PHYS 342/555 Introduction to solid state physics Instructor: Dr. Pengcheng Dai Professor of Physics The University of Tennessee (Room 407A, Nielsen, 974-1509) Chapter 5: Thermal properties Lecture in pdf format will be available at: http://www.phys.utk.edu 1
Dai/PHYS 342/555 Spring 2013 Chapter 5-2
Dai/PHYS 342/555 Spring 2013 Chapter 5-3
Chapter 5: Phonons and thermal properties In quantum theory of specific heat, the total thermal energy of the phonons can be written as the sum of the energies over all phonon modes: U n k, p k, p, Here n denotes the thermal equilibrium occupancy of phonons k, p k p of wavevector and polarization. is given by the Planck k p n k, p 1 distribution function: n k, p. exp( / ) 1 U The heat capacity is then defined as C V ( ) V. T Dai/PHYS 342/555 Spring 2013 Chapter 5-4
Planck distribution ib i function: Consider a set of identical harmonic oscillators in thermal equilibrium. 1 The harmonic oscillator has an energy of ( n ). 2 N / exp( / ), 1. n Nn kbt Thus the fraction of the total number of oscillators in the nth quantum state is N n N s s0 s0 exp( n / ) exp( s / ) The average excitation quantum # of an oscillator is: n sexp( n / ) 1 exp( s/ ) exp( / ) 1 Dai/PHYS 342/555 Spring 2013 Chapter 5-5
Normal mode enumeration The energy of a collection of oscillators in thermal equilibrium k, p U, If crysal had Dp ( ) d modes of exp( / ) 1 k p a given plarization p between and d. U d Dp ( ), where Dp ( ) is the number exp( / ) 1 p of modes per unit frequency, called the density of modes or density of states. Assume x / k T. Then U / T C k dd B lat B p p 2 x exp( x) ( ). 2 [exp( x ) 1] Dai/PHYS 342/555 Spring 2013 Chapter 5-6
Density of states in 1D If each normal vibrational mode of polarization p has the form of a standing wave, where u u (0)exp( i t )sin ska s k, p The wavevector k is restricted by the fixed-end boundary conditions to the values k, L 2 ( N 1),, L L Dai/PHYS 342/555 Spring 2013 Chapter 5-7
For 1D line there is one mode for each interval k / L, so that the number of modes per unit range of k is L / for k / a and 0 for k / a. The number of modes D( ) d in d at D( ) d Ldk d, d / dk d is the group velocity. Dai/PHYS 342/555 Spring 2013 Chapter 5-8
Density of states in three dimensions 3 In three dimensional lattice with N primitive cells side L, periodic condition requires exp[ ikx ( ky kz)] exp[ ik ( ( xl) k( yl) k ( zl))] x y z x y z 2 4 Therefore kx, ky, kz 0; N ; ; ; L L L 3 There is one allowed value of k per volume (2 / L) in k space or the total # of modes with wavevector less than k N L k 3 3 ( / 2 ) (4 / 3). The density of states for each polarization i is D( ) dn 2 2 / d ( Vk /2 )( dk / d ) Dai/PHYS 342/555 Spring 2013 Chapter 5-9
Dai/PHYS 342/555 Spring 2013 Chapter 5-10
Dai/PHYS 342/555 Spring 2013 Chapter 5-11
Debye Model for density of states In the Debye approximation, vk, v is sound velocity. 2 2 2 V D( ) dn / d ( Vk /2 )( dk/ d). 2 3 2 v If there are N primitive cells, the total # of acoustic modes is N. The cutoff frequency is 6 v N / V. 3 2 3 The cutoff frequency D is D 6 / The cutoff wavevector in k space: k / v(6 N / V) The thermal energy is 2 D V 0 2 3 / v e U d D ( ) n ( ) d 2 1 D D 2 1/3 Dai/PHYS 342/555 Spring 2013 Chapter 5-12
If the phonon velocity is independent of the polarization 2 4 4 3 D 3V 3Vk D BT x x U d dx, 0 2 3 / 2 3 3 2 1 2 0 x v e v e 1 where x/ / T the Debye temperature is 2 v 6 N kb V T x D x the total phonon energy U 9 NkBT dx, 0 e x 1 the heat capacity C V 3 4 x x T D x e 9 Nk 2. BT dx 0 x ( e 1) 1/3 3 3 2 4 / 3V D e d 2 3 2 / 2 2 0 v kt B ( e 1) Dai/PHYS 342/555 Spring 2013 Chapter 5-13
Dai/PHYS 342/555 Spring 2006 Chapter 5-14
Dai/PHYS 342/555 Spring 2013 Chapter 5-15
Debye T 3 law At low temperatures, 4 x 1 6, 4 1 n 15 3 nx dx 0 x x e dx e 0 n1 n1 4 12 T the heat capacity CV NkB. 5 3 Dai/PHYS 342/555 Spring 2013 Chapter 5-16
Einstein models of the density of states In the case of N oscillators of the same frequency in 1D, the Einstein density of states is D ( ) N ( ) N. e 1 U N n / The heat capacity 0 U CV N NkB T V 2 0 e ( e 1) / 2 Dai/PHYS 342/555 Spring 2013 Chapter 5-17
Anharmonic crystal interactions Two lattice waves do not interact; No thermal expansion; The elastic constants t are independent d of pressure and temperature; The heat capacity becomes constant at high temperatures. Thermal expansion The potential energy of the atoms displaced x from their equilibrium, U( x) cx gx fx x 2 3 4 x exp[ U ( x )] dx 3g exp[ U( x)] dx 4c 2 k T. B Dai/PHYS 342/555 Spring 2013 Chapter 5-18
Dai/PHYS 342/555 Spring 2013 Chapter 5-19
Dai/PHYS 342/555 Spring 2013 Chapter 5-20
Dai/PHYS 342/555 Spring 2013 Chapter 5-21
Dai/PHYS 342/555 Spring 2013 Chapter 5-22
Dai/PHYS 342/555 Spring 2013 Chapter 5-23
k k k 1 2 3 k k k K 1 2 3 Dai/PHYS 342/555 Spring 2013 Chapter 5-24
Dai/PHYS 342/555 Spring 2013 Chapter 5-25