nk G δ q+k k,g where we used the decomposition r = R + r, where r is within the unit cell V c. Together we find the Electron-phonon transition probability per time W nk n k = π δ k,q+k+g U (q,j n k,nk [n j (qδ (E n (k E n (k ω j (q qj G ] + (n j ( q + δ (E n (k E n (k + ω j ( q (. Here processes employing a finite vector G of the reciprocal lattice are called Umklapp processes (from the German word for flip, while normal processes restrict to G = 0. As k k can be uniquely decomposed into a vector G of the reciprocal lattice and a vector q within the Brillouin zone, one finds that Umklapp processes allow for transitions with a rather large momentum transfer k k. On the other hand only normal process are of relevance if the physical processes are limited to a small region of the Brillouin zone, such as a single energy minimum in a semiconductor. Frequently, Umklapp processes are entirely neglected for simplicity..3. Thermalization Consider two states = (n k and = (n k with occupations f, f and energies E = E + E where E = ω α (k k > 0 matches a photon energy with wavevector q 0 = k k. Then the net transition rate from to is given by f ( f W f ( f W = f ( f π qj G f ( f π qj G δ k,q+k G U (q,j n k,n k n j (q δ ( E ω j (q δ k,q+k G U (q,j n k,n k [ n j ( q + ]δ ( E + ω j ( q

28 0 A. Wacker, Lund University: Solid State Theory, VT 05 where Pauli blocking has been taken into account by the ( f i terms. Now the δ k,k picks q = q 0 in the first line and q = q 0 in the second line. Furthermore, only the phonon mode j = α remains due to energy conservation. As (U (q,α n k,n k = U ( q,α n k,n k the squares of the matrix elements are identical. Assuming that the phonon system is in thermal equilibrium with the lattice temperature T, n j (q 0 is given by the Bose distribution (.7 and we find ( f ( f W f ( f W f ( f e E/k BT f ( f e E/k BT + = ( f ( ( f f f e E/k BT e E/k BT f f This net transition vanishes if f i f i = Ae E i/k B T. Relating the proportionality constant A to the chemical potential µ int via A = e µ int/k B T, we obtain f i = f Fermi (E i. Thus the net transition rate vanishes if the electron system is in thermal equilibrium with the temperature of the phonon bath. Phonon scattering establishes the thermal equilibrium in the electron distribution..4 Boltzmann Equation In the following we restrict us to a single band and omit the band index n. We define the distribution function f(r, k, t by f(r, k, td 3 rd 3 k/(π 3 to be the probability to find an electron in the volume d 3 r around r and d 3 k around k. If f(r, k, t is only varying on large scales r k its temporal evolution is based on the semiclassical motion (.,. leading to the Boltzmann equation t f(r, k, t + v(k ( e f(r, k, t + (F + v(k B f(r, k, t = r k ( f t scattering (. Read section 5.9 of Snoke (008 or sections of Ibach and Lüth (003 for detailed information. The scattering term has the form ( f(r, k, t t scattering = k W k kf(r, k, t[ f(r, k, t] W k k f(r, k, t[ f(r, k, t] for phonon (or impurity scattering. Note that the scattering term is local in time and space. An overview on different scattering mechanisms can be found in chapter 5 of Snoke (008 or section 9.3 of Ibach and Lüth (003. In section (.3. it was discussed that on the long run scattering processes restore thermal equilibrium. This suggests the relaxation time approximation ( f = t scattering δf(r, k, t τ m (k with δf(r, k, t = f(r, k, t f Fermi (E(k where the variety of complicated scattering processes is subsumed in a scattering time τ m (k, which is typically somewhat shorter than a picosecond 7. 7 The time introduced here is actually the momentum scattering time, which is larger than the total scattering time, as forward scattering is less effective.

29 Chapter : Transport.4. Electrical conductivity Now we want to consider the current caused by a weak electric field F ( and vanishing magnetic field. As the distribution function becomes thermal for zero field, we may write f(r, k, t = f Fermi (E(k + O(F. In linear response (i.e., only terms linear in F are considered only the zeroth order of f(r, k, t enters, as this term is multiplied by F in the Boltzmann equation k (.. Specifically, we use f(r, k, t k k f Fermi(E(k = df Fermi(E(k v(k. (.3 de Then the stationary (i.e. f(r, k, t = 0 and spatially homogeneous (i.e. t Boltzmann equation in relaxation time approximation reduces to f(r, k, t = 0 r δf(r, k, t = τ m (kef v df Fermi(E(k de Thus we find for stationary transport in a homogeneous systems (for spin( e J = d 3 k v(kδf(k (.4 (π 3.Bz ( = e d 3 k v(k df Fermi(E(k τ 4π 3 m (kv(k F (.5.Bz de } {{ } =σ providing us with the tensor of the electrical conductivity σ. The conductivity becomes rather simple for an isotropic, parabolic band structures E(k = k /m eff with τ m (k = τ m (E(k. Here the density of states is given by D(E = D 0 E, see Eq. (.9. Using k = nk with the unit vector n we find σ = e 0 ded(e E ( τ m (E df Fermi(E m eff de 4π π dϕ d(cos θ nn } 0 {{ } =T The elements of the tensor T are given by T zz = 4π dϕ d(cos θ cos θ = and 3 T xz = 4π dϕ d(cos θ sin θ cos ϕ cos θ = 0. Altogether we find T =. Thus the conductivity 3 is a scalar and the current is parallel to the field. For metals we have df Fermi /de δ(e E F and we find σ = e τ m(e F n which gives the mobility µ = eτ m(e F m eff m eff in accordance with the simple model (.5. The same expression holds for larger temperatures k B T E F (typical for semiconductors, if τ m is constant Transport in inhomogeneous systems Now we assume that there are spatial variations of the density [or correspondingly the internal chemical potential µ int (r] and the temperature T (r in addition to the electric field. This is 8 For non-degenerate systems, this can be generalized to τ m E r and an r-dependent pre-factor appears in the mobility. In this is case the Hall mobility differs from the transport mobility, see chapter 4. of Seeger (989.

30 A. Wacker, Lund University: Solid State Theory, VT 05 approximated by the local equilibrium f 0 (E(k, r = exp ( E(k µint (r k B T (r + which replaces the Fermi function in the relaxation time approximation. In the left hand side of the Boltzmann equation (. we use in lowest order for the spatial variations of µ int (r and T (r r f(r, k, t r f 0(E(k, r = f ( 0(E(k, r E and find together with Eq. (.3 δf(r, k, t = τ m (k f 0(E(k, r v(k E to be inserted into Eq. (.4. ( µ int (r + E(k µ int(r T (r T (r ef + µ int (r + E(k µ int(r T (r T (r ( Diffusion and chemical potential At first we neglect temperature gradients. Using Eq. (.5, we find J = σf + e σ µ int(r (.7 As the internal chemical potential is a function of the density, the second term provides the electron diffusion J = ed n with the Einstein relation D = σ e dµ int dn Note that the right-hand side of Eq. (.7 can be written as σ µ(r with the e chemical potential µ = µ int eφ(r Thus (for constant temperature there is no current flow for a constant chemical potential. We follow here the notation of Kittel and Krömer (980. Our chemical potential is frequently referred to as electrochemical potential ζ or, in particular for semiconductors, as Fermi level E F. For inhomogeneous systems (e.g. pn diodes or semiconductor heterostructures it is convenient to plot the band edge E c eφ(r in combination with µ. Then, spatial variations of µ imply current flow and the difference between µ and the shifted band edge provides µ int, i.e. the actual electron density. The underlying idea is demonstrated in Fig... It can be directly extended to different band edges (such as conduction and valence band as well as to heterostructures with a spatial dependence E c (r, see Fig Thermoelectric effects The heat-current density is given by (see sections of Ibach and Lüth (003 for details! J Q = d 3 k [E(k µ 4π 3 int ] v(kδf(k.bz

31 Chapter : Transport 3 µ int (x φ(x E c -eφ E c -eφ E(k E c µ int k high doped (a (b (c (d diffusion J= e σ µ int <0 drift J=σF=-σ φ>0 total J= e σ µ equilibrium spatial position x E(k E c µ int k low doped E c -eφ µ=µ int -eφ Figure.: (a Internal chemical potential in a junction between a highdoped and a low-doped semiconductor. (b Electrical potential, which provides a drift current via the electric field. (c (Electro-Chemical potential µ = µ int eφ, which drives the total current. Note that the difference between the curves for µ and E c eφ(r is a measure for the local charge density. (d same as (c for a different electric potential, so that there is no net current. Figure.: Spatial variation of the band alignment E c eφ(r in a high-electronmobility-transistor (HEMT which is determined by the uniformity of the chemical potential µ (here denoted as E F. SI stands for semi-insulating, where the chemical potential is in the middle of the gap. i and n stand for intrinsic (undoped and n-doped regions respectively. The ionized donors in n- Al x Ga x As provide a positive space charge with a positive curvature of eφ(r. The electrons in the DEG provide a negative curvature. (From Wikipedia Commons This provides us with transport coefficients [using F = F + µ int (r/e = µ/e in Eq. (.6] J = L F + L ( T (.8 J Q = L F + L ( T (.9 These equations describe electrical conductance, heat conductance as well as thermoelectric effects, such as (see chapter 3 of Ashcroft and Mermin (979 for details Peltier effect: An electric current implies heat current J Q = ΠJ for constant temperature with the Peltier constant Π = L /L Seebeck effect: A temperature difference is associated with a bias for vanishing electric current: F = S T with the thermoelectric constant (thermopower S = L /L thermal conductivity: A temperature difference causes a heat current J Q = κ T for vanishing electric current with the thermal conductivity κ = L L L /L.

32 4 A. Wacker, Lund University: Solid State Theory, VT 05 L and L are positive, while L and L are typically (e.g., for parabolic band structure negative for electron transport and positive for hole transport. Thus: Particles flow from high to low density and from high to low temperature. The coefficients L ij are not independent: One finds T L = L, and consequently Π = T S, which reflects the general Onsager relation 9. For metals with E F k B T one obtains L /L = T π k B /3e. Approximating the thermal conductivity K L, this provides the Wiedemann- Franz law K/T σ =const, which was found experimentally already in Details for Phonon quantization and scattering See also Sections 4.+ of Snoke (008 for an overview..5. Quantized phonon spectrum We consider a crystal with ions at the equilibrium positions R + r α, where R is the lattice vector of the Bravais lattice and α the atom index, which counts the atoms of mass m α in each unit cell. The potential energy of the lattice has a minimum for the equilibrium positions and thus the potential is approximately quadratic in the elongations s(r, α from the equilibrium positions V ({s(r, α} = R, Rαα s(r, α D R α,α s(r + R, α We want to solve the 3NN α coupled equations of motion m α d s(r, α, t d t V ({s(r, α, t} = s(r, α = Rα D R α,α s(r + R, α, t. Due to the translational invariance of the lattice, the Ansatz s(r, α, t = e iq R e (j α (qq j (q, t Nmα gives us 3NN α uncoupled oscillators (see chap 4. of Ibach and Lüth (003 α R d Q j (q, t d t j,q = ω j (qq j (q, t (.0 where the frequencies ω j (q are obtained from the eigenvalue problem ( iq R e D R e (j α,α α (q = ω mα m j (qe (j α (q α 9 We consider the particle current J P = J/( e and the energy current J U = J Q + µj P as functions of their respective conjugate forces T and µ T, see, e.g., Kittel and Krömer (980: J P = ( L T J U = e L + L T µ e ( µ e T ( µ T ( L µ T + e L µt e ( L T + L µt e e T L µt + L T e T The Onsager relation states that this coefficient matrix is symmetric, requiring T L = L.

33 Chapter : Transport 5 Γ X R 40 K Σ Γ Γ Λ L Energy [mev] [q00] [0qq] [qqq] Figure.3: Phonon spectrum of GaAs with optical (blue and acoustic (red phonons. The full lines refer to longitudinal and the dashed lines the transversal phonons. [After J.S. Blakemore J. Appl. Phys 53, R3 (98] For each q we have 3N α eigenmodes (label j satisfying α [e(i α (q] e (j α (q = δ ij, e (i α ( q = [e (i α (q] and ω j ( q = ω j (q. As the elongations s are real we have Q j (q, t = Q j( q, t. An example for such a phonon spectrum is shown in Fig..3. The Hamilton operator corresponding to the classical equation of motion reads 0 : with the operators Ĥ phon = j,q ˆΠ j (qˆπ j (q + ω j (q ˆQ j (q ˆQ j (q ˆQ j (q = α,r mα N e iq R e (j α (q ŝ(r, α ˆΠj (q = α,r We define the lowering operator as usual b j (q = ωj (q [ ω j (q ˆQ ] j (q + iˆπ j (q Nmα e iq R e (j α (q ˆp(R, α As the operators ˆQ j (q = ˆQ j ( q and ˆΠ j (q = ˆΠ j ( q are non-hermitian, we have the corresponding raising operator b j (q = [ ω j (q ˆQ ] j ( q iˆπ j ( q ωj (q which can be inverted by ˆQ j (q = ω j (q [b j(q + b j ( q] ˆΠj (q = i ωj (q [b j (q b j ( q] (. Due to the commutation relation [ˆΠ j (q, ˆQ i (q ] = /i δ ij δ q, q we find the standard relations [b j (q, b j (q ] = δ ij δ q,q and [b j (q, b i (q ] = [b j (q, b i (q ] = 0 as well as Eq. (.6. From quantum mechanics we know the eigenstates and energies E n = ω j (q(n j (q + / for each phonon mode. Thus the set of numbers {n j (q} describes the state of the lattice vibrations. The time dependence of the operators is given by in the Heisenberg picture. b j (q, t = b j (qe iω j(qt b j (q, t = b j (qeiω j(qt (. 0 We follow R. Feynman Statistical Mechanics; G. Mahan Many Particle physics uses ˆP = ˆΠ

34 6 A. Wacker, Lund University: Solid State Theory, VT Deformation potential interaction with longitudinal acoustic phonons This is a common scattering process present in all materials. We follow Ferry (99 here. For acoustic phonons the elongations satisfy s(r, α s(r. Thus e (j mα α (q β m e (j Nm α (q = β ρ m V e(j (q Here ρ m = α m α/v c is the mass density. We can rewrite the elongations as a continuous field S(r = ρm V e(j (qe iq r Q j (q Now the band structure changes, when the crystal is compressed, which can be approximated by an energy shift δe n = Ξ n δv/v for each band n. The local compression due to the phonon mode is given by δv V = div S(r = ρm V iq e(j (qe iq r Q j (q, t and it becomes obvious that only longitudinal acoustic phonons contribute, i.e., j = LA. Thus we find the effective electron potential V Def.P ot. (r, t = [ Ξ n iq e (LA (q ] e [b iq r LA (qe iωla(qt + b LA(qt] LA ρ q m V ω LA (q ( qeiω ( Polar interaction with longitudinal optical phonons Polar scattering at optical phonons is the dominant phonon scattering mechanism in III/V semiconductors. For longitudinal optical phonons the average polarization of a unit cell is given by P(R, t = q α s(r, α = q V c V c s(r where s(r = α e iq R ẽ(qq LO (q. Nρm V c q Here ẽ(q and q are defined via qẽ(q/ ρ m V c = α q αe α (q/ m α and ẽ(q =. As there is no macroscopic charge we have ɛ 0 F + P = 0 and obtain the mechanical potential of an electron V (r, t = eφ(r, t = e dr P(r, t = e q s(r q ɛ 0 iɛ 0 V c q where we used ẽ(q q. This results in the potential V Polar optical phonon (r, t = [ ] g(qe iq r b LO (qe iωlo(qt + b LO ( qeiω LO( qt q (.4 with the Fröhlich-coupling g(q = V ie ω LO (q ɛ 0 ( ɛ( ( iẽ(q q ɛ(0 q } {{ } / q (.5

35 Chapter : Transport 7 satisfying g( q = g(q. Here ɛ(0 and ɛ( are the dielectric constants for zero frequency and for frequencies well above the optical phonon resonance, respectively. We used ( q = ω LO ɛ 0 V c ɛ( ρm (.6 ɛ(0 ɛ 0 resulting from the dielectric properties. See, e.g., section 6.8 of Seeger (989. A similar treatment is found in Ferry (99

36 8 A. Wacker, Lund University: Solid State Theory, VT 05

37 Chapter 3 Magnetism Electrodynamics of continua tells us, that the magnetic field B is related to the field H by B = µ 0 (H + M where the magnetization M (density of magnetic moments µ is a material property and µ 0 = 4π 0 7 Vs/Am is the vacuum permeability. In this chapter we want to provide a physical basis of M. Frequently, we have a linear relation M = χ µ 0 B (3. defining the magnetic susceptibility χ. Materials with χ > 0 are called paramagnetic, while materials with χ < 0 are called diamagnetic. In addition, M 0 is possible even for a vanishing magnetic induction. Such materials are called ferromagnetic. 3. Classical magnetic moments In classical electrodynamics the magnetic moment of a current distribution is defined by µ = d 3 r r j(r E.g., for an annual current I surrounding the oriented area A we find µ = IA. Now we consider a rotating body with a total mass m and a total charge q. We assume that the charge and the mass follow the same normalized spatial distribution P (r (corresponding to Ψ(r in quantum mechanics resulting in the charge distribution ρ(r = qp (r and the mass distribution ρ m (r = mp (r. Using the relation mv = p qa in the presence of a vector potential, we find µ = d 3 r r ρ(rv(r = q m d 3 r r P (rmv(r = q m L q m d 3 r r ρ(ra(r (3. Thus magnetic moments are intrinsically connected to the angular momentum L = r p with a ratio q. Qauntum mechanics provides the quantization of the angular momentum in (halfinteger units of. Thus the Bohr magneton µ B = e m e m = ev/t is a typical unit. We treat B as the primary magnetic field as it appears in the Lorentz force. Traditionally, H was denoted as magnetic field, as it is directly related to measurable free currents, and B was called magnetic induction. In the same spirit, the susceptibility is traditionally defined as M = χ H providing χ = χ/( χ, which in practice does not make a difference, as typically χ. See Chap. of Purcell and Morin Electricity and Magnetism (Cambridge University Press, 03 for an enlightening discussion. See, e.g., 9

38 30 A. Wacker, Lund University: Solid State Theory, VT 05 In addition there is a term A, which vanishes for point-like particles and is frequently not mentioned in the literature. The interaction of a magnetic moment with an external magnetic field B is described by the energy E = µ B (3.3 which can be proven by evaluating the force on a magnetic dipole in an inhomogeneous external magnetic field F = d 3 r j(r B(r difficult = [µ B(r] (3.4 as well as the torque on the magnetic moment in a constant external magnetic field τ = d 3 r r (j(r B difficult = µ B where in both cases the current distribution does not change in its own frame, if the magnetic moment is moved or rotated. For details see section 5.7 of Jackson ( Magnetic susceptibilities from independent electrons To simplify the notation, we redefine in this chapter both the orbital angular momentum ˆl = ˆr ˆp/ and the spin ŝ by dividing by, yielding dimensionless quantities. For a homogeneous constant magnetic field we can use the vector potential A(r = B r and the Hamilton operator (.0 becomes Ĥ = ˆp m e + V (r + B m ˆl } {{ e } =µ B B ˆl + e 8m e B r (3.5 where B = B and r is the projection of r onto the plane perpendicular to B. (Technically one has B r = (B r In addition, the electron has an intrinsic property, the spin, which can be described by a spinor wave function (Ψ(, t, Ψ(, t tr with an angular momentum, the spin, of /. The Spin operator is given by ŝ = σ where σ x = ( 0 0 σ y = ( ( 0 i 0 σ i 0 z = 0 are the Pauli matrices. The interaction with the magnetic induction is given by e Ĥ spin = g e ŝ B (3.6 m e where g e = is the Landé factor of the free electron. 3 Thus the corresponding magnetic moment is µ = gµ B s. The total Hamiltonian (3.5 provides the expectation value of the energy ˆp e Ĥ + Ĥspin = m + V (r + µ B B ˆl + g e ŝ + B r 8m e 3 This quantity could recently be measured with a remarkable accuracy of 7.6 parts in 0 3, B. Odom et al., Phys. Rev. Lett (006 and agrees well with calculations based on relativistic quantum electrodynamics

39 Chapter 3: Magnetism 3 Physical origin formula magnitude Thermal orientation of localized magnetic moments, paramagnetism χ = nµ 0 (gµ B J(J + 3k B T at 300 K A term for bound orbitals, Larmor diamagnetism χ = nµ 0 e 6m e Z a r a Second order perturbation theory for bound orbitals, Van Vleck paramagnetism χ = nµ 0 µ n ˆL z + g e Ŝ z 0 B E n E 0 n Spin splitting for band electrons Pauli paramagnetism (gµ B χ = µ 0 D(E F A further term for Bloch states in a magnetic field χ = χ P auli ( me Landau diamagnetism 3 m eff Table 3.: Overview for the different contribution to the susceptibility within the single electron model. For the numerical values the density n = 0 9 /m 3, g =, J = 0.5, Z a r a 0Å, n n ˆL z +g e Ŝ z 0 /(E n E 0 (5eV, and D(E F = (3π n /3 m eff /(π where m eff = m e are used. Read your textbook for details! [Detailed information for Van Vleck paramagnetism as well as Landau diamagnetism can be found in Marder (000 and Czycholl (004.] Now the magnetic moment can be defined as Ĥ / B providing µ = µ B ˆl + g e ŝ e r 4m e B (3.7 Except for the g e -factor of the electron spin, this is just the classical relation (3. with A(r = B r as considered here. As discussed in the following subsections, the first term µ B ˆl + g e ŝ causes paramagnetism and the second e r 4m e B brings about diamagnetism. A summary of the different contributions to the susceptibility in the independent electron model is given in Table 3.. The main general trend is that paramagnetism dominates if there are magnetic moments, which can align with the magnetic field. Otherwise, the diamagnetic contribution takes over. In particular, many molecules including water (χ = are diamagnetic (but oxygen O is paramagnetic. 3.. Larmor Diamagnetism The second term in Eq. (3.7 provides the susceptibility χ = nµ 0 e r /(4m e, which is negative, thus describing diamagnetism. Frequently one replaces r = r /3 which is appropriate

40 3 A. Wacker, Lund University: Solid State Theory, VT 05 for isotropic systems. 4 As energy is needed to establish a magnetic field in a diamagnetic substance, they experience a force at gradients of the magnetic field, see also Eq. (3.4. This may be used to levitate bodies, such as living frogs, see Paramagnetism by thermal orientation of spins We consider B = Be z and assume that µ = µ B ˆl z +g e ŝ z e z also points in z direction. A finite magnetic moment is thus related to the system occupying a state with a finite z-component of the angular momentum operator. For a symmetric system such states are part of multiplet which has the same energy for vanishing magnetic field. Thus, in thermal equilibrium all states of the multiplet have equal probability and there is no resulting magnetic moment. The situation is different for a finite magnetic field, which is studied for a spin-/ system here: For B > 0 the spin state with lowest energy, is the spin state = (0, tr which satisfies ŝ z = m with m = /. However, at finite temperature, the state with m = / can be occupied as well. From the Boltzmann distribution (canonical distribution we find m = ( exp ( exp gµ B B k B T gµ B B k B T exp ( ( + exp gµ B B k B T gµ B B k B T = tanh ( gµb B k B T gµ BB 4k B T Thus we find the magnetic moment µ = gµ B m e z = g µ B 4k B B and for a density n of spins, T the susceptibility becomes χ = nµ 0 g µ B /(4k BT (the Curie law 6, which is the dominant paramagnetic contribution. More generally, the total angular momentum J = L + S of an atom containing several valence electrons is related to its magnetic moment via ge J/(m e where g is the general Landé factor 7. Again, only the z-component (defined by the direction of B of the angular momentum Ĵ z is of relevance, which has the eigenvalues M J with M J = J, J,..., J. Correspondingly, the z-component of the magnetic momentum is µ z = gµ B M J. (See exercises and Table 3.. ( Pauli paramagnetism Now we consider metallic substances, where the Fermi energy is within a band. In a classical picture, the spin of the conduction electrons could be easily oriented to follow the magnetic field and one would expect a large paramagnetic contribution as discussed in section 3... However the Pauli principle does not allow for such an easy spin flip as both spins directions 4 This Larmor diamagnetism can be understood in a classical picture: Consider an electron oscillating in the potential of the nucleus. The Lorentz force ( ev B induces a rotation of the oscillation direction around B. Comparison with the Coriolis force mv ω 0 is a system which rotates with ω 0, shows that the presence of the magnetic field has the same implication as the pseudo force in a system rotating with the Larmor frequency ω 0 = ( eb/m. Thus the pendulum will rotate with ω 0 (like the Foucault pendulum creating a magnetic moment µ = eω 0 r / = e r B/4m in agreement with the result given above. Note however that the classical interpretation of magnetic effects is questionable as pointed out already in the PhD Thesis of N. Bohr, Copenhagen, 9. See also S.L. O Dell and R.K.P. Zia, Am. J. Phys. 54, 3 (986 5 See also M.D. Simon and A.K. Geim: Journal of Applied Physics 87, 600 ( after Pierre Curie, who is even better known for the work on radioactivity together with his wife Marie Curie [ ] 7 One finds g = 3 + S(S+ L(L+ J(J+, interpolating between g = for the total angular momentum L and g = for the total spin S, see chapter 3 of Ashcroft and Mermin (979.

41 Chapter 3: Magnetism 33 are occupied far below the Fermi energy. In contrast only the states close to the Fermi energy may contribute as outlined below. In metals, the Fermi energy is located within an electronic band, with density of state D(E (for both spin directions. A finite magnetic field B = Be z changes the energy E 0 (k of spin up/down states by ±g e µ B B/. Thus we obtain a spin resolved density of state D / (E = (π 3.Bz = D ( E g eµ B B ( d 3 k δ E E 0 (k g eµ B B (3.9 Filling up to the Fermi level, the total density of occupied spin-down states increases by D(E F g e µ B B/4 while the number of spin-up states decreases by the same amount, see Fig. 3.. The difference in occupation gives the magnetization M = g eµ B D(E F B/4 providing a parametric contribution χ = µ 0 g eµ B D(E F /4. D (E E D (E E F D(E/ D(E+ E/ D(E- E/ Figure 3.: Spin-resolved density of states for E = g e µ B B/ > 0. Indeed many metals such as sodium and aluminium are paramagnetic. However others such as copper and silver are diamagnetic. In particular bismuth has an extremely large value of χ = Ferromagnetism by interaction The interaction between magnetic moments can favor situations where all moments are aligned. The magnetic dipole-dipole interaction gives the energy E DD = µ 0 µ µ 50µeV for r = Å, µ 4π r 3 i = µ B which is far below the thermal energy (k B T = 5 mev at room temperature. It is found that an effective interaction in a quantum many-particle system, the exchange interaction, provides a strong interaction, which can favor parallel spin configurations Many-Particle Schrödinger equation The concept of many-particle wave functions can be introduced as a generalization of single particle states. Single particle wave function without spin: Ψ(r, where Ψ(r is the probability density to find the particle at r. Thus the normalization d 3 r Ψ(r = is needed. Single particle state with spin function ( a Ψ(r with a + b =. generalize: b ( Ψ(r, with Ψ(r, d 3 r Ψ(r, s = s=, Here Ψ(r, is the probability density to find the particle at r with spin.

42 34 A. Wacker, Lund University: Solid State Theory, VT 05 Two particle state Ψ(r, s, r, s, where Ψ(r, s, r, s is the probability density to find the first particle at r with spin s and the second particle at r with spin s. The normalization reads d 3 r s =, d 3 r s =, Ψ(r, s ; r, s = The dynamics is given by the Schrödinger equation: i ] t Ψ(r, s ; r, s, t = [ + V A (r, ŝ + V B (r, ŝ + V AB (r ; r Ψ m A m B where particle is of sort A and particle is of sort B and may have different masses, interaction parameters etc. It is often reasonable to consider product functions where also the spin function factorizes: Ψ Product (r, s ; r, s = φ a (r χ α (s φ b (r χ β (s where a, b denote the quantum state (e.g. quantum numbers. Note that only very few twoparticle wave function can be written in this way. For identical particles (A = B, the Hamilton operator is symmetric with respect to an exchange in the particle indices. This symmetry allows for a classification of the wave functions. Here all observed phenomena agree with the following Symmetry Postulate: For identical particles with half-integer spin (Fermions, e.g., electrons the states must be antisymmetric in the particle indices, i.e., Ψ(r, s ; r, s, t = Ψ(r, s ; r, s, t For more than two particles the same holds for all index combinations i, j. Product states can easily be anti-symmetrized as Ψ Slater (r, s ; r, s = [φ a (r χ α (s φ b (r χ β (s φ b (r χ β (s φ a (r χ α (s ] = ( φa (r χ α (s φ b (r χ β (s (3.0! φ a (r χ α (s φ b (r χ β (s where the Slater determinant of dimension N N with pre-factor / N! can be used for N- particle systems 8. We find directly that the function vanishes if a = b and α = β, i.e., two particle are put into the identical single-particle state. This is the Pauli principle The band model for ferromagnetism Now we study the joint probability density P (r, r = s s Ψ(r, s ; r, s for two plane waves e ik r / V, e ik r / V. We find from Eq. (3.0 P (r, r = ( e i(k k (r r + e i(k k (r r χ V α (s χ β(s s For α = β, e.g., both spins are, we find P (r, r = cos [(k k (r r ] V 8 We tacitly assume the orthonormality φ a φ b = δ ab, χ α χ β = δ αβ for the normalization properties here.

43 Chapter 3: Magnetism 35 and the probability to find both particles at the same place vanishes. In contrast, for α = and β = we find P (r, r =. As the Coulomb interaction between electrons is repulsive and V decreases as / r r we find The state with aligned spins has a reduced Coulomb interaction. The difference compared to the case with opposed spins is called exchange interaction. We approximate Ψ e 4πɛ 0 r r Ψ = d 3 r e d 3 r P (r, r 4πɛ 0 r r = UV c V IV c V δ αβ where I is the Stoner parameter which is of the order of ev. 9 Now we regard a Bloch state k with spin. The energy E 0 (k resulting from the periodic potential of the core ions does not dependent on the spin. Summing over the Coulomb interaction with all other valence electrons k (where n / V is the number of electrons with spin up/down, respectively we obtain an estimate for the total energy of the Bloch state E (k = E 0 (k + UV c (n + n IV c n As the total electron density n = n + n is fixed by the condition of charge neutrality, the corresponding term is a constant contribution to the energy and can be incorporated into E 0 (k. The corresponding expression with holds for E (k. In the exercises we show, that ferromagnetism occurs if ID(E F V c / >, which is the Stoner criterion. Thus, metals with a particular high density of states at the Fermi energy are ferromagnetic. This is the case for iron, cobalt, and nickel, where the localized d-shells are partially filled Singlet and Triplet states Alternatively one can symmetrize the spin-part and the spatial-part separately by Ψ Singlet = N ab (φ a (r φ b (r + φ b (r φ a (r (χ α (s χ β (s χ β (s χ α (s (3. Ψ Triplet = (φ a (r φ b (r φ b (r φ a (r N αβ (χ α (s χ β (s + χ β (s χ α (s (3. where N ab = for a = b and N ab = for a b to ensure the normalization. It is obvious that the triplet state has a reduced probability to find both particles at the same place. Thus we expect a lower energy of the triplet state due to the repulsive electron-electron interaction between the particles, which is quantified by a parameter J = E Singlet E Triplet. Now ŝ + ŝ is the operator for the total spin of both states and (ŝ + ŝ has the eigenvalue for the triplet state and 0 for the singlet state. Thus the effective Hamiltonian describing the interaction between the spins reads Ĥ eff = E Singlet J (ŝ + ŝ = E Singlet J (ŝ + ŝ ŝ + ŝ = E Singlet 3J 4 Jŝ ŝ (3.3 as the single electron states are eigenstates of ŝ i with eigenvalue 3/4. 9 See, e.g., J.F. Janak, Phys. Rev. B 6, 55 (977 for calculated values. 0 A detailed discussion including finite temperature can be found in section 8.4 of Ibach and Lüth (003. Concomitant, probability density is shifted away from the original states, where an external potential is low. This is a competing effect which can cause the singlet state to be lower is energy as well. Indeed, for a system of two particles, the ground state has always spin zero according to the Lieb-Mattis theorem, see Marder (000.

44 36 A. Wacker, Lund University: Solid State Theory, VT Heisenberg model For a lattice of localized magnetic moments (e.g., Mn atoms in a host material with total angular momentum Ŝn at lattice site n, Eq. (3.3 suggests the generalization Ĥ Heisenberg = J Ŝn Ŝm + gµ B B Ŝ n (3.4 (n,m where (n, m restricts the sums over n, m to those combinations of lattice sites forming next neighbors. Hereby each pair is counted twice. An extension to longer range interaction (describing ferrimagnetism, e.g. is straightforward. In order to find a simple solution we rewrite Ŝ n Ŝm = (Ŝn Ŝn (Ŝm Ŝm + Ŝn Ŝm + Ŝn Ŝm Ŝn Ŝm (3.5 Neglecting correlations of the form (Ŝn Ŝn (Ŝm Ŝm we receive the 3 Heisenberg model in Mean-Field approximation Ĥ MF Heis = n gµ B Ŝ n (B + B MF n + J Ŝn Ŝm with B MF n = (n,m δ n J gµ B Ŝn+δ (3.6 where each spin at position n is subjected to an additional effective magnetic field B MF (called mean field resulting from the average spins of its neighboring lattice sites n + δ. Now we assume that B = Be z and Ŝn = Ŝ = Ŝz e z is independent of the lattice site. Assuming ν neighboring spins, we have B MF = νj Ŝz /gµ B. Then we find from Eq. (3.8 for S = /: ( Ŝz = tanh gµ B B νj Ŝz k B T At T = 0 all spins are aligned and M = gµ B Ŝz /V c = ±gµ B /(V c = ±M 0 even for B = 0. Note that in the latter case the direction of M is not defined and depends on the history of the system. This is accompanied by hysteresis effects. For higher temperatures and small values of Ŝz and B we may expand tanh x x x 3 /3 providing ( ( Ŝz gµ B B νj Ŝz + 3 gµ B B νj Ŝz k B T 6 k B T This provides us with a finite magnetization M = gµ B Ŝz /V c for B = 0 below a critical temperature T c = νj/(4k B. Close to T c we find M(T gµ ( β B Tc T 3 for V c T c T < T c T T c Our definition of J agrees with Ashcroft and Mermin (979 and Marder (000. In contrast Ibach and Lüth (003; Kittel (996; Czycholl (004 use J = J/. Snoke (008 uses effectively J = J/8 as he represents spin / by the variable s i = ±. 3 I follow Ashcroft and Mermin (979 which is also consistent with Kittel (996. There is a different (in my opinion incorrect argumentation providing B eff = δ J/ Ŝn+δ /gµ B used in Ibach and Lüth (003; Czycholl (004. The conceptual difference is partially hidden by different definitions of J!

45 Chapter 3: Magnetism 37 <S z > cT.5 due to magnons Mean field solution Full result (qualitatively ~ (T c* -T 0.5 ~ (T c -T k B T c νj/4 k B T Figure 3.: Spin polarization for the Heisenberg model for B = 0. The mean-field solution provides a qualitative understanding of the temperature dependence. However, correlations in the fluctuations reduce the impact of the local field, and lead to a lower critical temperature and different critical exponents. This feature can be seen in the low temperature behavior, where the magnons (spin waves reduce the polarization, see Eq. (3.7. with β = /. For T > T c the susceptibility reads ( (gµ B γ Tc χ(t µ 0 for 4V c k B T c T T c T > T c T T c with γ =. Thus χ diverges for T T c. While this behavior is qualitatively right, the critical exponents are replaced by β = 0.34 and γ =.38, if the fluctuations are properly taken into account 4. In the same way the critical temperature T c and the pre-factors change, see Fig. 3.. The magnetization vanishes above the critical temperature T c, where the susceptibility shows a divergence χ(t T T c γ Spin waves The ground state 0 of the Heisenberg-Hamiltonian (for J > 0 is the state where all spins are aligned (in z-direction with the magnetization M = M 0 = gµ B S/V c. The flip of a single spin costs an energy of SνJ. The resulting state is n = Ŝ n 0 / S where Ŝ± n = Ŝx n ± iŝy n raises/lowers the z-component of the spin at site n by. This state is not an eigenstate of the Hamiltonian as Ĥ Heis n = S [ĤHeis, Ŝ n ] 0 + S Ŝ n ĤHeis 0 = S [ĤHeis, Ŝ n ] 0 + E 0 n Now the spin operators at different sites commute with each other and using the identity Ŝ n+δ Ŝ n = Ŝz n+δŝz n + Ŝ n+δŝ+ n + Ŝ+ n+δŝ n we find ( Ŝ n+δ[ŝz z n, Ŝ n ] + Ŝ n+δ [Ŝ+ n, Ŝ n ] + Ŝ+ n+δ [Ŝ n, Ŝ n ] [ĤHeis, Ŝ n ] = J δ Using the commutation relations [Ŝz, Ŝ ] = Ŝ, [Ŝ+, Ŝ ] = Ŝz, provides us with [ĤHeis, Ŝ n ] = J ( Ŝz n+δŝ n + Ŝ n+δŝz n. δ 4 This is done by renormalization group theory. The exponents are according to page 7 of D. Amit Field theory, the renormalization group, and critical phenomena.

46 38 A. Wacker, Lund University: Solid State Theory, VT 05 Then we find Ĥ Heis n = E 0 n + JS δ ( n n + δ. However we can construct eigenstates in form of a spin wave with energies k = E k = E 0 + JS e ik RnŜ n 0 SN n ( e ik R δ δ } {{ } k a for small k and a cubic 3 dim. lattice which dominate the excitation spectrum at low energies. These excitations can be approximately treated as independently from each other. In this sense they are similar to the phonons in the lattice and are called magnons. Thus their excitation probability follows the Bose distribution f k = ( exp E k E 0 k B T and we find in quadratic approximation the total excitation f k = k V (π 0 dk 4πk ( exp JSk a k B T = V ( 3/ kb T dx. JSa π 0 e x } {{ } As each magnon reduces the total magnetization by gµ B /V and we find for simple cubic lattices with V c = a 3 [ M = gµ BS ( ] 3/ kb T (3.7 V c S 5/ J Thus the magnetization drops as M(t = M 0 constt 3/, which constitutes a significant deviation from the mean field result, see Fig. 3.. Similarly, the magnons contribute to the specific heat 5 T 3/. 5 See Czycholl (004

47 Chapter 4 Introduction to dielectric function and semiconductor lasers 4. The dielectric function In this section physical quantities such as the frequency or electric fields are frequently allowed to be complex in order to simplify calculations. In this case I denote quantities which are complex by an additional tilde (such as ω. Maxwell s equations in material div D(r, t = ρ free (r, t (4. div B(r, t = 0 (4. rot F(r, t = B(r, t (4.3 t rot H(r, t = j free (r, t + D(r, t (4.4 t together with the material relations B(r, t = µ 0 ( H(r, t + M(r, t and D(r, t = ɛ 0 F(r, t + P(r, t (4.5 Here F is the electric field averaged over the atomic scale and ρ free, j free refer to macroscopic charges/currents, respectively. In this chapter we concentrate on the polarization P(t which is related to the electric field via P(t = t dt χ(t t ɛ 0 F(t with the susceptibility function χ(t t which takes into account the history. It is important to realize, that the time argument in χ(t refers to the time difference between field and polarization and not to a change of material properties with time. A possible spatial dependence is ignored here and we assume that the response is local in space. We also assume an isotropic material, otherwise χ(t t becomes a tensor. We use the following convention for Fourier-transforms in time: h(ω = dt h(te iωt and h(t = π dω h(ωe iωt The conventional static susceptibility is given by χ = χ(ω = const leading to χ(t = χδ(t. 39

48 40 A. Wacker, Lund University: Solid State Theory, VT 05 If we define χ(t = 0 for t < 0 we find P(ω = χ(ωɛ 0 F(ω. Furthermore we define the dielectric function ɛ(ω = + χ(ω such that D(ω = ɛ(ωɛ 0 F(ω. In materials the dielectric function is essentially determined by four different effects:. The interaction with lattice vibrations, associated with elongations of the ionic charges. See Section 4... Interaction of light with free carriers in the conduction band. See Section Electron-transitions between the bands, see Section Electron-electron interactions, see the discussion in Chapter Kramers-Kronig relation While χ(t is a real quantity by definition (both P(t and F(t physical quantities its Fourier transform χ(ω can be complex. Causality implies that χ(t vanishes for negative times. This provides a relation between imaginary and real part of χ(ω, which is derived in the following. Considering complex frequencies ω the Fourier transformation χ( ω = dt χ(te i ωt of the 0 susceptibility converges for all ω with Im{ ω} > 0 if χ(t is bounded. Therefore the complex function χ( ω is analytical in the upper half of the complex plane. The theory of complex functions tells us that this implies for real ω 0 χ( ω 0 = d ω ω ω 0 + i0 = χ(ω { } χ(ω dω + ω ω 0 + i0 = dωp iπχ(ω + 0 ω ω 0 upper half plane as ω 0 i0 + is outside the closed path of integration, where we used the relation (sometimes called Sokhotski-Weierstrass theorem { } x x 0 ± i0 = P iπδ(x x + 0 (4.6 x x 0 Separating imaginary and real part we obtain the Kramers-Kronig Relations Re{χ(ω 0 } = π { } Im{χ(ω} dωp, Im{χ(ω 0 } = ω ω 0 π { } Re{χ(ω} dωp ω ω 0 (4.7 relating the imaginary part to the real part for any dielectric function satisfying causality. Math used for Eq. (4.6: Lets consider the function x x 0 + iɛ = x x 0 (x x 0 + ɛ i ɛ (x x 0 + ɛ For ɛ 0, the real and imaginary part both develop a singularity at x 0. This singularity can often be cured by using the expression in an integral for finite ɛ > 0 and performing the limit ɛ 0 at a later stage. This procedure defines the principal value and the delta function as { } P x x 0 x x 0 = lim δ(x x ɛ 0 (x x 0 + ɛ 0 = lim ɛ 0 π ɛ (x x 0 + ɛ This limit lim ɛ 0 for ɛ > 0 (which has to be taken after an integration over one of the variables x, x 0 is specified by the notation ɛ 0 + providing Eq. (4.6

49 Chapter 4: Introduction to dielectric function and semiconductor lasers Connection to oscillating fields Assuming a homogeneous and isotropic material without macroscopic charges of currents, Maxwell s equations give us for a plane wave { F(r, t = Re { F0 e k r ωt} i( ke and B(r, t = Re B0 e k r ωt} i( ke I(r = F(r, t H(r, t = F 0 e i ke k r Re{ k} µ 0 ω = F 0 e αe k r ncɛ 0 e k (4.0 where n = Re{ñ} is the refractive index, which describes the change in wavelength in the material. Here α can be measured by the absorption of light with increasing thickness of a sample. The refractive index is commonly determined by the reflection of the light intensity given by R = ñ ñ + (4. In- for normal incidence from vacuum(air, see section 7.3 of D. Jackson Electrodynamics. 3 specting Eqs. (4.0,4., we find: This radiation intensity is zero if Re{ñ} = 0, i.e. ɛ(ω 0 and no radiation can propagate in the material. This is accompanied with total reflection R = at the surface of the material. For the longitudinal fields (e k F 0, Eq. (4.8 provides the condition ɛ(ω = 0, which may be satisfied at a discrete frequency ω 0. As D(ω = ɛ(ωɛ 0 F(ω this allows for the presence of a finite electric field F (t e iω 0t with zero displacement D(t. Eq. (4.5 shows that the polarization P(t = F(t/ɛ 0 oscillates in antiphase with the field providing a natural oscillation of the system at ω 0. The zeros of ɛ(ω determine natural oscillations of the system, where the electric field is longitudinal and the magnetic field vanishes. In most cases, however, n is positive and the situation becomes particularly simple, if we consider a small change in the dielectric properties in a host material with real ɛ host. Then it is convenient to redefine [see, e.g., Sec.8 of Chow and Koch (999] P change (ω = χ rel (ωɛ host ɛ 0 F(ω As the magnetic susceptibility is small for most materials we neglect all magnetic properties and set in the following µ r = for simplicity. This does not hold for specially designed metamaterials, where even µ r < 0 can be realized, see R.A. Shelby, D.R. Smith, and S. Schultz, Science 9, 77 (00. 3 In practice one considers incidence under an oblique angle, where the polarization changes (ellipsometry, see chapter 6 of Yu and Cardona (999. the relations e k B 0 = 0, B0 = k ω e k F 0, ɛ(ωe k F 0 = 0 and k [ F0 e k (e k F ] 0 = ω c ɛ(ω F 0. (4.8 For transversal fields (e k F 0 = 0, we find the common electromagnetic waves with the dispersion k = ωñ c = k + iα with ñ = ɛ(ω (4.9 where α is the absorption coefficient (in unit /length for the radiation intensity I given by temporal average of the Poynting vector over one oscillation period

50 4 A. Wacker, Lund University: Solid State Theory, VT 05 Assuming χ rel / the change in refractive index is given by δñ = ( + χ rel (ωɛ host ɛ host ɛ host χ rel (ω/ (4. where we used the Taylor expansion + x + x/ x /8.... With k host = ω ɛ host /c we find from Eq. (4.9 ( { } Re{χ rel (ω} χ rel δk = k k host k host + O 8 ( { } (4.3 χ α k host Im{χ rel rel (ω (ω} + O Thus the imaginary part of χ rel (ω provides absorption and the real part a change in the refractive index (or, equivalently, the wavelength. If Im{χ rel (ω} < 0, the absorption coefficient becomes negative, i.e., the radiation intensity increases while transversing the medium, which is called gain associated with the gain coefficient G = α. 4. Interaction with lattice vibrations The lattice is formed by ions. Thus lattice vibration can be associated with local charge transfer on the atomic scale resulting in a polarization, which is of particular significance for optical phonons As a model system we consider an harmonic oscillator with mass m c effective charge q, frequency ω 0 and damping γ. For a given electric field F(t we obtain the equation of motion for the elongations s m c s(t + m c γṡ(t + m c ω 0s(t = qf(t s(ω(ω 0 ω iγω = q m c F(ω Here we neglect the retroaction of the polarization on the electric field for simplicity. 4 The electric dipole moment p = qs(t is proportional to the elongation and thus, the dielectric function has the form A ɛ(ω = ɛ( + ω0 ω iγω where ɛ( is the dielectric constant well above the phonon resonance, and A describes the strength of the interaction. We obtain the static dielectric constant ɛ(0 = ɛ( +A/ω0. 5 Thus we may write ( [ ] ɛ(ω ɛ(0 ɛ( = + ω 0 ɛ( ω0 ω (ω0 ω + (γω + i γω (4.4 (ω0 ω + (γω We find that the imaginary part of ɛ(ω is positive and has a pole at the resonance frequency, while the real part exhibits a dramatic drop there, see Fig For a full treatment, see Ashcroft and Mermin ( This allows for a quantitative determination of q, as used in Eq. (.6 for the matrix element of polar optical phonon scattering, which however, requires a more detailed treatment, see section.3. of Marder (000.

51 Chapter 4: Introduction to dielectric function and semiconductor lasers K Re{ε} Im{ε} model Re{ε} model Im{ε} hω (mev Figure 4.: Dielectric function of GaAs from Eq. (4.4 with experimental data [from Properties of GaAs, edited by M.R. Brozel and G. E. Stillman (Inspec, 995]. We use ɛ( = 0.88, ɛ(0 =.9, and ω 0 = 33.3 mev. For γ = 0 the dielectric function becomes zero for ω = ω with ω = ω0. As discussed in ɛ( Sec. 4.., this constitutes an oscillation mode with frequency ω where the vibration of the ions is modified by the polarization field. For a finite phonon vector q, we find the microscopic polarization charge P iq P. Thus, the retroaction of the polarization on the oscillation is only possible for longitudinal phonon modes q s. Therefore ω = ω LO can be identified as the frequency of the longitudinal optical phonons. If the ionic elongations s are perpendicular to q, there is no polarization charge with the same spatial behavior e iq r and ω 0 = ω TO can be identified as the frequency of the transverse optical phonons. 6 This provides us with the Lyddane-Sachs-Teller relation ωlo = ɛ(0 ωto ɛ( Further note, that the real part of ɛ(ω is negative for ω TO < ω < ω LO, which is causing total reflectance at the surface of the material. The imaginary part of the dielectric function has a pronounced peak at the frequency of the transversal optical phonons, ω TO, while vanishing of the real part defines the longitudinal optical phonon frequency ω LO. Transverse polar optical phonons couple to the electromagnetic radiation, which is of particular significance for q ω TO /c. The coupled modes are called polaritons. Please consult your textbook for details! ɛ(0 4.3 Interaction with free carriers The Drude model (.5 provides a frequency dependent conductivity for the electrons in the conduction band with density n c and effective mass m eff σ(ω = n ce m eff iω. τ Now the polarization changes by microscopic currents transferring charge by Ṗ = J, thus iωp(ω = J(ω = σ(ωf(ω and we find χ rel (ω = iσ(ω/(ωɛ host ɛ 0. This yields χ rel (ω = n ce m eff ɛ host ɛ 0 ω + i ω τ (4.5 6 This is only true for q ω 0 /c otherwise one has to include the formation of polaritons due to the interaction of transverse phonons with the light field, which is always of transverse character. At q = 0, where one cannot distinguish between transverse and longitudinal character, the transverse phonons appear indeed with the frequency ω LO.

52 44 A. Wacker, Lund University: Solid State Theory, VT 05 Adding the host contribution, this provides the dielectric function [ ɛ(ω = ɛ host ω pl τ ( i ] with the plasma frequency ω ω τ pl = + ωτ n c e m eff ɛ host ɛ 0 For ω pl τ < (moderately doped semiconductors, the real part is always positive and we find absorption due to the positive imaginary part. This is the free carrier absorption α free carrier = σ(0 c ɛ host ɛ 0 ω τ + which is of importance for the THz physics in doped semiconductors, see Yu and Cardona (999 for details. If, on the other hand, ω pl τ (metals, we find a zero of ɛ(ω at the plasma frequency ω pl, which is a self-sustained longitudinal oscillation, as discussed in the exercise. Note, that for /τ ω < ω pl we observe a negative ɛ(ω, which implies total reflection. As ω pl 0eV for metals (n 0 3 /cm 3, m eff m e, ɛ host, this implies that metals reflect visible light. 4.4 Optical transitions The electromagnetic field is given in Coulomb gauge by A(r, t = ( F0 e iq r iωt F iω 0e iq r+iωt with q = nω/c and A 0 q = 0 (4.6 where n is the refractive index (n 3.6 for GaAs for ω E gap =.43 ev. Using Eq. (.0 this gives us the perturbation potential 7 ˆV (t = ˆF e iωt + ˆF e iωt + O(F 0 with ˆF = e iωm e e iq r F 0 ˆp providing the scattering matrix element Ψ n k ˆF Ψ nk = ef 0 d 3 r e i(q+k k R e i(k q r u n iωm e V c V c N k (rˆpeik r u nk (r R } {{ } =δ k,q+k δ k,k e δ k iωm,kf 0 P n n(k e (4.7 with the momentum matrix element from Eq. (.6. Here we used the fact that the q-vector of the photon ( /µm is negligible with respect to the extension of the Brillouin zone, which implies that k k holds. Using ˆp = im e [Ĥ0, ˆr]/, we find for normalizable (i.e. localized eigenstates a, b of Ĥ0 the electric dipole moment: e µ ab = e a r b = a [Ĥ0, ˆr] b = E a E b In the same spirit, we define the transition dipole-moment as ie m e (E a E b a ˆp b µ n n(k = ie m e E n (k E n (k P n n(k (4.8 7 Notation as in

53 Chapter 4: Introduction to dielectric function and semiconductor lasers 45 albeit the Bloch states nk are not-normalizable and the expression n k r nk is ill defined. 8 However, this comparison suggests, that µ n n(k is of the order of elementary charge times lattice constant. Putting things together and taking into account that the energy-conserving δ-function implies E n (k E n (k = ± ω we find the total transition rate from Fermi s golden rule W nk n k = π F 0 µ n n(k [δ(e n (k E n (k ω + δ(e n (k E n (k + ω] Now we investigate transitions between the valence and the conduction band in a semiconductor and we find the net transition rate R v c = π F 0 µ cv (k δ(e c (k E v (k ω [f v (k f c (k] (4.9 k where the transition c v are due to stimulated emission and v c are due to stimulated absorption of a photon. Both processes are related to a change of the number of photons which can be quantified as follows: The decrease in energy density of the radiation field is R v c ω/v, while the average energy flux density is given by Eq. (4.0. The ratio gives the absorption coefficient α(ω = R v c ω V I = πω µ cv (k δ(e c (k E v (k ω [f v (k f c (k] (4.0 cɛ 0 n V k which describes the decrease of radiation intensity by length (unit /cm. Here µ cv is the component of the electric dipole matrix element in the direction of the electric field. Using parabolic bands with E c (k = E c + k /m c and E v (k = E v k /m h, a transition at k corresponds to the frequency ω = E c (k E v (k = E gap + k m r with m r = m cm h m c + m h In order to simplify Eq. (4.0 we assume a constant dipole matrix element and that f c/v (k only depends on the respective energy E c/v (k. Then we set E k = k /m r and use V k de k D joint (E k with the joint density of states D joint (E k = m3/ r Ek Θ(E π 3 k in the continuum limit (spin is contained in the k-sum and obtain α(ω = πω µ cv cɛ 0 n D joint( ω E gap ( [f v E v ( ω E gap m ( r f c E c + ( ω E gap m ] (4. r m h m c For a semiconductor in the ground state with f c 0 and f v we thus find the absorption for a pure semiconductor α Θ( ω E gap ω E gap as shown in Fig. (4.. For conversion to wavelength (in vacuum/air, the wavelength in the semiconductor is reduced by the factor /n we have λ = πc/ω =.4µm/E, where the energy is in ev. One can memorize that green light is approximately 550 nm, 550 THz or.3 ev. A collection of data for the band gap of different semiconductors is shown in Fig See M.G. Burt, J. Phys. Condensed Matter 5, 409 (993

54 46 A. Wacker, Lund University: Solid State Theory, VT 05 Figure 4.: Absorption of GaAs from Eq. (4. using E gap =.4eV, m r = 0.06, µ cv = e 0.5nm in comparison with experimental data [from Properties of GaAs, edited by M.R. Brozel and G. E. Stillman (Inspec, 995]. The strong deviation between experiment and theory around E gap is due to the exciton peak discussed in section α [0 3 /cm] K exp model hω [ev] Band gap energy (ev 4 3,5 3,5,5 0,5 0 AlN (6. ev SiC GaN BP UV InN IR GaP ZnSe AlP AlAs CdS GaAs Si 4 4,5 5 5,5 6 6,5 7 Lattice parameter (Å InP Ge PbS AlSb ZnTe CdSe GaSb InAs SnTe PbSe HgTe CdTe InSb Band gap (nm 999 Figure 4.3: Band gap and lattice constant of different semiconductors, full/open circles indicate direct/indirect gaps, respectively. (At indirect gaps the minimum of the conduction band is at a different position in the Brillouin zone than the maximum of the valance band, which does not allow for direct optical Martin H. Magnusson, Lund University 4.5 The semiconductor laser In thermal equilibrium we have f c (k f v (k and Eq. (4.0 shows that there is only absorption of radiation. However, in the vicinity of a pn-junction in forward bias, there is a strong injection of conduction band electrons from the n region and of holes from the p-region, see Fig This allows for inversion f c (k > f v (k, in particular for those k-values close to the band edge. This provides negative α(ω, i.e. gain, for frequencies close to E gap /. The occurrence of gain in such a pn-diode can be used to fabricate semiconductor lasers. In practice, one combines the pn-

55 Chapter 4: Introduction to dielectric function and semiconductor lasers 47 E (a thermal equilibrium E (b d forward bias high p-doping E c +eφ E c e-diffusion µ electron E v +eφ µ high n-doping high p-doping eu µ hole Recombination (Lasing h-diffusion high n-doping E v z z Figure 4.4: Scheme of a semiconductor laser: a pn-diode in equilibrium. b pn-diode under forward bias eu = µ el µ hole E gap. There is inversion within the recombination zone. junction with a semiconductor heterostructure (the advantage is quantified in Sects. 4.5.,4.5., see Fig If the confinement is smaller than the thermal wavelength of the carriers (typically 0 nm, one speaks about quantum well lasers Phenomenological description of gain The occupation f c (k in the active region depends on the carrier density N in the conduction band, which is determined by a rate equation 9 dn(t dt = Jη ed Nγ eff V R c v (4. where η is the efficiency, with which the carriers reach the active region of thickness d. γ eff is the effective recombination rate. Gain, the increase of the radiation intensity per length, can be expressed in the active region by G = A g (N N g (4.3 where N g is the density, where stimulated emission and absorption compensate each other. For small optical fields, R c v can be neglected in Eq. (4. and we have the stationary solution N = N 0 = Jη eγ eff d, G = G 0 = A g (N 0 N g (4.4 In contrast, for large intensities I of the optical field Eq. (4. gives us N = N g + N 0 N g, G = G 0 with I + I I sat + I sat = ωγ eff I sat A g where R c v ω = V GI has been used 0. Thus the gain saturates at high lasing intensities. 9 I follow chapter of Chow and Koch (999 here. 0 Note that γ eff comprehends non-radiative recombination γ nr as well as radiative recombination by spontaneous optical transitions. The latter give a rate which is N or N 3/ (see, e.g., Fig..3 of Chow and Koch (999. Thus γ eff (N depends on density and γ eff (N 0 should be used in Eq. (4.4 to determine N 0. while γ eff (N g γ nr is more appropriate for I sat.

56 48 A. Wacker, Lund University: Solid State Theory, VT 05 E (a E c (z d heterostructure E (b E c +eφ d heterostructure laser µ electron undoped Al x Ga (-x As undoped GaAs undoped Al x Ga (-x As eu high p-doping Al x Ga (-x As - + GaAs high n-doping Al x Ga (-x As E v (z µ hole -+ E v +eφ z z 4.5. Threshold current Figure 4.5: Scheme of a heterostructure laser A laser combines gain material with a resonator cavity of length L, with reflectivity R, R. Here one may use the (parallel! facets of the semiconductor crystal, exploiting Fresnel reflection. The condition for lasing is that the radiation is enhanced during a round-trip in the cavity, i.e. R R e (ΓG α absl > where the confinement factor Γ < is the overlap of the optical mode with the active region and α abs denotes the absorption in the material. This provides us with J > J th = eγ { effd N g + [ α abs ]} η A g Γ L log(r R where the threshold current density J th is a key figure of merit. In order to minimize power consumption and lattice heating a low value is desirable. This implies Low recombination rate γ eff. As non-radiative recombination occurs at lattice imperfections, this implies a good crystal quality. High capture efficiency η of the injected carriers to the active zone. Small thickness d of active region. For standard pn-junctions the active region is of the order of the diffusion length, which is typically several µm. Therefore a crucial improvement was the use of semiconductors heterostructures (Nobel prize for Alferov and Kroemer, 000 in the late 60ies, which confine the carriers to a smaller region, see Fig Small N g, and large A g, describing the gain properties of the material. As the carriers are spread over a wider k-range for increasing temperature, N g increases with temperature. The temperature dependence is often written as J th e T/T 0 where T 0 is the characteristic temperature of the laser. Good confinement of the optical mode around the active region by a waveguide resulting in large values of Γ. See section.7 of Ibach and Lüth (003 I do not know the origin of the exponential behavior.

57 Chapter 5 Quantum kinetics of many-particle systems 5. Occupation number formalism The occupation number formalism (also called second quantization is the central tool for describing the quantum mechanics of many-particle systems. Here I give a short summary. Detailed proofs are given in the textbooks Schrieffer (983; Kittel (987; Czycholl (004. If we treat a system of many identical particles, it is extremely cumbersome to work with many-particle wave functions as in section In particular it is difficult to keep track of the antisymmetry. Furthermore, this formulation is only appropriate for a fixed number of particles. To overcome these problems, the occupation number representation is a convenient way, which is the standard tool of many-particle physics. 5.. Definitions Here we start with a fixed basis set of orthonormal single-particle states ϕ n (r where n =,,... labels the states. For a crystal one typically uses the Bloch states and the addition of spin is straightforward. For an N-particle state we write the Slater determinant (3.0 formed by the states with indices n, n... n N (with n < n... < n N as an abstract state 0 n=, 0, n=n, 0, 0, n=n, n=n 3, 0,... = o, o,... o n,.... Here the numbers o n are if n {n, n... n N } and 0 otherwise. They give the occupations of the states ϕ n in the many-particle state. Two such states differ unless all occupations are identical, and we have the orthonormality relation o, o,... o, o,... = δ o,o δ o,o... which can be verified by using the corresponding Slater determinants. Furthermore we define creation â n and annihilation operators â n of a particle in state n by their action on our states via â n o, o,... o n,... = δ on,0( Sn o, o,... o n +,... (5. â n o, o,... o n,... = δ on,( Sn o, o,... o n,... (5. 49

58 50 A. Wacker, Lund University: Solid State Theory, VT 05 where S n = n n = o n. The notation indicates, that the creation operator is the adjoint of the annihilation operator. This is indeed true as o â n o = ( S n δo n,0δ on, i n δ o i,o = i o â n o holds for arbitrary states o = o, o,... and o = o, o,.... Now we define the vacuum state 0 = 0, 0, 0,... (5.3 which does not contain any particles. Note that this state is a physical state with norm one, i.e. 0 0 = in contrast to the null-element null of the space, which has null null = 0. For example this null-element is obtained by annihilating a state which is not there, â 0, o,... = null, or by multiplying an arbitrary state with zero, 0 Ψ = null. The difference between null and 0 becomes particularly clear for the action of a creation operator: â 0 =, 0, 0,... while â null = null In this spirit any many-particle Slater state formed by the states with indices n, n... n N can be written as 0, 0,, 0, 0,,, 0,... = â n n= n=n n=n n=n â n... â n N 0 3 which provides a very easy notation. Note that the exchange of two indices gives a factor due to the definition (5. as appropriate for fermions. Example: The singlet state (3. reads for a b: Ψ Singlet (r, s ; r, s = [ (ϕ a (r χ α (s ϕ b (r χ β (s ϕ b (r χ β (s ϕ a (r χ α (s + ] (ϕ b (r χ α (s ϕ a (r χ β (s ϕ a (r χ β (s ϕ b (r χ α (s and thus can be written as Ψ Singlet = [ â aαâ bβ 0 + â bαâ aβ 0 ] in occupation number representation. 5.. Anti-commutation rules The fermionic creation and annihilation operators satisfy the Anti-commutation rules {â n, â n } = 0 {â n, â n } = 0 } where {Â, ˆB =  ˆB + ˆB } {â n, â n = δ n,n (5.4 {Â, ˆB} = 0 implies  ˆB = ˆBÂ. Thus exchanging the order of the annihilation or creation operators provides a minus sign for fermions. If both operators refer to the same state (n = n we find: â n â n = â nâ n = 0. Thus it is not possible to annihilate or create two particles in the same state, in accordance with the Pauli principle.

59 Chapter 5: Quantum kinetics of many-particle systems 5 â n â n = â nâ n as well as â nâ n = â n â n. Thus exchanging the order of an annihilation and a creation operator for the same state gives an additional. â nâ n o, o,... o n,... = δ on, o, o,... o n,.... Thus â nâ n is the number operator counting the number of particles in state n. Proof: (exemplary for â n â n = δ n,n â n â n with n n For n < n we find â n â n o,... o n,... o n,... =â n δ on,0( Sn o,... o n,... n=n,... =δ on,δ on,0( Sn ( S n o,... 0 n=n,... n=n,... =δ on,( Sn ( â n o,... 0 n=n,... o n,... =( â n â n o,... o n,... o n,... and for n = n â n â n o,... o n,... =â n δ on,0( Sn o,... n=n,... = δ on,0 o,... o n,... = o,... o n,... δ on, o,... o n,... = o,... o n,... δ on,â n ( Sn o,... 0 n=n,... = o,... o n,... â n â n o,... o n,... = ( â n â n o,... o n, Field operators We define the field operators ˆΨ(r = n ϕ n (râ n ϕ n(râ n (5.5 ˆΨ (r = n which satisfy { ˆΨ(r, ˆΨ(r } = 0, { ˆΨ (r, ˆΨ (r } = 0, and { ˆΨ(r, ˆΨ (r } = δ(r r. We find o, o,... ˆΨ (r ˆΨ(r o, o,... = ϕ n (rϕ n(r o, o,... â n â n o, o,... = } {{ } nn n o nδ n,n o n ϕ n (r Thus ˆΨ (r ˆΨ(r gives us the total particle density at r. We may interpret the field operators as follows ˆΨ (r creates a particle at position r ˆΨ(r annihilates a particle at position r Historically, the transition from the Schrödinger wave function Ψ(r to the field operator ˆΨ(r was considered as a quantization. Therefore the name second quantization is frequently used synonymously for occupation number formalism. For spin-dependent systems this is generalized by ˆΨ(r, s = n ϕ n(r, sâ n.

60 5 A. Wacker, Lund University: Solid State Theory, VT Operators Now we consider a systems of N identical particles. We have single-particle operators Ô single particle = N Ô(i i= which act on each particle i individually. Typical examples are the kinetic energy ˆT (i = ˆp /m, external potentials Û(i = U(r i, or the spin-orbit interaction Ô(i = f(r iˆl i Ŝi. In the same spirit there are two-particle operators Ô two particle = N N Ô(i, j i= j= where the Coulomb interaction ˆV ee e (i, j = 4πɛ 0 r i r j is a typical example. Starting with the spatial representations ÔSR (r, Ô SR (r, r of the individual operators, it is suggestive to write in occupation number formalism Ô single particle = d 3 r ˆΨ (rôsr (r ˆΨ(r Ô two particle = d 3 r d 3 r ˆΨ (r ˆΨ (r ÔSR (r, r ˆΨ(r ˆΨ(r Using the definition of the field operators we find the Operators in occupation number formalism Ô single particle = O nm â nâ m with O nm = n,m d 3 r ϕ n(rôsr (rϕ m (r (5.6 Ô two particle = O nn ;m mâ nâ n â m â m nn,mm with O nn ;m m = d 3 rd 3 r ϕ n(rϕ n (r ÔSR (r, r ϕ m (r ϕ m (r (5.7 These important relations can be proven by checking that the matrix elements for arbitrary many-particle states are identical. For the single-particle case, N i= ÔSR (r i has the same matrix elements with Slater determinants like Ôsingle particle with the corresponding occupation number states. The two-particle case is analogous but lengthy. Example: The single-particle Hamilton-operator of the crystal reads in occupation number formalism Ĥ 0 = nk E n (kâ nkânk (5.8 A perturbation of lattice periodicity, such as the presence of ionized impurities, provides a Hamilton operator with terms â nk â nk, see exercises. Note, that different notations for the two-particle matrix elements can be found in the literature. E.g., Kittel (987; Schrieffer (983 use O nn ;m m = n, n Ô(r, r m, m which provides a different ordering of indices in the matrix element.

61 Chapter 5: Quantum kinetics of many-particle systems Temporal evolution of expectation values If the system is in a quantum state φ we can evaluate expectation values of arbitrary operators  in the form  = φ  φ Typically we do not know exactly the quantum state, but can give a probability P r to find the system in each many-particle quantum state φ r (which form a complete orthonormal set. Then the expectation value of the operator reads  = r P r φ r  φ r (5.9 Using i φ r = Ĥ φ r, the time-dependence of the expectation values is given by d dt  = P r ( φ r  φ r + φ r  φ r = P n ( φ r i Ĥ φ r + φ r ÂĤ φ r r r Thus we find (5.0 d dt  = i [Ĥ, Â] (5. Thus the time dependence of the expectation value is given by the expectation value of the commutator with the Hamilton operator Density operator Now we want to provide a more general quantum description of a system, where we do not know the specific state, but only can give probabilities P r to find it in certain states Ψ r (which are orthonormal. First note that such a system can not be described by the state Ψ = r P r Ψ r, as such a ket state would be a complete description of a system (this would also not be normalized as r P r <. In contrast, a helpful tool to fully characterize the system is the Density operator ˆρ = s φ s P s φ s (5. E.g., one obtains the expectation value of the observable related to  by taking the trace over the product of ˆρ and any operator { } Â: Trace ˆρ = φ r φ s P s φ s  φ r = P r φ r  φ r =  r s r Note the essential difference between (i a quantum states Ψ = ( + and (ii a statistical mixture where the states and have both the probability 50%. The density operators (and corresponding matrices with respect to the basis {, } read ˆρ (i = ( + ( + ( ˆρ (ii = + ( We derived this equation in the conventional Schrödinger picture, i.e. the states φ n are time dependent. Alternatively we can consider the states (or the density operator to be fixed, while the operators are time dependent with d dtâh(t = i [Ĥ, ÂH]. This is called Heisenberg picture.

62 54 A. Wacker, Lund University: Solid State Theory, VT 05 The time-dependence of the density operator is given by the von Neumann equation: d dt ˆρ = ] [Ĥ i φ s P s φ s + ] Ĥ φ s P s [ φ s = {Ĥ φs P s φ s φ s P s φ s Ĥ} i i s s s = [Ĥ, ˆρ] i Note the difference in sign with respect to Eq. (5.. 4 In thermal equilibrium, the density operator reads for fixed particle number N ( ( ˆρ = E exp r Ĥ { ( } k B exp T k B T Ĥ φ r φ r = with Z = Trace exp Z Z k r B T If there is particle exchange with a bath of chemical potential µ, thermodynamics gives us (here N r is the number of particles in state φ r. ( ( ˆρ = µn exp r E r µ k B exp ˆN Ĥ { ( T k B T µ φ r φ r = with Y = Trace exp ˆN } Ĥ Z Y k r B T where ˆN = n â nâ n counts the number of particles. If the Hamilton operator separates, Ĥ = n E nâ nâ n, this provides us with the Fermi distribution â nâ n = ( exp E n µ k B T + ( Semiconductor Bloch equation Now we want to study the interaction of electrons in a semiconductor with a classical light field F (z, t. ( F (z, t = e x F (ze i[(k+δkz ωt] + F (ze i[(k+δkz ωt] (5.4 where K = ɛ host ω/c is the wave vector in the semiconductor without carrier injection. Using d F (z dz = α F (z, Eq. (4.3 relates F (z and δk to the change in susceptibility by the nonequilibrium electron distribution in an operating laser. Taking into account the coupling of the electric field to the dipole moment (compare Sec. 4.4 we find the Hamilton operator in two-band approximation Ĥ = E c (kâ ckâck + E v (kâ vkâvk ( µ k â ckâvk + µ kâ vkâck F (z, t + Ĥinteraction k k k where µ k = e x µ cv (k is the dipole moment of the transition between conduction and valence band from Eq. (4.8. Using the notation of holes (.4 we have to replace the electron creation operator â vk by the hole annihilation operator ˆd k, resulting in Ĥ = [( E g + k â k â k + k ˆd m k c m ˆd ( k k µ k â ˆd k k + µ ˆd ] k k â k F (z, t +Ĥinteraction h (5.5 4 All equations are in Schrödinger picture here.

63 Chapter 5: Quantum kinetics of many-particle systems 55 0 ϕ v (x ϕ c (x ϕ v (x ϕ c (x ϕ c (x ϕ v (x 0 0 x/lattice period Figure 5.: Sketch of Bloch functions for conduction and valence band (without the e ik r factor which is identical for both together with their absolute values and overlap. where we set E = 0 at the edge of the valence band and use effective mass approximation. Furthermore the index c is skipped and the energy k E v(k of the full valence band has been omitted. The goal is to evaluate the polarization P = P (te x. As motivated in Fig. 5., the electron densities from conduction or valence band states ϕ ck (r and ϕ vk (r do not provide any displacement of charge from the core positions (symmetry points. In contrast, the product between these states provides a displacement of charge from the core positions given by the dipole moment µ k = e d 3 rxϕ ck (rϕ vk(r (which is only well-defined for tightly bound states as shown in figure. The definition (4.8 is more general.. Thus the macroscopic polarization becomes P (t = ( µ k â k V ˆd k + µ k ˆd k â k (5.6 k in occupation number formalism in the two-band limit addressed here. In these equations the spin is included in the k-summations, in order to simplify the notation. In order to determine the polarization we evaluate the equation of motion for ˆd k â k, which is given by Eq. (5. d dt ˆd k â k = i ] [Ĥ, ˆd k â k For k k, we need four exchanges of fermionic annihilation/creation operators to change the order of ˆd k â k and the part of Ĥ. This provides a factor of ( 4 =, and thus the commutator vanishes for k k. For k = k we have â kâk ˆd k â k = ( ˆd k â kâkâ k = 0 and ˆd k â k â kâk = ˆd k ( â kâkâ k = ˆd k â k Thus [â kâk, ˆd ] k â k = ˆd k â k Similarly [ ˆd ˆd k k, ˆd ] k â k = ˆd k â k (5.7 [â k ˆd k, ˆd ] k â k =â kâk + ˆd ˆd [ k k ˆd k â k, ˆd ] k â k =0 (5.8 [ â ˆd ] k k, â kâk = â ˆd k k [â k ˆd k, ˆd k ˆd ] k = â ˆd k k (5.9

64 56 A. Wacker, Lund University: Solid State Theory, VT 05 With these commutation relations we find the Semiconductor Bloch equations where and d i dt p k(t = ω k p k µ k F (z, t (n ek + n h k + [Ĥinteraction, ˆd ] k â k d ] i dt n ek(t = F (z, t (µ k p k µ kp k + [Ĥinteraction, â kâk d i dt n h k(t = F (z, t (µ k p k µ kp k + [Ĥinteraction, ˆd ˆd ] k k p k = ˆd k â k, n ek = â kâk, n h k = ˆd k ˆd k ω k = E g + k m c + k m h (5.0 (5. ( Free carrier gain spectrum Now we want to discuss the case of free carriers, i.e., the details of Ĥ interaction are neglected. The key function of this interaction part is the decay of the polarization which may be treated as [Ĥinteraction, ˆd kâ k ] i γp k where T = /γ is the dipole lifetime. γ can be evaluated microscopically and it is approximately given by the mean total scattering rate of the electron and hole state involved. Typical values are T 0. ps. Then we find from Eq. (5.0: p k (t = i t dt e (iω k+γ(t t µ k F (z, t (n ek (t + n h k (t (5.3 The t -dependence of the different terms is sketched in Fig. 5.. The term e γ(t t ensures that the integrant vanishes for t t T. As the occupations n k (t typically vary on a timescale which is longer than ps, we may approximate n k (t n k (t. The electric field, see Eq. (5.4, has components with e iωt and e iωt. Now e i(ω+ω kt is always oscillating very fast on the time scale T and therefore only the component e iωt of F (z, t contributes to the final value of the integral. (The neglect of e iωt is called rotating wave approximation. Together we find: p k (t i µ k F (z ei(k+δkz iωt (n ek (t + n h k (t γ + i(ω k ω (5.4 Now the complex susceptibility is defined as χ rel (ω = P (ω/[ɛ 0 ɛ host F (ω]. From Eqs. (5.6,5.4 we have P (ω = π V µ k F (ze i(k+δkz (n ek (t + n h k (t ω ω k + iγ. k Identifying F (ω = π F (ze i(k+δkz from Eq. (5.4 we obtain the susceptibility χ rel = k ( µ k ω ω k (n ek (t + n h k (t V ɛ 0 ɛ host (ω ω k + γ i γ (ω ω k + γ (5.5

65 Chapter 5: Quantum kinetics of many-particle systems cos(ω+ω k t cos(ω ω k t ω=.3x0 5 /s ω k =.3x0 5 /s -γ (t -t e - n(t γ=0 3 /s t -t (fs Figure 5.: Sketch of the different parts of the integrant in Eq. (5.3 with ω k = E gap + 0 mev for E gap =.43eV (GaAs. It is assumed that n k (t varies on a ps timescale (probably even slower. Note that this satisfies the Kramers-Kronig relation. Using α = ω ɛ host Im{χ rel (ω}/c, the imaginary part provides us with the earlier result (4.0 for gain in the limit γ Quasi-equilibrium gain spectrum For low intensities F of the laser field, Eqs. (5.,5. show that the occupations of the electron and hole states are essentially determined by the interactions. As the electrons and holes dominantly interact with themselves this provides a local thermal equilibrium in each band n ek ( = f e (k, n hk ( = f h (k exp Ee(k µ e k B + exp Eh (k µ h T k B + T Assuming charge neutrality (if the active region is undoped the chemical potentials µ e, µ h are determined by N = f e (k = f h (k k k where N is determined by Eq. (4.. We find from Eq. (5.5 the material gain G = ω ɛ host Im{χ rel (ω}/c. The change in refractive index is given by δn = ɛ host Re{χ rel (ω}/, where we have to subtract the result for the bare semiconductor (i.e. f e (k = f h (k 0 in the evaluation of Re{χ rel }. Results are shown in Figure 5.3. We find: We have gain in the region E g ω µ h + µ e. The gain becomes weaker with increasing γ. Numerically, we find approximately G = A g (N N g as in Eq. (4.3 at the gain peak. Here N g increases and A g drops with temperature. Gain is associated with a change in the refractive index of the semiconductor.

66 58 A. Wacker, Lund University: Solid State Theory, VT 05 G (/cm δn increasing N N=0 8 /cm 3 N=3*0 8 /cm 3 N=5*0 8 /cm 3 T=300 K, γ=0 3 /s E gap hω (ev T=300 K, γ=0 3 /s increasing N N=0 8 /cm 3 N=3*0 8 /cm 3 N=5*0 8 /cm E gap hω (ev G (/cm G max (/cm γ=0 /s γ=5*0 /s γ=0 3 /s increasing γ γ=*0 3 /s T=300 K, N=3*0 8 /cm E gap.45 hω (ev T=00 K T=300 K T=400 K increasing T γ=0 3 /s N (0 8 /cm 3 Figure 5.3: Gain for a GaAs-laser material calculated from Eq. (5.5 using effective mass approximation with m e = 0.067, m h = 0.5 (the real valence band structure of GaAs is dominated by 3 bands, which are represented by a single parabolic band for simplicity here, µ k = e 0.473nm, E gap =.43eV, ɛ host = Spectral hole burning At high intensities I of the optical field, gain saturation occurs, as discussed in subsection The treatment tacitly assumes that even at high intensities n ek and n hk are given by Fermi distribution. In contrast, Eqs. (5.,5.,5.4 show, that n ek and n h k are diminished for those k-values satisfying ω ω k γ, which is called spectral hole burning. This effect reduces the inversion in particular for those states, which are responsible for the laser transition, and thus gain is stronger reduced than estimated by the treatment in On the other hand scattering processes effectively restore a quasi-equilibrium in the bands, and thus this effect is only prominent at very high intensities, in particular for short laser pulses.

67 Chapter 6 Electron-Electron interaction The electron-electron interaction is a two-particle operator. In the basis of Bloch states with Bloch vector k, l and band indices m, n, Eq. (5.7 provides the two-particle operator in occupation number representation. ˆV ee = V ml,m l,n k,nkâ mlâ m l â n k â nk (6. with the matrix elements V ml,m l,n k,nk = ml,m l,n k,nk d 3 r d 3 r ϕ ml(rϕ m l (r 4πɛ host ɛ 0 r r ϕ n k (r ϕ nk (r The calculation of the matrix elements is essentially simplified by applying the Fourier decomposition of the (screened Coulomb potential, which reads: e e λ r 4πɛ 0 ɛ host r = Vq λ e iq r = d 3 qv λ e iq r q with V λ e V (π 3 q = (6. ɛ host ɛ 0 ( q + λ q and is used at many places throughout this chapter. In particular we use the symbol V q in the limit λ 0 (no screening. Here we are considering a finite crystal of volume V, so that we have a discrete set of Bloch vectors, which facilitates the numbering. Inserting the discrete Fourier transformation, the two spatial integrals disentangle. Using Eqs. (.,.0 and neglecting Umklapp processes the r integral provides d 3 r ϕ ml(re iq r ϕ nk (r δ nm δ l,k+q (6.3 assuming u m(k+q (r u mk (r, which holds for small q. In the same way we find d 3 r ϕ m l (r e iq r ϕ nk (r δ n m δ l,k q Putting things together provides V ml,m l,n k,nk = V e V q δ nm δ l,k+q δ n m δ l,k q q which reduces the number of running indices in Eq. (6. dramatically and we obtain the Operator for electron-electron interaction in occupation number formalism ˆV ee V nk,n k V q â n(k+qâ n (k qân k â nk with V q = q e ɛ host ɛ 0 q (6.4 59

68 60 A. Wacker, Lund University: Solid State Theory, VT 05 This can be interpreted as follows: Two electron with k and k interact by transferring the crystal momentum q as indicated in the Feynman diagram on the right. Note that the sum of quasimomenta k + k is conserved by the interaction. Here the respective band index is kept by the interaction. This is actually an approximation, see Eq. (6.3, and transitions between bands (as happening in the Auger effect, e.g. are possible for q 0 albeit their likelihood is much smaller than the processes considered is Eq. ( Coulomb effects for interband transitions 6.. The Hamiltonian Restricting Eq. (6.4 to the valence and conduction band we find ˆV ee = V k,k q V q ( â (k+qâ (k qâk â k + ˆd (k+q ˆd (k q ˆd k ˆdk â (k+q ˆd (k q ˆd k â k (6.5 where we replaced k q k and k q k for the hole operators. (k, k contain also the spin here. The electron-electron interaction is a part of Ĥinteraction in Eqs. ( and we find e.g. [ ˆVee, ˆd k0â k0] = V V q (â ˆd (k +q k0 â k â k0 +q + ˆd ˆd (k q k â k0 ˆd k0 q k q â (k +q ˆd (q k0 â k â k0 ˆd (k q ˆd k0 ˆdk â k0 q + ˆd (q k0 â (k0 qδ k,q k 0 Thus the right-hand side of the equation of motion for the polarization (5.0 contains higher expectation values such as â (k +q ˆd k0 â k â k0 +q. The same holds for the equations for n e and n h. Thus the Eqs. ( constitute no longer a closed system as exploited in Sec In contrast we should take into account the equation of motion for the two-particle expectation values â (k +q ˆd k0 â k â k0 +q, which will itself generate three-particle expectation values on the right hand side. This process will generate an infinite hierarchy of many-particle expectation values. Thus, approximations have to be done on a certain stage. 6.. Semiconductor Bloch equations in HF approximation The simplest approximation is to assume that the expectation values for four operators factorize into products of expectation values of two operators. For simple states (such as slater states the expectation values vanish unless an electron creation operator is paired with an annihilation operator and we find â (k +q ˆd k0 â k â k0 +q â (k +qâk 0 +q ˆd k0 â k â (k +qâk ˆd k0 â k0 +q Additional terms k (â kâk + ˆd k ˆd k V 0 V k + V k,k V 0 k,q V q describe the electron-electron interaction of the full valence band and are subsumed in redefined single particle energies. V

69 Chapter 6: Electron-Electron interaction 6 where the minus sign is a result of the ordering of the operators. For spatially independent quantities, the respective expectation values vanish unless they have the same k ( k for hole operators, see Eq. (6.. Thus we find â (k +q ˆd k0 â k â k0 +q â (k +q ˆd k0 â k â k0 +q HF = δ k,k 0 n ek0 +qp k0 δ q,0 n ek p k0 (6.6 which is called Hartree-Fock approximation (HF. Using the same procedure for all two-particle expectation values we obtain the Semiconductor Bloch equations in Hartree-Fock approximation ( d dt p k(t = iωk ren pk p k iω k (t (n ek + n h k + (6.7 t col ( d dt n ek(t = iω k (tp k iω nek k(tp k + (6.8 t col ( d dt n h k(t = iω k (tp k iω nh k k(tp k + (6.9 t where the terms ω k and µ k F (z, t in Eqs. ( are replaced by ωk ren =E g + k + k V q (n ek+q + n h k q m c m h V q Ω k =µ k F (z, t + V q p k q V containing the exchange shift and the Coulomb field renormalization, respectively. q We find that the electron-electron interaction reduces the transition energies with increasing excitation (increasing N. Furthermore the polarizations of different k-values couple to each other. col 6..3 Excitons Let us now solve Eq. (6.7 in the limit of low densities n e 0, n h 0. Furthermore we assume that µ k = µ cv does not depend on k. The Fourier transformation of Eq. (6.7 in time reads: iωp k (ω = i (E g + k p k (ω + i m r µ cvf (ω + i V q p k q (ω γp k (ω V Performing a second Fourier transformation in space p k (ω = d 3 r p(r, ωe ik r (6.0 we obtain with Eq. (6. and the convolution theorem V q V qp k q (ω d 3 rv (rp(r, ω ] [E g e (ω + iγ p(r, ω = µ cv F (ωδ(r (6. m r 4πɛ host ɛ 0 r One might argue, that this is a double-counting. However, this can be formalized by the assumption, that the joint cumulant abcd c of the four-operator expectation value abcd vanishes. Using abcd = abcd c + abc d + abd c + acd b + bcd a ab c d ac b d ad b c cd a b bd a c bc a d + ab cd + ac bd + ad bc } {{ } +6 a b c d the underbraced terms provide the HF approximation (taking into account the sign changes by permutation of fermionic operators, while all other expectation values vanish due to the conservation of particle number. q

70 6 A. Wacker, Lund University: Solid State Theory, VT 05 In order to solve Eq. (6. we first solve the eigenvalue equation ] [ e Ψ α (r = E α Ψ α (r m r 4πɛ host ɛ 0 r which is called Wannier equation. It is identical with the equation for the hydrogen atom where m r = m e m p /(m e + m p is the reduced mass taking into account the proton motion as well. Thus, for E α < 0 we have a discrete set of levels with energies m r e 4 E α = 3π ɛ host ɛ 0 n α = Ryd n α where n α is the principal quantum number of the state α = (n, l, m. Ryd is the effective Rydberg constant taking into account the effective mass and the dielectric constant. These states are bound states between a conduction band electron and a hole in the valence band, which is called Wannier exciton. The wave function Ψ α (r describes the relative motion between the electron and the hole. In addition, we obtain a continuous set of states with energies E α 0, so that the functions Ψ α (r form a complete orthonormal system. Decomposing p(r, ω = p α (ωψ α (r α we find from Eq. (6. after multiplying with Ψ α (r and integrating over space: which provides us with Inserting into Eq. (6.0 we find p k (ω = α p α (ω = p(r, ω = α µ cv F (ωψ α (0 E g + E α (ω + iγ µ cv F (ωψ α(0ψ α (r E g + E α (ω + iγ µ cv F (ωψ α(0 E g + E α (ω + iγ d 3 kµ cvp k (ω and we find the suscepti- Finally, the polarization 3 is given by P (ω = bility: χ rel = µ cv ɛ 0 ɛ host α (for spin (π 3 d 3 r Ψ α (re ik r Ψ α (0 E g + E α (ω + iγ For γ 0 this provides absorption at sharp exciton peaks located at ω = E gap Ryd /n, as well as a continuum absorption spectrum above the band gap, see Fig The Hartree-Fock approximation In Eq. (6.6 we defined the Hartree-Fock approximation on the level of expectation values. Here we want to perform the equivalent approximation on the operator level. We start with the general two-particle operator Ô TP = â nâ n â m â m (6. 3 We neglect the additional term µ cv p k ( ω from Eq. (5.6, which is negligible for γ E g. This corresponds to the rotation wave approximation. 4 For a further discussion, see Sec. 3. of Chow and Koch (999.

71 Chapter 6: Electron-Electron interaction 63. x0 4 α [cm - ] K 96 K 90 K K Energy [ev] Figure 6.: Absorption in GaAs for different temperatures. Note that the band gap decreases with temperature. At low temperature an exciton peak becomes visible. After M.D. Surge, Physical Review 7, 768 (96. and look for an equivalent operator ÔHF containing only single-particle operators (and numbers which satisfies o, o,... ÔHF o, o,... = o, o,... ÔTP o, o,... (6.3 If this would hold for arbitrary states, the operators would be identical. Ô TP contains two annihilation and two creation operator. Thus its matrix elements vanish unless the states o, o,... and o, o,... (i are identical, (ii have an exchange in occupation [i.e. (o i, o j = (, 0 and (o i, o j = (0, ] for one pair of states (i, j, or (iii have an exchange in occupation for two different pairs of states. In contrast, a single-particle operator provides always zero for case (iii. Thus, the best approximation is obtained by the requirement, that the matrix elements agree for Slater states o, o,..., o, o,..., which are identical or differ by exchanging the occupation in an arbitrary pair of states. As shown below this requirement is satified by the Hartree-Fock approximation Ô TP = â nâ n â m â m ÔHF =â n â m â nâ m + â nâ m â n â m â n â m â nâ m â n â m â nâ m â nâ m â n â m + â n â m â nâ m (6.4 Here the expectation values have to be taken self-consistently with the true state of the system. The first line is commonly referred to as Hartree-term and the second as Fock-term. Note that the expectation value ÔHF is identical to the Hartree-Fock approximation for the expectation value ÔTP used in (6.6, showing the equivalence between both definitions. Indeed there are many different ways to define a Hartree-Fock approximation, which essentially boil down to the same thing. 6.. Proof We first consider the case (ii of different states, which can be written as o, o,... = â i â j o, o,... with i j. (Here we neglect a factor ( for an odd number of occupied states between i and j for simplicity. o, o,... â j â iâ nâ n â m â m o, o,... = δ om,δ om, (δ m,n δ n,jδ m,i δ m,nδ n,jδ m,i + δ m,n δ n,jδ m,i δ m,n δ n,j δ m,i

72 64 A. Wacker, Lund University: Solid State Theory, VT 05 For Ψ = o, o,... we may replace δ om,δ m,n = Ψ â nâ m Ψ 5 and find that the single-particle operator Ô HF SP = â n â m Ψ â nâ m Ψ â n â m Ψ â nâ m Ψ + â nâ m Ψ â n â m Ψ â nâ m Ψ â n â m Ψ satisfies Eq. (6.3 for arbitrary states which differ by exchanging the occupation of two states. Now consider the case (i of identical states o, o,... = o, o,..., which provides We find o, o,... â nâ n â m â m o, o,... = δ om,δ om,(δ n,m δ n,m δ n,m δ m,n o, o,... ÔTP o, o,... = o, o,... ÔHF SP o, o,... Ψ â n â m Ψ Ψ â nâ m Ψ + Ψ â nâ m Ψ Ψ â n â m Ψ } {{ } =O HF double counting As the state o, o,... is normalized this double counting correction Odouble HF counting can be established by adding this number to ÔHF SP. This does not affect the case (ii where o, o,... and o, o,... are orthogonal. Thus ÔHF = ÔHF SP + OHF double counting is the effective operator to approximate ÔTP, which provides Eq. ( Application to the Coulomb interaction The Coulomb interaction reads in occupation number representation ˆV ee = d 3 r d 3 r ˆΨ (r, s ˆΨ (r, s e 4πɛ 0 r r ˆΨ(r, s ˆΨ(r, s ss Now we want to replace this two-particle interaction by an effective mean field model, which operates on single particles, i.e. containing only one annihilation and one creation operator. The other two operators are replaced by their expectation value. Here we have four possibilities: ˆΨ (r, s ˆΨ(r, s and ˆΨ (r, s ˆΨ(r, s provide the same result exchanging the particle coordinates and yield the Hartree term ˆV Hartree = d 3 r ˆΨ (r, s ˆΨ(r, s d 3 r e 4πɛ s 0 r r ˆΨ (r, s ˆΨ(r, s s d 3 r d 3 r e 4πɛ 0 r r ˆΨ (r, s ˆΨ(r, s ˆΨ (r, s ˆΨ(r, s ss which describes the interaction of the electron (with charge e at r with the classical electric potential φ(r = s d 3 r e 4πɛ 0 r r n(r of the electron distribution. ˆΨ (r, s ˆΨ(r, s and ˆΨ (r, s ˆΨ(r, s provide the Fock term ˆV Fock = d 3 r d 3 r ˆΨ (r, s ˆΨ(r, s e 4πɛ s,s 0 r r ˆΨ (r, s ˆΨ(r, s + d 3 r d 3 r e 4πɛ 0 r r ˆΨ (r, s ˆΨ(r, s ˆΨ (r, s ˆΨ(r, s s,s 5 The following equations hold for the state Ψ = o, o,... as well. I believe that Ψ = α o, o,... + β o, o,... with α + β = is also possible, but could not proof it yet.

73 Chapter 6: Electron-Electron interaction 65 which describes the exchange interaction due to the antisymmetry of the electronic wave function. It constitutes a single-particle interaction with a nonlocal potential. In both cases we have subtracted the product of the expectation values as in the mean field model for the Heisenberg model, see Sec The remaining terms of the full interaction ˆV correlations = ˆV ee ˆV Hartree ˆV Fock are called correlation energy, and cannot be expressed as a single-particle potential. Now we insert ˆΨ(r, s = kσ ϕ ck(rχ σ (sâ kσ for a single conduction band with two spins, χ σ (s = δ σ,s. (In this paragraph we write the spin state σ explicitly, while it is tacitly included within k otherwise. If we restrict to a homogeneous electron gas [i.e. setting â k σ â kσ = δ k,kδ σσ â kσâkσ, see Eq. (6.] we obtain the Hartree-Fock part ˆV HF = [ V 0 â k V σ â k σ â kσâkσ ] â k σ â k σ â kσâkσ kσk σ V q [ â V (k+qσâ(k+qσ â kσâkσ ] (6.5 â (k+qσâ(k+qσ â kσâkσ kσq 6.3 The free electron gas Let us consider a free electron gas of density n = N/V and evaluate the expectation value of the total energy E G = Ψ G ˆT + ˆV Background + ˆV ee Ψ G where Ψ G is the N-particle ground state, ˆT = k k m â kâk is the kinetic energy and ˆV Background describes the interaction with a homogeneous positive background charge of the same density n (this is called jellium model. It can be shown that the Slater state Ψ F (F stands for Fermi sphere, where all k-states up to k F = (3π n /3 corresponding to E F = m (3π n /3 (6.6 are occupied, is the ground state of the Hamiltonian, if we restrict the Coulomb interaction to its Hartree-Fock part ˆV HF from Eq. (6.5. Then we find (see, e.g. Sec. 5.4 of Czycholl (004 The kinetic energy per particle reads Ψ F ˆT Ψ F = k F k k m V = (for spin (π 3 kf 0 dk 4πk k m = N 3 5 E F The Hartree part V kk V 0 exactly cancels with ˆV Background. [ ] â kâk â k â k â kâk â k â k The Fock part yields (after some tedious calculation Ψ F V kq V q (â kâk â kâk â (k+qâ(k+q Ψ F = N 3e 6π ɛ 0 k F

74 66 A. Wacker, Lund University: Solid State Theory, VT 05 The total energy per particle is now a function of the density n, which can be conveniently expressed using Ryd = m ee 4 3π ɛ 0 = 3.6eV a B = 4πɛ 0 m e e = 0.53Å n = 4 3 πa3 Br 3 s where r s is the mean distance between the electrons in units of the Bohr radius. Then we find the total energy per particle E HF G N =.099Ryd r s 0.963Ryd r s (6.7 in Hartree-Fock approximation. This energy has its minimum at r s = 4.83 corresponding to an electron density.43 0 /cm 3 with the value EG HF /N = 0.095Ryd =.3eV. These values are comparable with the densities and separation energies of alkali metals (4.7 0 /cm 3 and.63 ev for Li; /cm 3 and ev for Cs, see tables.4 and 3. of Kittel (996. Thus we have an approximate description of the binding in metals. For higher densities (lower r s too much kinetic energy is needed to satisfy the Pauli principle. On the other hand, for lower densities (higher r s the attractive interaction between the positive background and the electrons (which is not fully compensated by electronic repulsion due to the Fock part becomes too weak. Treating the correlation part ˆV ee ˆV HF within perturbation theory (a complex task one finds further terms 6 E G N =.099Ryd 0.963Ryd Ryd log r rs s 0.094Ryd + O(r s r s Metals have densities of the order of 5 0 /cm 3 (see table of Kittel (996, resulting in r s 3. Thus, the kinetic energy and the Fock part dominate the energy and correlation corrections are somewhat smaller, but become very important for densities less than 5 0 /cm A brief glimpse of density functional theory As seen above, the complicated many-particle interactions can be divided into the Hartree term, the exchange term, and further correlation effects. For a given particle, these result in effective potentials depending on all other particles. In density functional theory this interaction is described via the density n(r of the electron gas. For the Hartree term this provides the potential V H (r = d 3 r e n(r 4πɛ 0 r r, while corresponding functionals for the exchange and correlation terms, subsumed in V XC (r, can be shown to exist for the ground state, but are not explicitly known. A common approximation is the local density approximation (LDA: Considering a fictitious homogeneous electron gas with the sum of exchange and correlation energy E XC (n (as partially evaluated above and sets V XC (r = de XC(n(r. Thus one obtains the single-particle Hamiltonian dn Ĥ Kohn Sham {n(r} = m + V lattice(r + V H (r + V XC (r which is a functional of n(r via the effective potentials V H and V XC. Numerical solution of the Kohn-Sham equations ĤKohn Shamϕ n (r = E n ϕ n (r provides the eigenvalues E n and 6 according to Sec 5. of Mahan: Many Particle Physics (Plenum 993

75 Chapter 6: Electron-Electron interaction 67 eigenfunctions ϕ n (r. Occupying the N lowest energy-eigenstates until charge neutrality is reached, the density reads n(r = N n= ϕ n(r. Now one has to find a self-consistent solution, so that the evaluated density equals the input in ĤKohn Sham{n(r}, which is done iteratively. In this way one obtains the ground state energy and its spatial density distribution of the many-particle system. This allows, e.g., for the determination of the equilibrium configuration of a set of atoms, (i.e. the positions with lowest total energy, which is of high relevance both in chemistry and solid state physics. Frequently, the energies E n are also considered as electronic excitation energies, which is however problematic, as the heuristic justification given above can only be validated for ground states of the system. For more information see, e.g., the notes by K. Capelle The Lindhard-Formula for the dielectric function 6.4. Derivation If we consider spatially inhomogeneous electric fields, the polarization reads P(r, t = ɛ 0 d 3 r dt χ(r r, t t F(r, t which results in P(q, ω = ɛ 0 χ(q, ωf(q, ω or P(q, ω = χ rel (q, ωɛ 0 ɛ host F(q, ω after Fourier transformation in space and time. Now we have P(r, t = ρ ind (r, t and F(r, t = φ(r, t in the quasistatic case, where A(r, tis negligible. Thus t ρ ind (q, ω = iq P(q, ω = iq ɛ 0 ɛ host χ rel (q, ωf(q, ω =iq ɛ 0 ɛ host χ rel (q, ωiqφ(q, ω = q ɛ 0 ɛ host χ rel (q, ωφ(q, ω (6.8 and the ratio ρ ind (q, ω/φ(q, ω provides us with the susceptibility. 7 Now we want to evaluate this charge density induced by the potential for a free electron gas. Using Eq. (6.3 the electrical potential φ(r, t = V q eiq r φ(q, t results in the Hamiltonoperator 8 Ĥ = E n (kâ nkânk + ( e φ(q, tâ n(k+qânk (6.9 V nk nk q } {{ } } {{ } =Ĥ0 for the electrons in the crystal, where the Bloch states are the natural basis. The charge density, averaged over a unit cell, is given by ρ(r = e Ψ (rψ(r = e nk,n k ϕ n,k (rϕ nk(r â n k â nk (6.0 with the Fourier transformation using Eq. (6.3 ρ(q e nk â n(k qânk ρ(r e V =δĥ â n(k qânk e iq r (6. 7 This is actually the longitudinal part of the susceptibility tensor. A full calculation for the tensor structure can be found in S.L. Adler, Phys. Rev. 6, 43 (96. As discussed there, the tensor becomes diagonal for cubic crsytals and q 0, so that our result (6. is general in this case. 8 Note that φ(q, t will both contain external potentials and the potential of the electron gas itself. Thus this Hamilton operator treats the electron-electron interaction in Hartree approximation. q nk

76 68 A. Wacker, Lund University: Solid State Theory, VT 05 which holds for small q disregarding any charge fluctuations on the atomic scale. Now we consider the equation of motion for the expectation value â n(k qânk d dt â n(k qânk = i ] ( [Ĥ, â n(k qânk γ â n(k qânk δ q,0 f n (k where the latter term phenomenologically restores the homogeneity if δĥ vanishes. We find d dt â n(k qânk = i ( [E n(k q E n (k] â n(k qânk γ â n(k qânk δ q,0 f n (k + i ( e ( φ(q, t â n(k+q V qânk â n(k qân(k q q For q 0, the expectation value â n(k qânk is of order φ in the potential. Thus we have â n(k+q qânk = f n (kδ q,q + O{φ} and we find after Fourier-transformation in time ( i iω â n(k qânk ω = [E n(k q E n (k] γ â n(k qânk ω + πγδ(ωδ q,0 f n (k with the solution + i â n(k qânk ω = πδ(ωδ q,0 f n (k + e V Eq. (6. gives us the induced charge ρ ind (q, ω = e V φ(q, ω nk ( e V φ(q, ω (f n(k f n (k q + O{φ } f n (k f n (k q E n (k q E n (k + ω + i γ φ(q, ω + O{φ }. f n (k q f n (k E n (k q E n (k + ω + i γ + O{φ } by the potential φ and Eq. (6.8 provides us with the Lindhard formula for the dielectric function χ rel e f n (k q f n (k (q, ω = q ɛ host ɛ 0 V E n (k q E n (k + ω + i γ nk (6. which is the central result of this section. We find: Entirely filled or entirely empty bands do not contribute to χ(q, ω In the limit γ 0 we find Im{χ(q, ω} = e q ɛ host ɛ 0 V (f n (k f n (k + qπδ (E n (k + ω E n (k + q nk Thus Im{χ(q, ω} > 0 if there is a possibility for electronic transitions between an occupied state at k and an empty state k + q at a higher energy (difference ω. For a free electron gas E n (k = k /m which is filled up to k F this provides Im{χ(q, ω} = 0 for (q k F q/m ω (k F q + q /m, see Fig. 6.. Thus χ(q, ω is real both in the high-frequency and large q limit. Im{χ(q, ω} 0 for ω 0 as f n (k + q = f n (k for E n (k + q = E n (k in equilibrium.

77 Chapter 6: Electron-Electron interaction Im{χ}=0 Im{χ}>0 hω/e F ω plasmon Im{χ}= q/k F Figure 6.: General behavior of the Lindhard dielectric function. The ratio ω Plasmon /E F =.36 is appropriate for aluminum (n= /cm 3 corresponding to E F =.6eV, k F = /cm Plasmons Now we consider the dielectric function (6. in the limit of small q. Replacing k k + q in a part of the sum we find for γ 0 e Re{χ(q, ω} = q ɛ host ɛ 0 V nk ( f n (k E n (k E n (k + q + ω E n (k q E n (k + ω Using E n (k ± q = E n (k ± q E n (k + q /m n (k we find for small q e Re{χ(q, ω} q ɛ host ɛ 0 V In the limit q 0 we thus obtain nk ( f n (k ω q E n (k χ rel (0, ω = ω plasma ω with ω plasma = ( q m n (k + O{q4 } e ɛ host ɛ 0 V nk f n (k m n (k (6.3 (6.4 For ω = ω plasma we find ɛ tot = ( + χ rel ɛ host = and we have a natural oscillation of the system (see section 4.., which is called plasma oscillation. For a single band with constant mass we obtain ω plasma = e n/(mɛ host ɛ 0, which is the classical frequency of a homogeneous chargedensity oscillating with respect to a positive background. We find that ω plasma is of the order of 0 ev for metals with densities 0 3 /cm 3. Read Sec.9 of Ibach and Lüth (003 for more details! Correspondingly, the equation χ rel (q, ω = provides us with the plasma frequencies 9 ω plasma (q for wavelike oscillations with finite q, as indicated in Fig. 6.. For ω < ω plasma we have ɛ tot < 0 and the refractive index ñ(ω is purely imaginary. As discussed in section 4.. this implies total reflection of electromagnetic waves. Therefore metals with ω plasma > ω light reflect visible light. ] 9 For parabolic bands one finds ω plasma (q = ω plasma [ + 3v F q +..., after Kittel (996. 0ωplasma

78 70 A. Wacker, Lund University: Solid State Theory, VT Static screening In thermal equilibrium we have f n (k = f F (E n (k and for small q we can expand f n (k q f n (k + df F (E n (k [E n (k q E n (k] de Then Eq. (6. provides us for ω = 0 and γ 0 with χ rel (q, 0 λ q for small q with λ = e ɛ host ɛ 0 V nk ( df F (E n (k de (6.5 Thus we find ɛ tot = ɛ host ( + λ /q. Now ρ free (r, t = D(r, t = d 3 r dt ɛ tot (r r, t t F(r, t and F(r, t = φ(r, t provides after Fourier transformation which corresponds to ρ free (q, ω = q ɛ 0 ɛ tot (q, ωφ(q, ω φ(q, 0 = φ(r = d 3 r ρ free(r e λ r r 4πɛ 0 ɛ host r r ρ free (q, 0 ɛ 0 ɛ host (q + λ where Eq. (6. has been used. Thus we obtain a screened potential of an external test charge. For a nondegenerate semiconductor we have df F (E f de F (E/(k B T and Eq. (6.5 provides us with the inverse Debye screening length λ Debye = ne ɛ 0 ɛ host k B T (6.6 For a degenerate metal we have df F (E δ(e de n (k E F providing us with the inverse Thomas- Fermi screening length λ TF = e ɛ host ɛ 0 D(E F (6.7 Assuming a parabolic band structure, we find with Eqs. (.9,. the screening length λ TF = 0.6 Å for a typical metallic electron density of 5 0 /cm 3 and m eff = m e. Thus, the electronelectron interaction is only effective on a very short range in metals. Read also Sec 6.5 of Ibach and Lüth (003, or Chapter 0 of Kittel (996.

79 Chapter 7 Superconductivity 7. Phenomenology Plenty of materials exhibit an intriguing effect at low temperatures, as first observed by H. Kammerlingh Onnes (Leiden 9 for Hg below 4. K. For many metals the resistance drops abruptly to zero, i.e. below any measurement resolution, at the critical temperature T c (typically a few K. See Fig. 7.. Data for different materials is given in Fig. 7.. The magnetic field satisfies B = 0 inside the material. Thus, the magnetization M exactly compensates the magnetizing field H (Meißner-Ochsenfeld effect. Superconductivity vanishes above a critical field H c. The heat capacity exhibits a discontinuity at the transition temperature T c. This indicates a phase transition (of second order between a normal and a superconducting phase. Superconductivity is an effect due to the electron gas of the solid. There is no change of lattice structure at the transition. There is an energy gap of the order of mev in the excitation spectrum, which can be manifested by the Figure 7.: Original measurement of the resistance of Hg as a function of temperature. At 4. K, the resistance drops abruptly below the resolution of the measurement.(from Wikipedia Commons absorption of microwaves. Electron tunneling (see Fig. 7.3 and the exponential behavior e /k BT in the specific heat 3 for T T c show the gap as well. Consult your textbook for details (e.g. Sects. 0.+ of Ibach and Lüth (003! For history, see and D. van Delft and P. Kes, Physics Today 63(9, 38 (00 Note that for type- superconductors there is a finite magnetic induction B within the superconductor for fields H c < H < H c, while the material is still superconductive. 3 page 344 of Kittel (996. 7

80 7 A. Wacker, Lund University: Solid State Theory, VT 05 Figure 7.: Critical temperature and critical magnetic fields of several materials (from Wikipedia Commons -4 4 Cu normal Nb superconductor Figure 7.3: Tunneling current from copper to niobium at 0.38 K, see M.G. Castellano et al. IEEE Trans. Appli. Supercond. 7, 35 (997. The vanishing current for V mv indicates the presence of a gap in the single particle spectrum, as indicated on the right hand side. U (mv I (ma - Energy (mev E F I eu E F Density of states The experimental results can be interpreted by the existence of superconducting electron density n s establishing a superconducting current density j s. The vanishing resistance indicates that there is no friction term in the acceleration of electrons, thus m v s = ef or the. London equation: d dt j s = nse m F (7. In addition there is a normal part of the electron density n n = n n s, which contributes to a normal current density j n = σ n F. This is called the two-fluid model. The key point is the gap (T for creating normal fluid excitation from the superconducting component. This gap vanishes for T T c and the distinction between normal and superconducting component vanishes similar to the difference between a gas and a liquid at the critical point. With Maxwell s equation F = Ḃ and Eq. (7. we find d ( j dt s + nse B = 0 and thus m j s + nse B = const. The observation of zero magnetic induction within a superconductor m suggests, that the constant is zero and thus we obtain the

81 Chapter 7: Superconductivity 73. London equation: j s = n se m B (7. Together with the Maxwell equation B = µ 0 j (stationary case we find B = µ 0n se B. m This provides a spatial decay B(x j(x = B(0e z e x/λ L = B(0 λ L µ 0 e y e x/λ with the penetration length λ L L = at the surface (x = 0 of superconducting material. m µ 0 n s e ( BCS Theory The BCS theory was developed by Bardeen, Cooper, and Schrieffer in 957 to describe superconductivity. The theory is rather general for a system of interacting fermions with an attractive interaction ˆV a = V k,k (qâ V σ(k+qâ σ (k qâσ k â σk (7.4 kk σσ q (In this chapter k does not include spin. Next to the electron gas, it is also of importance for the structure of nuclei and atomic Fermi gases. Here we assume for simplicity { V0 for E V k,k (q = k E k+q < ω D and E k E k q < ω D (7.5 0 otherwise This can be motivated by an effective interaction with the lattice, where the exchange of virtual acoustic phonons with ω < ω D (the Debye cut-off frequency provides an attractive interaction as shown in Sec The idea is that the first electron creates a distortion of the lattice, which attracts the second electron, similar to two heavy balls on a spring mattress. The following treatment given here follows the lines given in Sec of Ibach and Lüth (003 and Chapter of Schrieffer (983. While the former is easier to read, the latter provides a much deeper discussion of the matter. 7.. The Cooper pair We consider a sea of electrons Ψ F occupying the states k < k F and add two electrons. Their wave function can be written as Ψ Pair (r, s, r, s = ϕ(re ik R χ(s, s with R = r + r and r = r r (7.6 in relative and center-of-mass coordinates. Decomposing into plane waves we have ϕ(r = g(ke ik r = g(ke ik r e ik r V V k where we are only allowed to use states with k > k F due to the Pauli principle. In the following we neglect the center of mass motion (which will increase the energy and set K = 0 for simplicity. Then we may write the state as Ψ = S σ σ g(kâ σ kâ σ k Ψ F (7.7 σ σ k 4 See, e.g., Sec. 0.3 of Ibach and Lüth (003 for a qualitative discussion and Chap. 8 of Kittel (987 or Sec.. of Czycholl (004 for details. k

82 74 A. Wacker, Lund University: Solid State Theory, VT 05 Then the stationary Schrödinger equation ( ˆT + ˆV a Ψ = E Ψ with the interaction of Eq. (7.4 provides us with (E k + E k g(k + V k V, k (k k g(k = Eg(k k where we neglected all contributions from the homogeneous Fermi sea Ψ F, which would affect the energy of all possible two-particle excitations in a similar way. Then the approximation (7.5 provides the relation g(k = V Ek + ω D 0 de k E k E Min{E F,(E k ω D } D(E g(k where D(E is the density of states for both spin directions. Setting D(E D(E F and assuming that g(k 0 for E k > E F + ω D we obtain the relation g(k = D(E F which requires the self-consistency condition = V 0D(E F EF + ω D V 0 E k E EF + ω D de k E F E k E = V 0D(E F 4 Min{E F de k g(k providing us with the ground state energy ( 4 E E F ω D exp V 0 D(E F in the limit V 0 D(E F where ω D E F E. We find ( EF + ω D E log E F E (7.8 The attractive interaction provides a bound state between electron pairs, which is called Cooper pair. The ground state energy of the Cooper pair is well separated from excited energies in the relative motion, which are above E F. In contrast, the center-of-mass motion with finite K in Eq. (7.6 provides a continuous spectrum 5. As g(k = g( k the wave function ϕ(r r is symmetric in the particle indices. To guarantee antisymmetry, χ(s, s in Eq. (7.6 must be antisymmetric, i.e. a singlet state. This means that S σ σ = 0 for equal spins in Eq. (7.7. Using g(k = g( k the Cooper pair (7.7 can be written as Ψ Cooper Pair = k g(kâ kâ k Ψ F which differs essentially from a simple Slater state â nâ m Ψ F. In particular it is isotropic in space, while exhibiting strong correlations between k and k. The binding energy of the Cooper pair (7.8 is a non-analytic function in the variable V 0 at V 0 = 0. This means that it cannot be represented by a Taylor series in the vicinity around V 0 = 0. (This can be verified by the vanishing of all derivatives with respect to V 0. Thus, its energy (7.8 cannot be obtained by standard perturbation theory in the interaction which would provide a Taylor series in V 0. 5 See Sec.. of Schrieffer (983 for a general treatment of these issues.

83 Chapter 7: Superconductivity The BCS ground state As the formation of the Cooper pair lowers the energy, the Fermi sea Ψ F is unstable in the presence of the attractive interaction. Thus we consider a new trial state ( sin Θ k + e iϕ k cos Θ k â kâ k 0 (7.9 Ψ BCS = k describing internal correlations 6. Note that this constitutes only an approximation for the exact ground state, which is in practice impossible to determine. If we set Θ k = 0 for k < k F and Θ k = π/ for k > k F we recover the Fermi sea Ψ F as a special case. For calculations we define Ψ k 0 BCS = k k 0 ( sin Θ k + e iϕ k cos Θ k â kâ k 0 which has zero occupation of the states k 0 and k 0. Thus â k0 Ψ k 0 BCS = â k 0 Ψ k 0 BCS [( ] = 0. As sin Θ k + e iϕ k cos Θk â kâ k, (sin Θ k + e iϕ k cos Θ k â k â k = 0 due to the even number of permutations for the fermionic operators a, the order in the product can be chosen arbitrarily and we find Ψ BCS Ψ BCS = Ψ k BCS ( ( sin Θ k + e iϕ k cos Θ k â k â k sin Θ k + e iϕ k cos Θ k â kâ k Ψ k BCS = Ψ k BCS sin Θ k + cos Θ k Ψ k BCS. Repeating this operation, we proof the normalization Ψ BCS Ψ BCS =. Furthermore we find the single-particle expectation value Ψ BCS â kâ k Ψ BCS = Ψ k BCS ( ( sin Θ k + e iϕ k cos Θ k â k â k â kâ k sin Θ k + e iϕ k cos Θ k â kâ k Ψ k BCS (7.0 = cos Θ k Ψ k BCS â k â k â kâ kâ kâ k Ψk BCS = cos Θ k and similarly Ψ BCS â kâ k Ψ BCS = cos Θ k. Thus the expectation value of the total particle number is N = k cos Θ k. The total energy is given by the expectation value E = Ψ BCS ˆT + ˆV a Ψ BCS. We note, that all elements are zero, unless the operators in the ˆV a are pairs of the form â kâ k, â kâ k, â kâ k, or â kâ k. Together we find E = E k cos Θ k + V k, k (qe i(ϕ k ϕ k+q cos Θ k+q sin Θ k+q sin Θ k cos Θ k V k kq 0 + V k,k (0 cos Θ k cos Θ k V k,k+q (q cos Θ k cos Θ k+q V V kk kq } {{ } Hartree-Fock terms (7. The occurrence of expressions â kâ k and â kâ k provide the Hartree-Fock terms, which describe the interaction of single-particle states (with occupation cos Θ k with each other. For 6 According to Sec..4 of Schrieffer (983 this can be written as an anti-symmetrized product of N/ identical two-particle wave functions ϕ(r r for ϕ k =const. However such a state would have a constant particle number in contrast to Ψ BCS.

84 76 A. Wacker, Lund University: Solid State Theory, VT 05 a fixed particle number N these provide a contribution to the energy, which does not crucially depend on the actual values of Θ k. Thus we neglect these terms in the following in accordance with the literature. This corresponds to the effective interaction ˆV eff a = V k, k (qâ V k+qâ k qâ kâ k = V k q 0 with V eff k,k = V 0Θ( ω D E k E k see, e.g. Schrieffer (983. kk V eff k,k â k â k â k â k, (7. The combination of terms â kâ k and â k â k describe the interaction process annihilating one pair of particles (called Cooper pair and creating another. This provides a term proportional to cos Θ k sin Θ k sin Θ k cos Θ k in the energy (7.. Eq. (7.0 shows that this term is finite if the states k and k are neither fully occupied (Θ = 0 or empty (Θ = π/. For the case of attractive interaction (negative matrix element it provides a reduction of energy for partially occupied paired states (i.e. 0 Θ k π/ and identical phase ϕ k = ϕ =const so that the attractive interaction adds up with positive prefactors. This shows that a fixed phase relation between all pairs is crucial for a low energy of the BCS state. ϕ is the phase of the BCS state and is important for the Josephson effect, e.g. We are looking for the minimum of E for a fixed particle number N = k cos Θ k. This can be achieved by considering the variation of E Nµ with respect to any Θ k, where µ is the chemical potential (the Lagrange multiplier, which can be determined at a later stage to fix the particle number N. Thus we obtain the conditions: (E Nµ Θ k = (E k µ cos Θ k sin Θ k } {{ } =0 =sin Θ k + V k, k (q cos Θ k+q sin Θ k+q (cos Θ V } {{ } k sin Θ k } {{ } q = sin Θ =cos Θ k+q k using V k, k,q = V k, k, q. Now we set k = k + q and use the approximation (7.5 implying the restriction E k E k < ω D. This provides us with (E k µ tan Θ k = k with k = V 0 Using tan Θ = sin Θ/ sin Θ we find sin Θ k = Ek + ω D de k E k ω D D(E k sin Θ k (7.3 k k + (E k µ (7.4 and with sin θ = cos Θ cos Θ we obtain a minimum for E at ( cos Θ k = E k µ (Ek µ + k (7.5 Now cos Θ k is the probability to find a pair k. Fig. 7.4(a shows: For E k < µ k, the states are essentially occupied For E k > µ + k, the states are essentially empty There is a smooth transition between and 0 around E k = µ with a width of k

85 Chapter 7: Superconductivity 77 f k, cos Θ k (a BCS State,T=0 Fermi sea, T=0 Fermi sea, k B T= / µ E k Excitation energy δe-µδn 0 (b hole excitation of Fermi sea electron excitations of Fermi sea BCS State µ E k Figure 7.4: (a occupation of single-particle states/cooper pairs as a function of the single particle energy E k. (b Excitation energies of the BCS state in comparison with excitations of the Fermi sea at zero temperature. Fig. 7.4(a, shows that cos Θ k resembles a Fermi distribution with k B T = /. However, this similarity is misleading: At first, we are considering the ground state of a quantum system, which is most relevant for zero temperature here. Secondly, the values cos Θ k are the expansion coefficients of a many-particle quantum state (7.9, where a defined phase relation between all parts exists. In contrast, the Fermi occupation function describes the average occupation of a level in a statistical ensemble, where all phase relations between different states are washed out. Provided k ω D, Eq. (7.3 shows that k = becomes independent of E k for E k µ ω D. Inserting Eq. (7.4 gives V µ+ ωd ( 0D(µ de k 4 µ ω D + (E k µ = V 0D(µ ωd sinh with the solution 7 = in the limit V 0 D(µ. ( ω D arcsinh[/v 0 D(µ] ω D exp V 0 D(µ ( Excitations from the BCS state Now we consider a single-particle excitation of the BCS state: ( k 0 = â k sin Θ 0 Ψ BCS = â k 0 sin Θ k + cos Θ k â kâ k 0 (7.7 k0 k k 0 Applying the effective interaction (7. we find the energy k 0 ˆT eff + ˆV a k 0 = E BCS + E k0 ( cos Θ k0 V and the number of electrons k k 0 V eff k 0 k cos Θ k sin Θ k } {{ } = k0 k 0 ˆN k 0 = N BCS + ( cos Θ k0. 7 The additional solution = 0 provides the normal state Ψ F which exhibits a larger energy. sin Θ k0 cos Θ } {{ k0 } sin(θ k0

86 78 A. Wacker, Lund University: Solid State Theory, VT 05 Thus, the state k 0 corresponds to a single-particle excitation for E k0 µ and a hole excitation for E k0 µ, while its nature is a mixture of both for E k0 µ. Now, for the effective energy of the excitation, one has to take into account the change in the particle reservoir as well. Thus we obtain with Eqs. (7.4,7.5 δe(k 0 µδn = k 0 ˆT + ˆV eff a k 0 E BCS µ( k 0 ˆN k 0 N BCS = (E k0 µ + k 0 (7.8 Thus creating an excitation containing a single-particle state k 0 costs at least the energy as shown in Fig. 7.4(b. This explains the gap observed in the tunneling experiment from Fig In comparison for the non-interacting Fermi sea Ψ F we have electron excitations for â σk 0 Ψ F for E k0 > µ = E F and hole excitations for â σk0 Ψ F for E k0 < µ = E F with δe(k 0 µδn = E k0 µ. Here excitations are possible at arbitrary small energies, which allows for absorption at low frequencies, as obtained by the Lindhard theory in Fig. 6.. The BCS state from Eq. (7.9 is the ground state, which is taken by the system at zero temperature. With increasing temperature, single-particle excitations become more frequently. Thereby the superfluid density decreases and so does the energy gap, which finally vanishes at k B T c 0.57 (T = 0 (7.9 as shown in Sec. 0.6 of Ibach and Lüth (003 or Sec...4 of Snoke ( Electron transport in the BCS state The presence of an electric field causes a shift in the Bloch vector according to k = ef. This will cause a center of mass motion K = ef for the Cooper pair from Eq. (7.6 and correspondingly the BCS state will be shifted by K/ ( sin Θ k + cos Θ k â (K/+kâ (K/ k 0 Ψ K BCS = k carrying the current density j s = en s K/m for a parabolic band E k = k /m as depicted in Fig. 7.5(b. This justifies the London equation (7.. The state Ψ K BCS exhibits a fixed phase relation between pairs with total crystal momentum K = k + k and has thus similar properties as the K = 0 BCS ground state (7.9. In particular there is the same gap (provided Kk F /m ω D. For a single particle scattering process as indicated in Fig. 7.5(c we have the energy balance δe = E k + E k E kf K/ E kf +K/ + (7.0 as the breaking of the pair (k, (K k and adding the electron at another place k corresponds to two single particle excitations. Thus, for small K, any single particle scattering event to restore the equilibrium costs energy in contrast to the simple Fermi sea, where is zero. Therefore the probability to change the entire state Ψ K BCS is negligible small. The latter would either imply N subsequent single-particle excitations with the cost of a total energy of N or the collective scattering of all particles within a single process. Both scenarios become extremely unlikely for a macroscopic system. This explains, that scattering does not restore the zero-current state as in a normal metal. However the energy balance (7.0 becomes negative for K > m/ k F, when the gain in kinetic energy by single-particle excitations can compensate the gap energy, so that a macroscopic number of electrons will leave the BCS condensate, causing a breakdown of superconductivity. Thus there is a

87 Chapter 7: Superconductivity 79 Equilibrium Current carrying state restoring scattering event k y /k F 0 k k - (a (b K/ (c - 0 k x /k F - 0 k x /k F - 0 k x /k F Figure 7.5: The superconducting state (a in equilibrium, (b after acceleration by a pulse with finite electric field in x direction, (c a possible scattering process tho restore equilibrium. critical current density j c en s k F (7. which provides us with a critical magnetic field H c λ L j c from Eq. (7.3. See also exercises as well as Sec. 0.6 of Ibach and Lüth (003. With Eq. (7.9 we thus understand the approximately linear increase of H c with T c displayed in Fig Justification of attractive interaction In order to justify Eq. (7.5 we consider a model of free electrons with phonon interaction described by the Fröhlich Hamiltonian Ĥ = E k â kâk + ω qˆb qˆbq + g q â k+qâk(ˆb q + ˆb q k q k,q } {{ } } {{ } =Ĥ0 =Ĥi (7. A canonical transformation changes all states via X X = Û a and operators as Ô Ô = ÛÔÛ with an unitarian operator Û (i.e. Û = Û. Such a transformation maintains all physical properties such as equations of motion and expectation values. A convenient choice is Û = exp( iŝ, where Ŝ is hermitian (i.e. Ŝ = Ŝ. Then we find ( Ĥ = exp( iŝĥ exp(iŝ = iŝ Ŝ Ĥ ( + iŝ Ŝ + O(S 3 =Ĥ + i[ĥ, Ŝ] + [Ŝ, [Ĥ, Ŝ]] + O(S3 The idea is to find an operator Ŝ, which cancels the electron-phonon interaction Ĥi in order g. This is the case if i[ĥ0, Ŝ] = Ĥi holds, which is achieved by setting Ŝ = i ( â k g +q â k ˆbq â k q E k,q k +q E k ω + +q â k ˆb q q E k +q E k + ω q Using g q = g q and ω q = ω q this provides us with Ĥ = Ĥ0 + i [Ĥi, Ŝ] + O(g3 = Ĥ0 + ˆV a + ˆV Polaron + ˆV sp + O(g 3

88 80 A. Wacker, Lund University: Solid State Theory, VT 05 Here ˆV a = k,k,q ω q g q (E k q E k ( ω q â k+qâ k qâk â k is the interaction used in Eq. (7.4. It is attractive if E k q E k < ω q, which suggests the approximation (7.5, as ω D is an estimate for the maximum frequency of the acoustic branch. ˆV Polaron = ( g q â kâk E k q k+q E k + ω q is the polaron shift, which renormalizes the single particle energies and can be tacitly incorporated into Ĥ0. The remaining term ( ˆV sp = (ˆbq + g q g ˆb qˆb q ˆb q (ˆb q E k qq k +q E k ω + q + ˆb ( q q E k +q E k + ω â k +q +qâk â k +q â k q q also modifies the single particle levels, if there are specific excitations of the phonon spectrum. However it does not constitute an interaction between the electrons.