Chapter 4: Summary. Solve lattice vibration equation of one atom/unitcellcase Consider a set of ions M separated by a distance a,

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1 Chapter 4: Summary Solve lattice vibration equation of one atom/unitcellcase case. Consider a set of ions M separated by a distance a, R na for integral n. Let u( na) be the displacement. Assuming only neighboring ions interact, we have 1 harm U Cu( na) u([ n1] a), n Newton's second law F Ma or harm du( na) U M ( ) ([ 1] ) ([ 1] ) ( ) C u na dt u na u n a u n a Dai/PHYS 34/555 Spring 013 Review 1-1

2 For each of the N values of k there are thus two solutions, leading to a total of N normal modes. The two vs k curves are two branches of the dispersion relation. Acoustic and optical branches. Dai/PHYS 34/555 Spring 013 Review 1-

3 Prob. 3, Consider a longitudinal wave u ucos( tska) which propagates in a monatomic linear lattice of atoms of mass M, spacing a, and nearest neighbor interaction C. a) show that the total energy of the wave is 1 1 E Mdus / dt Cus us 1, s where s runs over all atoms. s b)by substitution of u in this expression, show that the time average s total energy per atom is M u C 1cos Ka u M u, 4 where the last step we have used the dispersion relation for this problem. s Dai/PHYS 34/555 Spring 013 Review 1-3

4 Chapter 5: Summary Planck distribution function: The average excitation ti quantum # of an oscillator is: n sexp( n/ ) 1 exp( s / ) exp( / ) 1 At low temperatures, 4 x 3 nx 1 dx 6, x x e dx 4 e 1 n1 n1 n T the heat capacity CV NkB. 5 3 Dai/PHYS 34/555 Spring 013 Review 1-4

5 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 0 e ( e 1) / Dai/PHYS 34/555 Spring 013 Review 1-5

6 Thermal conductivity The thermal conductivity coefficient K of a solid is defined as, dt ju K, where ju is the flux of thermal energy, and dx x is distance. From the kinetic theory of gases we find 1 1 K Cvl Cv, where C is the heat capacity per volume, 3 3 v is the average particle velocity, and l is the mean free 1 path of a particle between collisions, is the phonon collision rate. Dai/PHYS 34/555 Spring 013 Review 1-6

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8 Chapter 6: Free electron Fermi gas Under quantum theory and the Pauli exclusion principle, we consider N of single electron is ( r ), then 3 noninteracting electrons confined to a volume V( L ). If the wave function ( r ) ( r ) ( r ). m x y z m Applying boundary condition ( xyz,, L) ( xyz,, ); ( xy, Lz, ) ( xyz,, ); ( x Lyz,, L) ( xyz,, ). The solutions are 1 ik r k ( r K ) e, ( k ). Note the probability of V m finding the electron somewhere in the volume is 1 dr ( r ). Dai/PHYS 34/555 Spring 013 Review 1-8

9 Note that K ( r ) is an eigenstate of the momentum operator, ik r ik r p, e ke i r i i r an electron in the level K ( r) has a momentum p k and a velocity v p / mk / m, where / k. Periodic boundary condition requires ik xl ikyl ikzl n n x y nz e e e 1 or k x, k y, k z L L L Thus in a 3-D k-space, the allowed wavevectors are those along the three axes given integer mutiples of. L Dai/PHYS 34/555 Spring 013 Review 1-9

10 To calculate the allowed states in a region of k-space volume, V ( / L) ( ) 3 3 or the number of allowed k-values per unit volume of k-space (known as the k-space density of levels) is V. Because the electrons are noninteracting we can built up 3 ( ) the N-electron ground state by placing electrons into the allowed one-electron levels. Pauli exclusion principle allows each wavevector to have electronic levels with spins up and down. Dai/PHYS 34/555 Spring 013 Review 1-10

11 Since the energy of a one-electron electron level is directly proportional to k, when N is enormous the occupied region will be indistinguishable from a sphere. The radius of this sphere is called k F (F for F ermi), 3 and its volume is 4 k / 3. The # of allowed k within the sphere is: 4 k V k F F 3 F V. Since each allowed k-value leads to two 3 3 kf kf one-electron levels, we must have N V V If electron density is n N / V, then we have n k / 3. F Dai/PHYS 34/555 Spring 013 Review 1-11

12 The sphere of radius k F containing the occupied one eletron levels is called the Fermi sphere. The Surface of the Fermi sphere, which separate the occupied form the unoccupied levels is called the Fermi surface. 3 N The momentum pf kf= of the occupied V one-electron levels of highest energy is the Fermi momentum. 1/3 F 3 N kf / m is the Fermi energy; m V /3 3 N v F p F / m( ) m V 1/3 is the Fermi velocity. Dai/PHYS 34/555 Spring 013 Review 1-1

13 Experimental heat capacity of metals 3 At sufficient low temperatures, CV T AT. Where is the Sommerfeld parameter. The ratio of the observed to the free electron values of the electronic heat capacity is related to thermal effective mass as: mth (observed) m (free) Dai/PHYS 34/555 Spring 013 Review 1-13

14 Electrical l conductivity it and Ohm s law Considering Newton's second law, we have dv dk 1 F m e ( E v B ) dt dt c The displacement of the Fermi sphere, k eet/. If collision time is, the incremental velocity is v ee / m. In a constant e lectric field E and n electrons per volume,the electric current density is j nqv ne E / m E. The electrical conductivity ne / m. Dai/PHYS 34/555 Spring 013 Review 1-14

15 Thermal conductivity of metals Wiedemann-Franz law Thermal conductivity for a Fermi gas 1 nk T nk T Kel Cvl v l 3 3 mv 3 B B F F m The Wiedemann-Franz law states that for metals at not too low temperatures the ratio of the thermal conductivity to the electrical conductivity is directly proportional to the temperature, independent of the particular metal. Kel nkbt /3m kb T ne / m 3 e LT. Lorenz number L.4510 watt-ohm/deg 8 Dai/PHYS 34/555 Spring 013 Review 1-15

16 Chapter 7: Summary Bloch s theorem The eigenstates of the one-electron Hamiltonian H U( r), where U( r R) U( r) for all R in m a Bravias lattice, can be chosen to have the form of a plane wave times a function with the periodicity of the Bravias lattice: ik r ( r) e u ( r), where u ( r R) u ( r) nk nk nk nk for all R in the Bravias lattice. or ik R ( r R) e ( r) Dai/PHYS 34/555 Spring 013 Review 1-16

17 The effect of a periodic potential The periodic potential has form: x U U0 U1cos, where a is lattice parameter a and U U. If U 0 then we have the free k electron gas case where m Dai/PHYS 34/555 Spring 013 Review 1-17

18 Wave equation of electron in periodic potential U( x) U e U ( e e ) U cosgx igx igx igx G G G G G0 G0 1 H p U e x x igx ( G ) ( ) ( ) m G ikx ( x) C( k) e, k n/ L. k 1 ikx p ( x) k C( k) e, m m k igx igx ikx ( U e ) ( x ) U e C ( k ) e, G G G k ikx igx ikx kcke ( ) Ue G Cke ( ) C k m G k k k G ( ke ) ikx ( ) Ck ( ) UCk G ( G ) 0. k k / m. m G Dai/PHYS 34/555 Spring 013 Review 1-18

19 Crystal momentum of an electron Under a crystal lattice translation we have ( ) ik T ik r ( ) ik r T e e u r T e T ( r). k k k If the lattice potential vanishes, the result recovers to that of free electron gas. k is called the crystal momentum of an electron. If an electron k absorbs in a collision a phonon of wavevector q, the selection rule is k q k ' G. Dai/PHYS 34/555 Spring 013 Review 1-19

20 Approximate solution near a zone boundary At the zone boundary the kinetic energy of the waves 1 k G are equal. ( ) C( k) UGC( kg) 0. ( ) CG ( / ) UC( G/ ) 0 ( ) C ( G /) UC ( G /) 0 1 ( ) U ; U ( G) U. m Thus the potential energy has created an energy gap U at the zone boundary. G Dai/PHYS 34/555 Spring 013 Review 1-0

21 Chapter 8: Summary A solid with an energy gap will be nonconducting at T 0 unless electric breakdown occurs or unless the AC field is of such high frequency that exceeds the energy gap. However, when T 0 some electrons will be thermally excited to unoccupied bands (conduction bands). If the enegy gap E the gap is of order g E e 0.5 ev, the fraction of electrons across g /k T B 10, and observable conductivity will occur. These materials are semiconductors. Dai/PHYS 34/555 Spring 013 Review 1-1

22 Effective Mass 1 The group velocity vg d/ dk, /, so vg d / dk. dvg 1 d 1 d dk 1 d dk 1 d F dt dkdt dk dt dk dt dk F 1 1 d ma, then we have. * m dk Effective Mass in Semiconductors The angular rotation frequency c of the current carriers eb * is: c, where m is the effective mass. * mc Dai/PHYS 34/555 Spring 013 Review 1-

23 In an intrinsic semiconductor, n p, E E E, 3/ kt B 3/4 n m m exp[( E / k T]. e h g B g c v Dai/PHYS 34/555 Spring 013 Review 1-3

24 Chapter 9: Reduced zone scheme ik ' r For a Bloch function written as k' ( r) e uk' ( r), with k ' outside the first Brillouin zone, we have k k ' G. ' ( ) ik r ( ) ik r ig r e u r e e r u ( r) e ik r u ( r) ( r) k' k' k' k k Dai/PHYS 34/555 Spring 013 Review 1-4

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26 Electron orbits, hole orbits, and open orbits An electron on the Fermi surface will move in a curve on the Fermi surface, because it is a surface of constant energy. Dai/PHYS 34/555 Spring 013 Review 1-6

27 Tight banding method for energy bands Tight banding approximation deals with the case in which theovelapofatomicwavefunctionsisenough of atomic functions is enough to require corrections to the picture of isolated atoms, but not so much as to render the atomic description irrelevant. Dai/PHYS 34/555 Spring 013 Review 1-7

28 Free electrons in a uniform magnetic field The orbital energy levels of an electron in a cubic box with sides of length L parallel to the x-, y-, and z-axes are determined in the presence of a uniform magnetic field H along the z-direction by two quantum numbers, and k : 1 eh kz kz ( ) c, c. m mc runs through all nonmagnetic integers, and kz takes on the same values as in the absence of a magnetic field: kz nz / L for any integral nz. The energy of motion perpendicular p to the field, which would be ( kx ky) / m if no field were present, is quantized in steps of ( eh / mc). This is orbit quantization. c c z Dai/PHYS 34/555 Spring 013 Review 1-8

29 Origin of the oscillatory phenomena Most electronic properties of metals depend on the density of levels g at the Fermi energy, g. It follows that will be singular F whenever the value of H causes an extramal orbit on the Fermi surface to satisfy the quantization condition ( + ) A A e ( F ). 1 e 1 e H c A mck K / G. e F B Dai/PHYS 34/555 Spring 013 Review 1-9 F

30 For a system of N electrons at absolute zero the Landau levels are filled to s. Orbitals at the next higher level s 1 will be partially filled. The Fermi level will lie between s and s1. As the magnetic field is increased the electrons move to lower levels because the area between successive circles are increased ( k ) ( k )( k ) eb/ c. Dai/PHYS 34/555 Spring 013 Review 1-30