FYS Vår 2015 (Kondenserte fasers fysikk)

Transcription

1 FYS Vår 2015 (Kondenserte fasers fysikk) Pensum: Introduction to Solid State Physics by Charles Kittel (Chapters 1-9 and 17, 18, 20, and Appendix D) Andrej Kuznetsov delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO Tel: , e-post: andrej.kuznetsov@fys.uio.no visiting address: MiNaLab, Gaustadaleen 23a

2 Condensed Matter Physics

3 Condensed Matter Physics Solid State Physics of Crystals

4 Condensed Matter Physics Solid State Physics of Crystals Properties of Waves in Periodic Lattices

5 Condensed Matter Physics Solid State Physics of Crystals Properties of Waves in Periodic Lattices Elastic waves Vibrations Phonon DOS Planck distribution Electron waves Free electrons Electron DOS Fermi-Dirac distribution

6 FYS3410 Lectures (based on C.Kittel s Introduction to SSP, Chapters 1-9, 17,18,20) Module I Periodic Structures and Defects (Chapters 1-3, 20) 26/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 2h 27/1 Laue condition, Ewald construction, interpretation of a diffraction experiment Bragg planes and Brillouin zones 4h 28/1 Elastic strain and structural defects in crystals 2h 30/1 Atomic diffusion and summary of Module I 2h Module II Phonons (Chapters 4 and 5) 09/2 Vibrations, phonons, density of states, and Planck distribution 2h 10/2 Lattice heat capacity: Dulong-Petit, Einstien and Debye models. Planck distribution. Comparison of different models 4h 11/2 Thermal conductivity 2h 13/2 Thermal expansion and summary of Module II. 2h Module III Electrons (Chapters 6, 7, 18 pp , and Appendix D) 23/2 Free electron gas (FEG) versus free electron Fermi gas (FEFG) 2h 24/2 Effect of temperature Fermi- Dirac distribution. Heat capacity of FEFG. DOS in nanostructures. Origin of the band gap 4h 25/2 Nearly free electron model. Kronig-Penney model. Empty lattice approximation. 2h 27/2 Number of orbitals in a band. Summary of Module III. 2h Module IV Semiconductors and Metals (Chapters 8, 9, and 17) 09/3 Recap of the energy bands. Metals versus isolators 2h 10/3 Semiconductors: effective mass method, intrinsic and extrinsic carrier generation 4h 12/3 Carrier statistics in semiconductors 2h 13/3 p-n junctions. Examples of optoelectronic devices. Summary of Module IV. 2h

7 Lecture 10: Free electron gas versus free electron Fermi gas Free electron gas (FEG) - Drude model - FEG explaing Ohm s law - FEG expaling Hall effect - FEG explaing Wiedemann-Franz law - FEG heat capacity Free electron Fermi gas (FEFG) a gas of electrons subject to Pauli principle One electron system wave functions orbits; FEFG in 1D in ground state FEFG in 3D in ground state Fermi-Dirac distribution and electron occupancy at T>0

8 Lecture 14: Free electron gas and free electron Fermi gas Free electron gas (FEG) - Drude model - FEG explaing Ohm s law - FEG expaling Hall effect - FEG explaing Wiedemann-Franz law - FEG heat capacity Free electron Fermi gas (FEFG) a gas of electrons subject to Pauli principle One electron system wave functions orbits; FEFG in 1D in ground state FEFG in 3D in ground state Fermi-Dirac distribution and electron occupancy at T>0

9 Free electron gas (FEG) - Drude model There could be different opinions what particular discovery was the main breakthrough for the acceleration of the condensed matter physics but a very prominent kick-off was by the discovery of the electron by J.J. Thompson in Soon afterwards (1900) P. Drude used the new concept to postulate a theory of electrical conductivity. Drude was considering why resistivity in different materials ranges from 10 8 m (Ag) to m (polystyrene)? Drude was working prior to the development of quantum mechanics, so he began with a classical model, specifically: (i) positive ion cores within an electron gas that follows Maxwell-Boltzmann statistics; (ii) following the kinetic theory of gases - the electrons ae in form of free electron gas (FEG), so that individual electrons move in straight lines and make collisions only with the ion cores; electron-electron interactions are neglected; (iii) Electrons lose energy gained from the electric field in colisions and the mean free path was approximately the inter-atomic spacing. Drude (or FEG) model successfully exlpained Ohm amd Wiedemann-Franz laws, but failed to explain, e.g., electron heat capacity and the magnetic susceptibility of conduction electrons.

10 Free electron gas (FEG) - Drude model Velocity of electrons in FEG e.g. at room temperature

11 Free electron gas (FEG) - Drude model #atoms per volume calculate as #valence electrons per atom density atomic mass

12 FEG explaining Ohm s law Apply an electric field. The equation of motion is integration gives and if is the average time between collisions then the average drift speed is for we get remember:

13 FEG explaining Ohm s law number of electrons passing in unit time current of negatively charged electrons current density Ohm s law and with we get

14 FEG explaining Ohm s law

15 FEG explaining Ohm s law line

16 FEG explaining Hall effect Accumulation of charge leads to Hall field EH. Hall field proportional to current density and B field is called Hall coefficient

17 FEG explaining Hall effect for the steady state

18 FEG explaining Hall effect

19 FEG explaining Wiedemann-Franz law Wiedemann and Franz found in 1853 that the ratio of thermal and electrical conductivity for ALL METALS is constant at a given temperature (for room temperature and above). constant Later it was found by L. Lorenz that this constant is proportional to temperature

20 FEG explaining Wiedemann-Franz law estimated thermal conductivity (from a classical ideal gas) the actual quantum mechanical result is this is 3, more or less...

21 FEG explaining Wiedemann-Franz law at 273 K metal 10-8 Watt Ω K -2 Ag 2.31 Au 2.35 Cd 2.42 Cu 2.23 Mo 2.61 Pb 2.47 Pt 2.51 Sn 2.52 W 3.04 Zn 2.31 L = Watt Ω K -2

22 FEG heat capacity an average thermal energy of an ideal gas particle, e.g. an electron from FEG, moving in 3D at some temperature T is: E 3 2 k B T Then for a total nr of N electrons the total energy is: Nk And the electronic heat capacity would then be: U 3 2 du dt If FEG approximation is correct this C el should be added to phonon-related heat capacitance, however, if we go out and measure, we find the electronic contribution is only around one percent of this, specifically at high temperature is still Dulong- Petit value,, that is valid. B T 3 Cel 2 Nk B

23 Free electron gas (FEG) - Drude model Why does the Drude model work so relatively well when many of its assumptions seem so wrong? In particular, the electrons don t seem to be scattered by each other. Why? How do the electrons sneak by the atoms of the lattice? What are mysterious positive charges realed by Hall effect measuremt in semiconductors? Why do the electrons not seem to contribute to the heat capacity?

25 Free electron Fermi gas a gas of electrons subject to Pauli principle At low temperature, free mean path of a conduction electron in metal can be as long as 1 cm! Why is it not affected by ion cores or other conduction electrons? (30 seconds discussions) Motion of electrons in crystal (matter wave) is not affected by periodic structure such as ion cores. Electron is scattered infrequently by other conduction electrons due to the Pauli exclusion principle

27 One electron system wave functions - orbits Neglect electron-electron interaction, infinite potential well, simple QM solution Standing wave B. C. n = 1, 2, - The Pauli exclusion principle n: quantum number m(=1/2 and -1/2): magnetic quantum number - degeneracy: # of orbitals with the same energy - Fermi energy (E F ): energy of the topmost filled level in the ground state of the N electron system In this simple system, every quantum state holds 2 electrons => n F = N/2 Fermi energy: Great, if we know the electron density, we know the Fermi energy!

29 FEFG in 3D Only if we know how much space one k point occupies (dk x dk y dk z = (2π/L) 3 ) Invoking periodic boundary condition instead of the infinite potential wall (standing wave) boundary condition, we get traveling waves as solutions: Due to spin Fermi wave vector Fermi energy Fermi velocity

30

32 Fermi-Dirac distribution and electron occupancy at T>0 Describes the probability that an orbit at energy E will be occupied in an ideal electron gas under thermal equilibrium is chemical potential, f( )=0.5; at 0K, F In comparison, High energy tail approximation E E F 3k B T Boltzmann-Maxwell distribution

33 Fermi-Dirac distribution and electron occupancy at T>0 # of states between E and E+dE Note here: N N( ) D(E) has a unit of ev -1.

35 FYS3410 Lectures (based on C.Kittel s Introduction to SSP, Chapters 1-9, 17,18,20) Module I Periodic Structures and Defects (Chapters 1-3, 20) 26/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 2h 27/1 Laue condition, Ewald construction, interpretation of a diffraction experiment Bragg planes and Brillouin zones 4h 28/1 Elastic strain and structural defects in crystals 2h 30/1 Atomic diffusion and summary of Module I 2h Module II Phonons (Chapters 4 and 5) 09/2 Vibrations, phonons, density of states, and Planck distribution 2h 10/2 Lattice heat capacity: Dulong-Petit, Einstien and Debye models. Planck distribution. Comparison of different models 4h 11/2 Thermal conductivity 2h 13/2 Thermal expansion and summary of Module II. 2h Module III Electrons (Chapters 6, 7, 18 pp , and Appendix D) 23/2 Free electron gas (FEG) versus free electron Fermi gas (FEFG) 2h 24/2 Effect of temperature Fermi- Dirac distribution. Heat capacity of FEFG. DOS in nanostructures. Origin of the band gap 4h 25/2 Nearly free electron model. Kronig-Penney model. Empty lattice approximation. 2h 27/2 Number of orbitals in a band. Summary of Module III. 2h Module IV Semiconductors and Metals (Chapters 8, 9, and 17) 09/3 Recap of the energy bands. Metals versus isolators 2h 10/3 Semiconductors: effective mass method, intrinsic and extrinsic carrier generation 4h 12/3 Carrier statistics in semiconductors 2h 13/3 p-n junctions. Examples of optoelectronic devices. Summary of Module IV. 2h

36 Lecture 11: FEFG at T>0 and DOS in nanostructures (i) Repetition of DOS for FEFG (ii) Fermi-Diract distribution (iii) Deriviation and estimate for the FEFG heat capacity (iv) DOS in nanostructures

38 Repetition of DOS for FEFG Consider electrons as quantum particles in a box FEFG model means that

39 Repetition of DOS for FEFG ~ boundary conditions provide restrictions the wavevector k

40 Repetition of DOS for FEFG

41 Repetition of DOS for FEFG The volume of k x1 k y1 k z1 = (2π/L) 3 corresponds to only one k-state, accommodating 2 electrons k x1 k y1 k z1 = (2π/L) 3 While k max or k F accommodate N/2!

42 Repetition of DOS for FEFG For any E

44 The Fermi-Dirac distribution μ At T=0 all the states are filled up to the highest occupied state. This state is called the Fermi energy E F. It is equal to the chemical potential μ at T=0.

45 The Fermi-Dirac distribution Describes the probability that an orbit at energy E will be occupied in an ideal electron gas under thermal equilibrium is chemical potential, f( )=0.5; at 0K, F High energy tail approximation E E F 3k B T Boltzmann-Maxwell distribution

46 The Fermi-Dirac distribution ~ k B T

48 Estimate for the heat capacity of FEFG One of the greatest successes of the free electron model and FD statistics is the explanation of the T dependence of the heat capacity of a metal. k B T To calculate the heat capacity, we need to know how the internal energy of the Fermi gas, E t (T), depends on temperature. By heating a Fermi gas, we populate some states above the Fermi energy E F and deplete some states below E F. This modification is significant within a narrow energy range ~ k B T around E F. The fraction of electrons that we transfer to higher energies ~ k B T/E F, the energy increase for these electrons ~ k B T. Thus, the increase of the internal energy with temperature is C dq T V V const dt de T t dt proportional to n (k B T/E F ) (k B T) ~ n (k B T) 2 /E F. Note, E F = k B T F C e 2 nk 2 B kt B E F compare C V 3 2 nk B C for an ideal gas V de T t dt 2 kt B N E F The Fermi gas heat capacity is much smaller (by k B T/E F <<1) than that of a classical ideal gas with the same energy and pressure. The small heat capacity is a direct consequence of the Pauli principle: most of the electrons cannot change their energy, only a small fraction k B T/E F or T/T F of the electrons are excited out of the ground state.

49 Heat capacity of a metal: lattice + electrons two contributions: lattice and electrons electrons unimportant at high T but dominating and sufficiently low T

51 DOS in nanostructures (i) Motivation: intriguing carrier recombination in QW resulting in photons with energies above the band gap; (ii) DOS for low-dimentional FEFG (iii) DOS in quantum wells (iv) DOS in nanowires (v) DOS in quantum dots

52 Motivation: intriguing carrier recombination in QW resulting in photons with energies above the band gap Single quantum well - repetitions of ZnO/ZnCdO/ZnO periods MQWs

53 DOS for low-dimentional FEFG

54 DOS in quantum wells

55 DOS in quantum wells

56 DOS in nanowires and quantum dots Density of states per unit volume and energy for a 3-D semiconductor (blue curve), a 10 nm quantum well with infinite barriers (red curve) and a 10 nm by 10 nm quantum wire with infinite barriers (green curve).

57 DOS in nanowires and quantum dots

59 FYS3410 Lectures (based on C.Kittel s Introduction to SSP, Chapters 1-9, 17,18,20) Module I Periodic Structures and Defects (Chapters 1-3, 20) 26/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 2h 27/1 Laue condition, Ewald construction, interpretation of a diffraction experiment Bragg planes and Brillouin zones 4h 28/1 Elastic strain and structural defects in crystals 2h 30/1 Atomic diffusion and summary of Module I 2h Module II Phonons (Chapters 4 and 5) 09/2 Vibrations, phonons, density of states, and Planck distribution 2h 10/2 Lattice heat capacity: Dulong-Petit, Einstien and Debye models. Planck distribution. Comparison of different models 4h 11/2 Thermal conductivity 2h 13/2 Thermal expansion and summary of Module II. 2h Module III Electrons (Chapters 6, 7, 18 pp , and Appendix D) 23/2 Free electron gas (FEG) versus free electron Fermi gas (FEFG) 2h 24/2 Effect of temperature Fermi- Dirac distribution. Heat capacity of FEFG. DOS in nanostructures. Origin of the band gap 4h 25/2 Nearly free electron model. Kronig-Penney model. Empty lattice approximation 2h 27/2 Number of orbitals in a band. Summary of Module III. 2h Module IV Semiconductors and Metals (Chapters 8, 9, and 17) 09/3 Recap of the energy bands. Metals versus isolators 2h 10/3 Semiconductors: effective mass method, intrinsic and extrinsic carrier generation 4h 12/3 Carrier statistics in semiconductors 2h 13/3 p-n junctions. Examples of optoelectronic devices. Summary of Module IV. 2h

60 Lecture Origin of energy bands and nearly free electron model Recap of the one-electron approximation Intuitive picture Bragg-reflection for nearly free electrons Gap opening for nearly free electrons Solution of the Schrödinger equation 2. Kronig-Penney Model 3. Empty lattice approximation

61 Recap of one electron approximation Solving the Schrödinger equation for an all-electron wave function in a rigid lattice is still hopeless. We assume an effective one electron potential. we know for sure that...but not much more

62 The idea of energy bands: Na Consider one atom of Na: 11 electrons

63 The idea of energy bands: Na consider tow Na atoms / a Na 2 molecule: 22 electrons big separation Focus only on the valence (outer) electrons (3s). What happens when we move them together?

64 The idea of energy bands: Na consider two Na atoms / a Na 2 molecule: 2 3s electrons molecular energy levels capacity: 2x2=4 electrons Levels split up in bonding and anti bonding molecular orbitals and are occupied according to the Pauli principle. The distance between the atoms must be such that there is an energy gain.

65 The idea of energy bands: Na consider many (N) Na atoms (only 3s level) N levels with very similar energies, like in a super-giant molecule. We can speak of a band of levels. Every band has N levels. We can put 2N electrons into it (but we have only N electrons from N Na atoms).

66 The idea of energy bands: Si 4 electrons per atom 2 atoms per unit cell sp 3 : 8 states 4 sp 3 per cell 8 states per atom 16 states per unit cell

67 free electron model naive band picture

68 Bragg-reflection for nearly free electrons electron traveling perpendicular to crystal planes

69 Bragg-reflection for nearly free electrons consider only one direction (x) free electron wave function with a de Broglie wavelength Bragg condition with this gives a Bragg condition for electron waves: 69

70 Bragg-reflection for nearly free electrons Bragg condition for electron waves: Bragg reflection results in standing, not traveling electron waves two possible linear combinations of

71 Gap opening for nearly free electrons

72 General form of Bloch theory Schrödinger equation for a periodic potential solutions have the form of Bloch waves plane wave lattice-periodic function

73 Kronig-Penney Model Kronig and Penney assumed that an electron experiences an infinite one-dimensional array of finite potential wells. Each potential well models attraction to an atom in the lattice, so the size of the wells must correspond roughly to the lattice spacing.

74 Kronig-Penney Model An effective way to understand the energy gap in semiconductors is to model the interaction between the electrons and the lattice of atoms. R. de L. Kronig and W. G. Penney developed a useful one-dimensional model of the electron lattice interaction in 1931.

75 Kronig-Penney Model Since the electrons are not free their energies are less than the height V 0 of each of the potentials, but the electron is essentially free in the gap 0 < x < a, where it has a wave function of the form where the wave number k is given by the usual relation:

76 Kronig-Penney Model In the region between a < x < a + b the electron can tunnel through and the wave function loses its oscillatory solution and becomes exponential:

77 Kronig-Penney Model The left-hand side is limited to values between +1 and 1 for all values of K. Plotting this it is observed there exist restricted (shaded) forbidden zones for solutions.

78 Kronig-Penney Model Matching solutions at the boundary, Kronig and Penney find Here K is another wave number. Let s label the equation above as KPE Kronig-Penney equation

79 Kronig-Penney Model (a) Plot of the left side of Equation (KPE) versus ka for κ 2 ba / 2 = 3π / 2. Allowed energy values must correspond to the values of k for for which the plotted function lies between -1 and +1. Forbidden values are shaded in light blue. (b) The corresponding plot of energy versus ka for κ 2 ba / 2 = 3π / 2, showing the forbidden energy zones (gaps).

80 Lecture 18: Energy bands in solids Origin of the band gap and Bloch theorem Kronig-Penney model Empty lattice approximation Number of states in a band and filing of the bands Interpretation of the effective mass Effective mass method for hydrogen-like impurities

81 Empty lattice approximation Suppose that we have empty lattice where the periodic V(x)=0. Then the e - s in the lattice are free, so that

82 Empty lattice approximation in 3-dim.

83 Empty lattice approximation In simple cubic (SC) empty lattice

84 Empty lattice approximation Everything can be described within the 1 st B.Z.

86 FYS3410 Lectures (based on C.Kittel s Introduction to SSP, Chapters 1-9, 17,18,20) Module I Periodic Structures and Defects (Chapters 1-3, 20) 26/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 2h 27/1 Laue condition, Ewald construction, interpretation of a diffraction experiment Bragg planes and Brillouin zones 4h 28/1 Elastic strain and structural defects in crystals 2h 30/1 Atomic diffusion and summary of Module I 2h Module II Phonons (Chapters 4 and 5) 09/2 Vibrations, phonons, density of states, and Planck distribution 2h 10/2 Lattice heat capacity: Dulong-Petit, Einstien and Debye models. Planck distribution. Comparison of different models 4h 11/2 Thermal conductivity 2h 13/2 Thermal expansion and summary of Module II. 2h Module III Electrons (Chapters 6, 7, 18 pp , and Appendix D) 23/2 Free electron gas (FEG) versus free electron Fermi gas (FEFG) 2h 24/2 Effect of temperature Fermi- Dirac distribution. Heat capacity of FEFG. DOS in nanostructures. 4h 25/2 Origin of the band gap. Nearly free electron model. Kronig-Penney model. 2h 27/2 Number of orbitals in a band. Summary of Module III. 2h Module IV Semiconductors and Metals (Chapters 8, 9, and 17) 09/3 Recap of the energy bands. Metals versus isolators 2h 10/3 Semiconductors: effective mass method, intrinsic and extrinsic carrier generation 4h 12/3 Carrier statistics in semiconductors 2h 13/3 p-n junctions. Examples of optoelectronic devices. Summary of Module IV. 2h

87 Lecture 13: Energy bands in solids (continuation) and recap of Module III Origin of the band gap and Bloch theorem Number of states in a band and filing of the bands Recap of FEG, FEFG, and fundamentals of energy band structure

89 Why do we get a gap? Let us start with a free electron in a periodic crystal, but ignore the atomic potentials for now At the interface (BZ), we have two counter-propagating waves e ikx, with k = /a, that Bragg reflect and form standing waves y E Its periodically extended partner - /a /a k

90 Why do we get a gap? y + ~ cos( x/a) peaks at atomic sites y - ~ sin( x/a) peaks in between y + y - E Its periodically extended partner - /a /a k

91 Let s now turn on the atomic potential The y + solution sees the atomic potential and increases its energy The y - solution does not see this potential (as it lies between atoms) Thus their energies separate and a gap appears at the BZ This happens only at the BZ where we have standing waves U 0 y + y - - /a /a k

92 Bloch theorem If V(r) is periodic with the periodicity of the lattice, then the solutions of the one-electron Schrödinger eq. Where u k (r) is periodic with the periodicity of the direct lattice. u k (r) = u k (r+t); T is the translation vector of lattice. The eigenfunctions of the wave equation for a periodic potential are the product of a plane wave exp(ik r) times a function u k (r) with the periodicity of the crystal lattice.

94 E E Number of states in a band k n d Sin k o d Sin90 k o d Sin 45 [100] [110] Effective gap d k 2 d k

95 Filing of the bands Divalent metals Monovalent metals Monovalent metals: Ag, Cu, Au 1 e in the outermost orbital outermost energy band is only half filled Divalent metals: Mg, Be overlapping conduction and valence bands they conduct even if the valence band is full Trivalent metals: Al similar to monovalent metals!!! outermost energy band is only half filled!!!

96 Filing of the bands SEMICONDUCTORS and ISOLATORS Band gap Elements of the 4 th column (C, Si, Ge, Sn, Pb) valence band full but no overlap of valence and conduction bands Diamond PE as strong function of the position in the crystal Band gap is 5.4 ev Down the 4 th column the outermost orbital is farther away from the nucleus and less bound the electron is less strong a function of the position in the crystal reducing band gap down the column