FYS Vår 2015 (Kondenserte fasers fysikk)

Transcription

1 FYS410 - Vår 015 (Kondenserte fasers fysikk) Pensum: Introduction to Solid State Physics by Charles Kittel (Chapters 1-9 and 17, 18, 0) Andrej Kuznetsov delivery address: Department of Physics, PB 1048 Blindern, 016 OSLO Tel: , e-post: andrej.kuznetsov@fys.uio.no visiting address: MiNaLab, Gaustadaleen a

2 FYS410 Lectures (based on C.Kittel s Introduction to SSP, chapters 1-9, 17,18,0) Module I Periodic Structures and Defects (Chapters 1-, 0) 6/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space h 7/1 Laue condition, Ewald construction, interpretation of a diffraction experiment Bragg planes and Brillouin zones 4h 8/1 Elastic strain and structural defects in crystals h 0/1 Atomic diffusion and summary of Module I h Module II Phonons (Chapters 4 and 5) 09/ Vibrations, phonons, density of states, and Planck distribution h 10/ Lattice heat capacity: Dulong-Petit, Einstien and Debye models Comparison of different models 4h 11/ Thermal conductivity h 1/ Thermal expansion and summary of Module II h Module III Electrons (Chapters 6, 7 and 18) / Free electron gas (FEG) versus Free electron Fermi gas (FEFG) h 4/ Effect of temperature Fermi- Dirac distribution FEFG in D and 1D, and DOS in nanostructures 4h 5/ Origin of the band gap and nearly free electron model h 7/ Number of orbitals in a band and general form of the electronic states h Module IV Semiconductors and Metals (Chapters 8, 9, and 17) 09/ Energy bands; metals versus isolators h 10/ Semiconductors: effective mass method, intrinsic and extrinsic carrier generation 4h 1/ Carrier statistics h 1/ p-n junctions and optoelectronic devices h

3 Lecture 6: 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-D; Collective crystal vibrations phonons;

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-D; Collective crystal vibrations phonons;

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-D; Collective crystal vibrations phonons;

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-D Collective crystal vibrations phonons;

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-D; Collective crystal vibrations phonons;

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-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 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.

24 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

25 FYS410 - Vår 015 (Kondenserte fasers fysikk) Pensum: Introduction to Solid State Physics by Charles Kittel (Chapters 1-9 and 17, 18, 0) Andrej Kuznetsov delivery address: Department of Physics, PB 1048 Blindern, 016 OSLO Tel: , e-post: andrej.kuznetsov@fys.uio.no visiting address: MiNaLab, Gaustadaleen a

26 FYS410 Lectures (based on C.Kittel s Introduction to SSP, chapters 1-9, 17,18,0) Module I Periodic Structures and Defects (Chapters 1-, 0) 6/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space h 7/1 Laue condition, Ewald construction, interpretation of a diffraction experiment Bragg planes and Brillouin zones 4h 8/1 Elastic strain and structural defects in crystals h 0/1 Atomic diffusion and summary of Module I h Module II Phonons (Chapters 4, 5 and 18) 09/ Vibrations, phonons, density of states, and Planck distribution h 10/ Lattice heat capacity: Dulong-Petit, Einstien and Debye models Comparison of different models 4h 11/ Thermal conductivity h 1/ Thermal expansion and summary of Module II h Module III Electrons (Chapters 6 and 7) / Free electron gas (FEG) versus Free electron Fermi gas (FEFG) h 4/ Effect of temperature Fermi- Dirac distribution FEFG in D and 1D, and DOS in nanostructures 4h 5/ Origin of the band gap and nearly free electron model h 7/ Number of orbitals in a band and general form of the electronic states h Module IV Semiconductors and Metals (Chapters 8, 9, and 17) 09/ Energy bands; metals versus isolators h 10/ Semiconductors: effective mass method, intrinsic and extrinsic carrier generation 4h 1/ Carrier statistics h 1/ p-n junctions and optoelectronic devices h

27 Lecture 7: 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 Comparison of different models

28 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 Comparison of different models

29 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

30 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

31 Lecture 7: 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 Comparison of different models

32 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).

33 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

34 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!

35 Temperature dependence of experimentally measured heat capacity

36 Lecture 7: 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 Comparison of different models

37 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

38 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.

39 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:

40 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

41 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

42 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

43 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!

44 Lecture 7: 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 Comparison of different models

45 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

46 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

47 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

48 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

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

50 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

51 Lecture 7: 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 Comparison of different models

52 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

53 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

54 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

55 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

56 FYS410 - Vår 015 (Kondenserte fasers fysikk) Pensum: Introduction to Solid State Physics by Charles Kittel (Chapters 1-9 and 17, 18, 0) Andrej Kuznetsov delivery address: Department of Physics, PB 1048 Blindern, 016 OSLO Tel: , e-post: andrej.kuznetsov@fys.uio.no visiting address: MiNaLab, Gaustadaleen a

57 FYS410 Lectures (based on C.Kittel s Introduction to SSP, chapters 1-9, 17,18,0) Module I Periodic Structures and Defects (Chapters 1-, 0) 6/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space h 7/1 Laue condition, Ewald construction, interpretation of a diffraction experiment Bragg planes and Brillouin zones 4h 8/1 Elastic strain and structural defects in crystals h 0/1 Atomic diffusion and summary of Module I h Module II Phonons (Chapters 4 and 5) 09/ Vibrations, phonons, density of states, and Planck distribution h 10/ Lattice heat capacity: Dulong-Petit, Einstien and Debye models Comparison of different models 4h 11/ Thermal conductivity h 1/ Thermal expansion and summary of Module II h Module III Electrons (Chapters 6, 7 and 18) / Free electron gas (FEG) versus Free electron Fermi gas (FEFG) h 4/ Effect of temperature Fermi- Dirac distribution FEFG in D and 1D, and DOS in nanostructures 4h 5/ Origin of the band gap and nearly free electron model h 7/ Number of orbitals in a band and general form of the electronic states h Module IV Semiconductors and Metals (Chapters 8, 9, and 17) 09/ Energy bands; metals versus isolators h 10/ Semiconductors: effective mass method, intrinsic and extrinsic carrier generation 4h 1/ Carrier statistics h 1/ p-n junctions and optoelectronic devices h

58 Lecture 8: 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 Comparison of temperature dependence of κ in crystalline and amorphous solids

59 Lecture 8: 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 Comparison of temperature dependence of κ in crystalline and amorphous solids

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

61 Lecture 8: 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 Comparison of temperature dependence of κ in crystalline and amorphous solids

62 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)

63 Lecture 8: 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 Comparison of temperature dependence of κ in crystalline and amorphous solids

64 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)

65 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?

66 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?

67 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).

68 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)

69 Lecture 8: 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 Comparison of temperature dependence of κ in crystalline and amorphous solids

70 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

71 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

72 Explanation for κ exp(θ D /T) at high temperature limit 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. 1 C V v n 1 / kt ph e 1 T 1 low T high T q 1 q a q a

73 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.

74 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).

75 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)

76 Lecture 8: 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 Comparison of temperature dependence of κ in crystalline and amorphous solids

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

78 FYS410 - Vår 015 (Kondenserte fasers fysikk) Pensum: Introduction to Solid State Physics by Charles Kittel (Chapters 1-9 and 17, 18, 0) Andrej Kuznetsov delivery address: Department of Physics, PB 1048 Blindern, 016 OSLO Tel: , e-post: andrej.kuznetsov@fys.uio.no visiting address: MiNaLab, Gaustadaleen a

79 FYS410 Lectures (based on C.Kittel s Introduction to SSP, chapters 1-9, 17,18,0) Module I Periodic Structures and Defects (Chapters 1-, 0) 6/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space h 7/1 Laue condition, Ewald construction, interpretation of a diffraction experiment Bragg planes and Brillouin zones 4h 8/1 Elastic strain and structural defects in crystals h 0/1 Atomic diffusion and summary of Module I h Module II Phonons (Chapters 4 and 5) 09/ Vibrations, phonons, density of states, and Planck distribution h 10/ Lattice heat capacity: Dulong-Petit, Einstien and Debye models Comparison of different models 4h 11/ Thermal conductivity h 1/ Thermal expansion and summary of Module II h Module III Electrons (Chapters 6, 7 and 18) / Free electron gas (FEG) versus Free electron Fermi gas (FEFG) h 4/ Effect of temperature Fermi- Dirac distribution FEFG in D and 1D, and DOS in nanostructures 4h 5/ Origin of the band gap and nearly free electron model h 7/ Number of orbitals in a band and general form of the electronic states h Module IV Semiconductors and Metals (Chapters 8, 9, and 17) 09/ Energy bands; metals versus isolators h 10/ Semiconductors: effective mass method, intrinsic and extrinsic carrier generation 4h 1/ Carrier statistics h 1/ p-n junctions and optoelectronic devices h

80 Lecture 9: Thermal expansion and repetition of Module II Thermal expansion Repetition

81 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 equilibrium position: x xe e U ( x)/ kt U ( x)/ kt dx dx

82 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 The numerator is zero! 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.

83 Potential Energy U (arb. units) Potential Energy of Anharmonic Oscillator (c = 1 g = c/10 f = c/100) U = cx - gx - fx4 U = cx Displacement x (arbitrary units)

84 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