FYS Vår 2014 (Kondenserte fasers fysikk)

Transcription

1 FYS410 - Vår 014 (Kondenserte fasers fysikk) Pensum: Solid State Physics by Philip Hofmann (Chapters 1-7 and 11) Andrej Kuznetsov delivery address: Department of Physics, PB 1048 Blindern, 016 OSLO Tel: , e-post: andrej.kuznetsov@fys.uio.no visiting address: MiNaLab, Gaustadaleen c

2 Lecture schedule (based on P.Hofmann s Solid State Physics, chapters 1-7 and 11) Module I Periodic Structures and Defects 0/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 4h 1/1 Laue condition, Ewald construction, interpretation of a diffraction experiment h /1 Bragg planes and Brillouin zones (BZ) h /1 Elastic strain and structural defects h /1 Atomic diffusion and summary of Module I h Module II - Phonons 0/ Vibrations, phonons, density of states, and Planck distribution 4h 04/ Dulong-Petit versus Debye models for heat capacity of the lattice h 05/ Einstein model; comparison of different models h 06/ Thermal conductivity h 07/ Thermal expansion and summary of Module II h Module III Free Electron Gas 4/ Free electron Fermi gas (FEFG), density of states in the ground state 4h 5/ Effect of temperature Fermi- Dirac distribution h 6/ Heat capacity of FEFG h 7/ Electrical and thermal conductivity in metals h 8/ FEFG in nanostructures h Module IV Nearly Free Electron Model 10/ Bragg reflection of electrons at the boundary of BZ, energy bands 4h 11/ Empty lattice approximation; number of orbitals in a band h 1/ Semiconductors, effective mass method, intrinsic carriers h 1/ Impurity states in semiconductors and carrier statistics h 14/ p-n junctions and optoelectronic devices h

3 Lecture 10: Vibrations and phonons Examples of phonon-assisted processes Infinite 1D lattice with one or two atoms in the basis; Examples of dispersion relations in D; Finite chain of atoms, Born von Karman boundary conditions; Phonon density of states in 1-, -, and -D; Collective crystal vibrations phonons; Thermal equilibrium occupancy of phonons Planck distribution.

4 Lecture 10: Vibrations and phonons Examples of phonon-assisted processes Infinite 1D lattice with one or two atoms in the basis; Examples of dispersion relations in D; Finite chain of atoms, Born von Karman boundary conditions; Phonon density of states in 1-, -, and -D; Collective crystal vibrations phonons; Thermal equilibrium occupancy of phonons Planck distribution

5 Diffraction k K k G k k G hkl -k

6 Photoluminescence E g CB E D hn hn E A hn VB EXCITATION Photo generation Electrical injection Photons

7 Photoluminescence

8 Lecture 10: Vibrations and phonons Examples of phonon-assisted processes Infinite 1D lattice with one or two atoms in the basis; Examples of dispersion relations in D; Finite chain of atoms, Born von Karman boundary conditions; Phonon density of states in 1-, -, and -D; Collective crystal vibrations phonons; Thermal equilibrium occupancy of phonons Planck distribution

9 Vibrations of crystals with monatomic basis longitudinal wave transverse wave

10 Vibrations of crystals with monatomic basis Spring constant, g Mass, m M Equilibrium Position a Deformed Position x n-1 u s-1 u x n x n+1 s u s+1 u s : displacement of the s th atom from its equilibrium position

11 Vibrations of crystals with monatomic basis Force on s th plane = F C u u C u u s s s1 s s1 (only neighboring planes interact ) du s1 s1 s s Equation of motion: M C u u u s dt i t s M u C u u u u t u e us s s1 s1 s ik as i K a i K a u0 e M C e e 1cos Ka C M Dispersion relation 4C 1 sin M Ka 4C 1 sin M Ka

12 Vibrations of crystals with monatomic basis Group velocity: v g K 1-D: v G d dk Ca M 1 cos Ka 4C 1 sin M Ka v G = 0 at zone boundaries

13 Vibrations of crystals with two atoms per basis dus 1 s s1 s M C v v u dt dvs s1 s s M C u u v dt us vs ue ve isk ait isk ait i K a M1 u Cv 1 e Cu i K a M v Cu 1 e Cv 1 1 C M C e i K a C 1 e C M i K a 0

14 Vibrations of crystals with two atoms per basis M M C M M C 1cos Ka Ka π: (M 1 >M ) Ka 0: = C/ M C/ M 1 optical acoustical 1 1 C M1 M = C Ka M1 M optical acoustical

15 Lecture 10: Vibrations and phonons Examples of phonon-assisted processes Infinite 1D lattice with one or two atoms in the basis; Examples of dispersion relations in D; Finite chain of atoms, Born von Karman boundary conditions; Phonon density of states in 1-, -, and -D Collective crystal vibrations phonons; Thermal equilibrium occupancy of phonons Planck distribution

16 Vibrations of crystals with two atoms per basis p atoms in primitive cell d p branches of dispersion. d = acoustical : 1 LA + TA (p ) optical: (p 1) LO + (p 1) TO E.g., Ge or KBr: p = 1 LA + TA + 1 LO + TO branches Ge KBr Number of allowed K in 1 st BZ = N

17 Phonon dispersion in real crystals: aluminium FCC lattice with 1 atom in the basis In a -D atomic lattice we expect to observe different branches of the dispersion relation, since there are two mutually perpendicular transverse wave patterns in addition to the longitudinal pattern we have considered. Along different directions in the reciprocal lattice the shape of the dispersion relation is different. But note the resemblance to the simple 1-D result we found.

18 Phonon dispersion in real crystals: FCC lattice with 1 (Al) and (Diamond) atoms in the basis Characteristic points of the reciprocal space Γ, X, K, and L points are introduced at the center and bounduries of the first Brillouin zone

19 Lecture 10: Vibrations and phonons Examples of phonon-assisted processes Infinite 1D lattice with one or two atoms in the basis; Examples of dispersion relations in D; Finite chain of atoms, Born von Karman boundary conditions; Phonon density of states in 1-, -, and -D; Collective crystal vibrations phonons; Thermal equilibrium occupancy of phonons Planck distribution.

20 Calculating phonon density of states DOS in 1-D A vibrational mode is a vibration of a given wave vector k (and thus ), frequency, and energy E. How many modes are found in the interval between (, E, k ) and ( d, E de, k dk )? # modes dn N( ) d N( E) de N( k) d k We will first find N(k) by examining allowed values of k. Then we will be able to calculate N() and evaluate C V in the Debye model. First step: simplify problem by using periodic boundary conditions for the linear chain of atoms: x = sa L = Na x = (s+n)a s+n-1 s s+1 s+ We assume atoms s and s+n have the same displacement the lattice has periodic behavior, where N is very large.

21 Lecture 10: Vibrations and phonons Examples of phonon-assisted processes Infinite 1D lattice with one or two atoms in the basis; Examples of dispersion relations in D; Finite chain of atoms, Born von Karman boundary conditions; Phonon density of states in 1-, -, and -D; Collective crystal vibrations phonons; Thermal equilibrium occupancy of phonons Planck distribution.

22 Calculating phonon density of states DOS in 1-D Since atoms s and s+n have the same displacement, we can write: u i( ksat ) i( k( sn) at ) s u s N ue ue ikna 1 e This sets a condition on n allowed k values: kna n k n 1,,,... Na So the separation between allowed solutions (k values) is: k n Na Na independent of k, so the density of modes in k-space is uniform Thus, in 1-D: # of modes interval of k space 1 Na L k

23 Calculating phonon density of states DOS in -D Now for a -D lattice we can apply periodic boundary conditions to a sample of N 1 x N x N atoms: N c # of modes volume of k space N1a Nb Nc V N( k) 8 N 1 a N b Now we know from before that we can write the differential # of modes as: dn N( ) d N( k) d k V 8 d k We carry out the integration in k-space by using a volume element made up of a constant surface with thickness dk: d k ( surface area) dk ds dk

24 Calculating phonon density of states DOS in -D Rewriting the differential number of modes in an interval: dn V N( ) d 8 ds dk We get the result: V dk V N( ) ds 8 d 8 ds 1 k A very similar result holds for N(E) using constant energy surfaces for the density of electron states in a periodic lattice!

25 Lecture 10: Vibrations and phonons Examples of phonon-assisted processes Infinite 1D lattice with one or two atoms in the basis; Examples of dispersion relations in D; Finite chain of atoms, Born von Karman boundary conditions; Phonon density of states in 1-, -, and -D; Collective crystal vibrations phonons; Thermal equilibrium occupancy of phonons Planck distribution.

26 Energy level diagram for one harmonic oscillator Energy level diagram for a chain of atoms with one atom per unit cell and a lengt of N unit cells

27 Lecture 10: Vibrations and phonons Examples of phonon-assisted processes Infinite 1D lattice with one or two atoms in the basis; Examples of dispersion relations in D; Finite chain of atoms, Born von Karman boundary conditions; Phonon density of states in 1-, -, and -D; Collective crystal vibrations phonons; Thermal equilibrium occupancy of phonons Planck distribution.

28 Einstein model for heat capacity accounting for quantum properties of oscillators constituting a solid Planck (1900): vibrating oscillators (atoms) in a solid have quantized energies n n 0,1,,... E n [later showed E n n 1 is actually correct] occupation of energy level n: (probability of oscillator being in level n) f ( E n ) e n0 E e n E / kt n / kt classical physics (Boltzmann factor)

29 Boltzmann factor determines Planck distribution e E i / kt Boltzmann factor is a weighting factor that determines the relative probability of a state i in a multi-state system in thermodynamic equilibrium at tempetarure T. Where k B is Boltzmann s constant and E i is the energy of state i. The ratio of the probabilities of two states is given by the ratio of their Boltzmann factors.

30 FYS410 - Vår 014 (Kondenserte fasers fysikk) Pensum: Solid State Physics by Philip Hofmann (Chapters 1-7 and 11) Andrej Kuznetsov delivery address: Department of Physics, PB 1048 Blindern, 016 OSLO Tel: , e-post: andrej.kuznetsov@fys.uio.no visiting address: MiNaLab, Gaustadaleen c

31 Lecture schedule (based on P.Hofmann s Solid State Physics, chapters 1-7 and 11) Module I Periodic Structures and Defects 0/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 4h 1/1 Laue condition, Ewald construction, interpretation of a diffraction experiment h /1 Bragg planes and Brillouin zones (BZ) h /1 Elastic strain and structural defects h /1 Atomic diffusion and summary of Module I h Module II - Phonons 0/ Vibrations, phonons, density of states, and Planck distribution 4h 04/ Lattice heat capacity: Dulong-Petit, Einstien and Debye models h 05/ Comparison of different models h 06/ Thermal conductivity h 07/ Thermal expansion and summary of Module II h Module III Free Electron Gas 4/ Free electron Fermi gas (FEFG), density of states in the ground state 4h 5/ Effect of temperature Fermi- Dirac distribution h 6/ Heat capacity of FEFG h 7/ Electrical and thermal conductivity in metals h 8/ FEFG in nanostructures h Module IV Nearly Free Electron Model 10/ Bragg reflection of electrons at the boundary of BZ, energy bands 4h 11/ Empty lattice approximation; number of orbitals in a band h 1/ Semiconductors, effective mass method, intrinsic carriers h 1/ Impurity states in semiconductors and carrier statistics h 14/ p-n junctions and optoelectronic devices h

32 Lecture 11: Lattice heat capacity: Dulong-Petit, Einstien and Debye models Repetition of phonon DOS Classical theory for heat capacity of solids treating atoms as classical harmonic oscillators - Dulong-Petit model success and problems Einstein model for heat capacity considering quantum properties of oscillators constituting a solid success and problems Debye model

33 Lecture 11: Lattice heat capacity: Dulong-Petit, Einstien and Debye models Repetition of phonon DOS Classical theory for heat capacity of solids treating atoms as classical harmonic oscillators - Dulong-Petit model success and problems Einstein model for heat capacity considering quantum properties of oscillators constituting a solid success and problems Debye model

34 Calculating phonon density of states DOS in 1-D Since atoms s and s+n have the same displacement, we can write: u i( ksat ) i( k( sn) at ) s u s N ue ue ikna 1 e This sets a condition on n allowed k values: kna n k n 1,,,... Na So the separation between allowed solutions (k values) is: k n Na Na independent of k, so the density of modes in k-space is uniform Thus, in 1-D: # of modes interval of k space 1 Na L k

35 Energy level diagram for one harmonic oscillator Energy level diagram for a chain of atoms with one atom per unit cell and a lengt of N unit cells

36 Lecture 11: Lattice heat capacity: Dulong-Petit, Einstien and Debye models Repetition of phonon DOS Classical theory for heat capacity of solids treating atoms as classical harmonic oscillators - Dulong-Petit model success and problems Einstein model for heat capacity considering quantum properties of oscillators constituting a solid success and problems Debye model

37 Classical (Dulong-Petit) theory for heat capacity For a solid composed of N such atomic oscillators: E NE 1 Nk B T Giving a total energy per mole of sample: E n Nk T B N AkBT RT n So the heat capacity at constant volume per mole is: d E CV R 5 dt n V J mol K This law of Dulong and Petit (1819) is approximately obeyed by most solids at high T ( > 00 K).

38 Temperature dependence of experimentally measured heat capacity

39 Lecture 11: Lattice heat capacity: Dulong-Petit, Einstien and Debye models Repetition of phonon DOS Classical theory for heat capacity of solids treating atoms as classical harmonic oscillators - Dulong-Petit model success and problems Einstein model for heat capacity considering quantum properties of oscillators constituting a solid success and problems Debye model

40 Einstein model for heat capacity accounting for quantum properties of oscillators constituting a solid Planck (1900): vibrating oscillators (atoms) in a solid have quantized energies n n 0,1,,... E n [later showed E n n 1 is actually correct] Einstein (1907): model a solid as a collection of N independent 1-D oscillators, all with constant, and use Planck s equation for energy levels occupation of energy level n: (probability of oscillator being in level n) f ( E n ) e n0 E e n E / kt n / kt classical physics (Boltzmann factor) Average total energy of solid: E U N n0 f ( E n ) E n N n0 E n0 n e e E E n n / kt / kt

41 Boltzmann factor determines Planck distribution e E i / kt Boltzmann factor is a weighting factor that determines the relative probability of a state i in a multi-state system in thermodynamic equilibrium at tempetarure T. Where k B is Boltzmann s constant and E i is the energy of state i. The ratio of the probabilities of two states is given by the ratio of their Boltzmann factors.

42 Einstein model for heat capacity accounting for quantum properties of oscillators constituting a solid 0 / 0 / n kt n n kt n e e n N U Using Planck s equation: Now let kt x 0 0 n nx n nx e ne N U n n x n n x n nx n nx e e dx d N e e dx d N U Which can be rewritten: Now we can use the infinite sum: x for x x n n / kt x x x x x e N e N e e e e dx d N U To give: x x x n n x e e e e So we obtain:

43 Einstein model for heat capacity accounting for quantum properties of oscillators constituting solids Differentiating: Now it is traditional to define an Einstein temperature : Using our previous definition: So we obtain the prediction: 1 / kt A V V e N dt d n U dt d C / / / / 1 1 kt kt kt kt kt kt A V e e R e e N C k E / / 1 ) ( T T T V E E E e e R T C

44 Einstein model for heat capacity accounting for quantum properties of oscillators constituting solids Low T limit: These predictions are qualitatively correct: C V R for large T and C V 0 as T 0: High T limit: 1 T E R R T C T T T V E E E ) ( 1 T E T T T T T V E E E E E e R e e R T C / / / ) ( R C V T/ E

45 Correlation with energy level diagram for a harmonic oscillator Energy level diagram for one harmonic oscillator E High T limit: 1 T E Low T limit: 1 T Energy level diagram for a chain of atoms with one atom per unit cell and a lengt of N unit cells

46 Problem of Einstein model to reproduce the rate of heat capacity decrease at low temperatures High T behavior: Reasonable agreement with experiment Low T behavior: C V 0 too quickly as T 0!

47 Lecture 11: Lattice heat capacity: Dulong-Petit, Einstien and Debye models Repetition of phonon DOS Classical theory for heat capacity of solids treating atoms as classical harmonic oscillators - Dulong-Petit model success and problems Einstein model for heat capacity considering quantum properties of oscillators constituting a solid success and problems Debye model

48 More careful consideration of phonon occupancy modes as a way to improve the agreement with experiment Debye s model of a solid: N normal modes (patterns) of oscillations Spectrum of frequencies from = 0 to max Treat solid as continuous elastic medium (ignore details of atomic structure) This changes the expression for C V because each mode of oscillation contributes a frequency-dependent heat capacity and we now have to integrate over all : max CV ( T) D( ) CE (, T) 0 # of oscillators per unit, i.e. DOS d Distribution function

49 Lattice heat capacity: Debye model ) ( v V d dn D v V v L k L N k k 1 6 V N k v k B B D Density of states of acoustic phonos for 1 polarization Debye temperature θ 6 v V N D L k N: number of unit cell N k : Allowed number of k points in a sphere with a radius k k vk ) ( phonon dispersion relation k

50 Thermal energy U and lattice heat capacity C V : Debye model D D D x x x B V B B B V V B e e x dx T Nk C T k T k d T k v V T U C T k v V d n D d U ) ( 9 1] ) / [exp( ) / exp( 1 ) / exp( ) ( ) ( polarizations for acoustic modes

51 Debye model Better agreement than Einstein model at low T Universal behavior for all solids! Debye temperature is related to stiffness of solid, as expected

52 Debye model Quite impressive agreement with predicted C V T dependence for Ar! (noble gas solid)

53 More careful consideration of phonon occupancy modes as a way to improve the agreement with experiment Energy level diagram for a chain of atoms with one atom per unit cell and a lengt of N unit cells

54 FYS410 - Vår 014 (Kondenserte fasers fysikk) Pensum: Solid State Physics by Philip Hofmann (Chapters 1-7 and 11) Andrej Kuznetsov delivery address: Department of Physics, PB 1048 Blindern, 016 OSLO Tel: , e-post: andrej.kuznetsov@fys.uio.no visiting address: MiNaLab, Gaustadaleen c

55 Lecture schedule (based on P.Hofmann s Solid State Physics, chapters 1-7 and 11) Module I Periodic Structures and Defects 0/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 4h 1/1 Laue condition, Ewald construction, interpretation of a diffraction experiment h /1 Bragg planes and Brillouin zones (BZ) h /1 Elastic strain and structural defects h /1 Atomic diffusion and summary of Module I h Module II - Phonons 0/ Vibrations, phonons, density of states, and Planck distribution 4h 04/ Lattice heat capacity: Dulong-Petit, Einstien and Debye models h 05/ Comparison of different models h 06/ Thermal conductivity h 07/ Thermal expansion and summary of Module II h Module III Free Electron Gas 4/ Free electron Fermi gas (FEFG), density of states in the ground state 4h 5/ Effect of temperature Fermi- Dirac distribution h 6/ Heat capacity of FEFG h 7/ Electrical and thermal conductivity in metals h 8/ FEFG in nanostructures h Module IV Nearly Free Electron Model 10/ Bragg reflection of electrons at the boundary of BZ, energy bands 4h 11/ Empty lattice approximation; number of orbitals in a band h 1/ Semiconductors, effective mass method, intrinsic carriers h 1/ Impurity states in semiconductors and carrier statistics h 14/ p-n junctions and optoelectronic devices h

56 Lecture 1: Comparison of different models N independent harmonic oscillators modelling vibrations in a solid; Repetition of phonon DOS Debye model

57 Lecture 1: Comparison of different models N independent harmonic oscillators modelling vibrations in a solid; Repetition of phonon DOS Debye model

58 Energy Ensemble of N independent harmonic oscillators modeling vibrations in a solid Classical oscillators Any energy state is accessible for any oscillator in form of k B T, i.e. no distribution function is applied and the total energy is E NE 1 Nk B T Quantum oscillators Energy level diagram for a chain of atoms with one atom per unit cell and a lengt of N unit cells

59 Energy Ensemble of N independent harmonic oscillators modeling vibrations in a solid Classical oscillators Any energy state is accessible for any oscillator in form of k B T, i.e. no distribution function is applied and the total energy is E NE 1 Nk B T E Quantum oscillators Not all Any energies energy are state accessible, is accessible but only for those in quants any of oscillator ħωn, and in Planck form of distribution k B T, i.e. is employed no distribution to calculate the function occupancy is at temperature necessary, T, so so that that Energy level diagram for a chain of atoms with one atom per unit cell and a E N lengt of N unit cells En / kbt E ne n0 N f ( En) En N N / n0 E / k T e 1 n0 e n B 1 kbt n

60 Ensemble of N independent harmonic oscillators modeling vibrations in a solid Dulong-Petit model is valid only at high temperatures Einstein model is in a good agreement with the experiment, except for that at low temperatures

61 Energy level diagram for one harmonic oscillator Energy level diagram for a chain of atoms with one atom per unit cell and a lengt of N unit cells E N n E dd( ) n max min

62 Lecture 11: Lattice heat capacity: Dulong-Petit, Einstien and Debye models Classical theory for heat capacity of solids treating atoms as classical harmonic oscillators - Dulong-Petit model success and problems; Einstein model for heat capacity considering quantum properties of oscillators constituting a solid success and problems; Repetition of phonon DOS Debye model

63 Calculating phonon density of states DOS in 1-D Since atoms s and s+n have the same displacement, we can write: u i( ksat ) i( k( sn) at ) s u s N ue ue ikna 1 e This sets a condition on allowed k values: kna n n k n 1, Na,,... k 1 Na Na 1 k 4 Na Na k N N Na a only one value of k in this amount of k-space or in other only one excited phonon mode available arbitrarily «unit» of k Thus, DOS in k-space in 1-D: 1mode unit k 11 an L interval of k space an

64 Lecture 11: Lattice heat capacity: Dulong-Petit, Einstien and Debye models Classical theory for heat capacity of solids treating atoms as classical harmonic oscillators - Dulong-Petit model success and problems; Einstein model for heat capacity considering quantum properties of oscillators constituting a solid success and problems; Repetition of phonon DOS Debye model

65 More careful consideration of phonon occupancy modes as a way to improve the agreement with experiment Debye s model of a solid: N normal modes of oscillations Spectrum of frequencies from = 0 to max Treat solid as continuous elastic medium, so that ω = v k, where v is the velocity of sound (which is probably fine since the target is to improve the understanding of crystall vibrations at low temperature, i.e primaraly for low energy acoustic phonons). E max dd( ) n 0

66 Debye model 4 k L N k 1 6 V N k v k B B D Density of states of acoustic phonos for 1 polarization Debye temperature θ 6 v V N D N: number of unit cell N k : Allowed number of k points in a sphere with a radius k k v / 6 4 ) ( v V v L N ) ( ) ( v V d dn D

67 Thermal energy U and lattice heat capacity C V : Debye model D D D x x x B V B B B V V B e e x dx T Nk C T k T k d T k v V T U C T k v V d n D d U ) ( 9 1] ) / [exp( ) / exp( 1 ) / exp( ) ( ) ( polarizations for acoustic modes

68 Debye model Better agreement than Einstein model at low T Universal behavior for all solids! Debye temperature is related to stiffness of solid, as expected

69 Debye model Quite impressive agreement with predicted C V T dependence for Ar! (noble gas solid)

70 FYS410 - Vår 014 (Kondenserte fasers fysikk) Pensum: Solid State Physics by Philip Hofmann (Chapters 1-7 and 11) Andrej Kuznetsov delivery address: Department of Physics, PB 1048 Blindern, 016 OSLO Tel: , e-post: andrej.kuznetsov@fys.uio.no visiting address: MiNaLab, Gaustadaleen c

71 Lecture schedule (based on P.Hofmann s Solid State Physics, chapters 1-7 and 11) Module I Periodic Structures and Defects 0/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 4h 1/1 Laue condition, Ewald construction, interpretation of a diffraction experiment h /1 Bragg planes and Brillouin zones (BZ) h /1 Elastic strain and structural defects h /1 Atomic diffusion and summary of Module I h Module II - Phonons 0/ Vibrations, phonons, density of states, and Planck distribution 4h 04/ Lattice heat capacity: Dulong-Petit, Einstien and Debye models h 05/ Comparison of different models h 06/ Thermal conductivity h 07/ Thermal expansion and summary of Module II h Module III Free Electron Gas 4/ Free electron Fermi gas (FEFG), density of states in the ground state 4h 5/ Effect of temperature Fermi- Dirac distribution h 6/ Heat capacity of FEFG h 7/ Electrical and thermal conductivity in metals h 8/ FEFG in nanostructures h Module IV Nearly Free Electron Model 10/ Bragg reflection of electrons at the boundary of BZ, energy bands 4h 11/ Empty lattice approximation; number of orbitals in a band h 1/ Semiconductors, effective mass method, intrinsic carriers h 1/ Impurity states in semiconductors and carrier statistics h 14/ p-n junctions and optoelectronic devices h

72 Lecture 1: Thermal conductivity We understood phonon DOS and occupancy as a function of temperature, but what about transport properties? Phenomenological description of thermal conductivity Temperature dependence of thermal conductivity in terms of phonon properties Phonon collisions: N and U processes

73 Lecture 1: Thermal conductivity We understood phonon DOS and occupancy as a function of temperature, but what about transport properties? Phenomenological description of thermal conductivity Temperature dependence of thermal conductivity in terms of phonon properties Phonon collisions: N and U processes

74 Understanding phonons as «harmonic waves» can not explain thermal restance since harmonic wafes perfectly move one through another

75 Lecture 1: Thermal conductivity We understood phonon DOS and occupancy as a function of temperature, but what about transport properties? Phenomenological description of thermal conductivity Temperature dependence of thermal conductivity in terms of phonon properties Phonon collisions: N and U processes

76 Phenomenological description of thermal conductivity When thermal energy propagates through a solid, it is carried by lattice waves or phonons. If the atomic potential energy function is harmonic, lattice waves obey the superposition principle; that is, they can pass through each other without affecting each other. In such a case, propagating lattice waves would never decay, and thermal energy would be carried with no resistance (infinite conductivity!). So thermal resistance has its origins in an anharmonic potential energy. low T high T Classical definition of thermal conductivity 1 C V v Thermal energy flux (J/m s) J dt dx C V v heat capacity per unit volume wave velocity mean free path of scattering (would be if no anharmonicity)

77 Lecture 1: Thermal conductivity We understood phonon DOS and occupancy as a function of temperature, but what about transport properties? Phenomenological description of thermal conductivity Temperature dependence of thermal conductivity in terms of phonon properties Phonon collisions: N and U processes

78 Temperature dependence of thermal conductivity in terms of phonon prperties Mechanisms to affect the mean free pass (Λ) of phonons in periodic crystals: 1. Interaction with impurities, defects, and/or isotopes. Collision with sample boundaries (surfaces) deviation from translation symmetry. Collision with other phonons deviation from harmonic behavior C V (T) To understand the experimental dependence, consider limiting values of and (since v does not vary much with T). 1 / C T low T kt V e 1 R high T nph kt low T high T 1) Please note, that the temperature dependence of T -1 for Λ at the high temperature limit results from considering n ph, which is the total phonon occupancy, from 0 to ω D. However, already intuitively, we may anticipate that low energy phonons, i.e. those with low k-numbers in the vicinity of the center of the 1st BZ may have quite different appearence conparing with those having bigger k-naumbers close to the edges of the 1st BZ. 1)

79 Temperature dependence of thermal conductivity in terms of phonon prperties Thus, considering defect free, isotopically clean sample having limited size D C V low T T, but then n ph 0, so D (size) T high T R 1/T 1/T How well does this match experimental results?

80 Temperature dependence of thermal conductivity in terms of phonon prperties Experimental (T) T estimation for κ the low temperature limit is fine! T T -1? However, T -1 estimation for κ in the high temperature limit has a problem. Indeed, κ drops much faster see the data and the origin of this disagreement is because when estimating Λ we accounted for all excited phonons, while a more correct approximation would be to consider high energetic phonons only. But what is high in this context?

81 Better estimation for Λ in high temperature limit k 1 Na Na 1 k 4 Na Na k N N Na a ω 1 1/ ω ω D «insignificant» modes «significant» modes estimate in terms of affecting Λ! The fact that «low energetic phonons» having k-values << π/a do not participate in the energy transfer, can be understood by considering so called N- and U-phonon collisions readily visualized in the reciprocal space. Anyhow, we account for modes having energy E 1/ = (1/)ħω D or higher. Using the definition of θ D = ħω D /k B, E 1/ can be rewritten as k B θ D /. Ignoring more complex statistics, but using Boltzman factor only, the propability of E 1/ would of the order of exp(- k B θ D / k B T) or exp(-θ D /T), resulting in Λ exp(θ D /T).

82 Temperature dependence of thermal conductivity in terms of phonon prperties Thus, considering defect free, isotopically clean sample having limited size D C V low T T, but then n ph 0, so D (size) T high T R exp(θ D /T) exp(θ D /T)

83 Lecture 1: Thermal conductivity We understood phonon DOS and occupancy as a function of temperature, but what about transport properties? Phenomenological description of thermal conductivity Temperature dependence of thermal conductivity in terms of phonon properties Phonon collisions: N and U processes

84 Phonon collisions: N and U processes How exactly do phonon collisions limit the flow of heat? -D lattice 1st BZ in k-space: q 1 q q a a q 1 q q No resistance to heat flow (N process; phonon momentum conserved) Predominates at low T << D since and q will be small

85 Phonon collisions: N and U processes What if the phonon wavevectors are a bit larger? -D lattice 1st BZ in k-space: q 1 q q G a a q q 1 q q 1 q G Two phonons combine to give a net phonon with an opposite momentum! This causes resistance to heat flow. (U process; phonon momentum lost in units of ħg.) Umklapp = flipping over of wavevector! More likely at high T >> D since and q will be larger

86 FYS410 - Vår 014 (Kondenserte fasers fysikk) Pensum: Solid State Physics by Philip Hofmann (Chapters 1-7 and 11) Andrej Kuznetsov delivery address: Department of Physics, PB 1048 Blindern, 016 OSLO Tel: , e-post: andrej.kuznetsov@fys.uio.no visiting address: MiNaLab, Gaustadaleen c

87 Lecture schedule (based on P.Hofmann s Solid State Physics, chapters 1-7 and 11) Module I Periodic Structures and Defects 0/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 4h 1/1 Laue condition, Ewald construction, interpretation of a diffraction experiment h /1 Bragg planes and Brillouin zones (BZ) h /1 Elastic strain and structural defects h /1 Atomic diffusion and summary of Module I h Module II - Phonons 0/ Vibrations, phonons, density of states, and Planck distribution 4h 04/ Lattice heat capacity: Dulong-Petit, Einstien and Debye models h 05/ Comparison of different models h 06/ Thermal conductivity h 07/ Thermal conductivity (continuation) and thermal expansion h Module III Free Electron Gas 4/ Free electron Fermi gas (FEFG), density of states in the ground state 4h 5/ Effect of temperature Fermi- Dirac distribution h 6/ Heat capacity of FEFG h 7/ Electrical and thermal conductivity in metals h 8/ FEFG in nanostructures h Module IV Nearly Free Electron Model 10/ Bragg reflection of electrons at the boundary of BZ, energy bands 4h 11/ Empty lattice approximation; number of orbitals in a band h 1/ Semiconductors, effective mass method, intrinsic carriers h 1/ Impurity states in semiconductors and carrier statistics h 14/ p-n junctions and optoelectronic devices h

88 Lecture 14: Thermal conductivity (continuation) and thermal expansion Recap of N and U processes; explanation for κ exp(θ D /T) at high temperature limit Comparison of temperature dependence of κ in crystalline and amorphous solids Thermal expansion

89 Lecture 14: Thermal conductivity (continuation) and thermal expansion Recap of N and U processes; explanation for κ exp(θ D /T) at high temperature limit Comparison of temperature dependence of κ in crystalline and amorphous solids Thermal expansion

90 Recap of N- and U-processes; explanation for κ exp(θ D /T) at high temperature limit 1 C V v The temperature dependence of T -1 for Λ results from considering the total phonon occupancy, from 0 to ω D. However, interactions of low energy phonons, i.e. those with low k- numbers in the vicinity of the center the 1st BZ, are not changing energy. These are so called N-processes having little impact on Λ. n 1 / kt ph e 1 T 1 low T high T q 1 q a q a

91 Recap of N- and U-processes; explanation for κ exp(θ D /T) at high temperature limit A more correct approximation for Λ (in high temperature limit) would be to consider high energetic phonons only, i.e those participating in U- processes. a q q 1 q 1 q G a q U-process, i.e. to turn over the wavevector by G, from the German word umklappen.

92 Recap of N- and U-processes; explanation for κ exp(θ D /T) at high temperature limit k 1 Na Na 1 k 4 Na Na k N N Na a ω 1 1/ ω ω D «insignificant» modes «significant» modes estimate in terms of affecting Λ! The fact that «low energetic phonons» having k-values << π/a do not participate in the energy transfer, can be understood by considering so called N- and U-phonon collisions readily visualized in the reciprocal space. Anyhow, we account for modes having energy E 1/ = (1/)ħω D or higher. Using the definition of θ D = ħω D /k B, E 1/ can be rewritten as k B θ D /. Ignoring more complex statistics, but using Boltzman factor only, the propability of E 1/ would of the order of exp(- k B θ D / k B T) or exp(-θ D /T), resulting in Λ exp(θ D /T).

93 Temperature dependence of thermal conductivity in terms of phonon prperties Thus, considering defect free, isotopically clean sample having limited size D C V low T T, but then n ph 0, so D (size) T high T R exp(θ D /T) exp(θ D /T)

94 Lecture 14: Thermal conductivity (continuation) and thermal expansion Recap of N and U processes; explanation for κ exp(θ D /T) at high temperature limit Comparison of temperature dependence of κ in crystalline and amorphous solids Thermal expansion

95 Comparison of temperature dependence of κ in crystalline and amorphous solids

96 Lecture 14: Thermal conductivity (continuation) and thermal expansion Recap of N and U processes; explanation for κ exp(θ D /T) at high temperature limit Comparison of temperature dependence of κ in crystalline and amorphous solids Thermal expansion

97 Thermal expansion In a 1-D lattice where each atom experiences the same potential energy function U(x), we can calculate the average displacement of an atom from its T=0 equilibrium position: x xe e U ( x)/ kt U ( x)/ kt dx dx

98 I Thermal Expansion in 1-D Evaluating this for the harmonic potential energy function U(x) = cx gives: x xe e cx cx / kt / kt dx dx Now examine the numerator carefully what can you conclude? x 0! independent of T! Thus any nonzero <x> must come from terms in U(x) that go beyond x. For HW you will evaluate the approximate value of <x> for the model function U( x) cx gx fx 4 ( c, g, f 0 and gx, fx 4 kt) Why this form? On the next slide you can see that this function is a reasonable model for the kind of U(r) we have discussed for molecules and solids.

99 Potential Energy U (arb. units) Potential Energy of Anharmonic Oscillator (c = 1 g = c/10 f = c/100) U = cx - gx - fx4 U = cx Do you know what form to expect for <x> based on experiment? Displacement x (arbitrary units)

100 Lattice Constant of Ar Crystal vs. Temperature Above about 40 K, we see: a( T) a(0) x T Usually we write: L L T 1 T 0 0 = thermal expansion coefficient