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Introduction To Modern Solid State Physics(477s).pdf |
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Figure 1.6: Primitive vectors for bcc (left panel) and (right panel) lattices. procedure to choose the cell which reflects the symmetry of the lattice. The procedure is as follows: 1. Draw lines connecting a given lattice point to all neighboring points. 2. Draw bisecting lines (or planes) to the previous lines....
Now we can replace k ρ cos(ρ, R) by k ρ where k is the scattered vector in the direction of R. Finally, the phase equal to k R − ρ∆k, ∆k = k − k . e...
Diatomic Chain. Acoustic and Optical branches.
We use this case to discuss vibrations of compound lattices. Let us consider the chain shown in Fig. 2.5 One can see that the elementary cell contains 2 atoms. If we assume the...
and introduce the dielectric function according to the electrostatic equation for the displacement vector D = E + 4π P = E . (2.55) This function is dependent on the vibration frequency ω . We get P= −1 E. 4π (2.56)...
Quantization of Atomic Vibrations: Phonons
The quantum mechanical prescription to obtain the Hamiltonian from the classical Hamilton function is to replace classical momenta by the operators: i ˙ ˆ Qj (q) = Pj (q) → Pj (q) = ∂ . ∂ Qj (q) (2.83)...
(the current of a completely filled band is zero!). So we can express the current of a partly full band as the current of holes with the charge +e > 0. To get a more deep analogy let us calculate the energy current due to the flux of electrons and holes. Characterizing the energy current in a state k as v[ε(k) − eϕ] where ϕ is the electric potential we get k k νn (k, s)[ε(k) − eϕ]v(k) = [1 − νp (k, s)][ε(k) − eϕ]v(k) = w=
,s ,s...
Now we briefly discuss the way to describe many-electron states by the occupation numbers. Such a procedure was introduced for phonons which are Bose particles and we first generalize that procedure. In general case, the total wave function of bosons is symmetric in replacement of the particles. Thus it can be expressed as a symmetric product of individual wave functions ΦN1 N2 ... = N
1 !N 2 ! . . ....
4.4. MAGNETIC PROPERTIES OF ELECTRON GAS. where we keep only linear in the field H1 terms. Solving this equation we get χx = where χ0 = Mx χ0 = Hx 1 − (ω /ω0 )2 M 0z , Hz...
95 semiconductor physics. Substituting the Hamiltonian p2 /2m + eϕ = p2 /2m − e2 / r with ˆ ˆ the effective mass m to the SE we get the effective Bohr radius aB = m0 2 m0 ˚ = a0 = 0.53 , A. B 2 me m m...
For a typical metal p ≈ /a, and we get a. There is another important criterion which is connected with the life-time τϕ with the respect of the phase destruction of the wave function. The energy difference ∆ε which can be resolved cannot be greater than /τϕ . In the most cases ∆ε ≈ kB T , and we have kB T τ.
ϕ...
The conductivity tensor σ can be calculated with the help of the relation (6.16) if the ˆ diffusivity tensor Dik = vi I −1 vk ε is known. Here the formal ”inverse collision operator” is introduced which shows that one should in fact solve the Boltzmann equation....
The function K cannot be derived from classical considerations because the typical spatial scale of the potential variation appears of the order of the de Broglie wave length /p. We will come back to this problem later in connection with the quantum transport. The function K (r) reads (in the isotropic case) c . p3 os(2kF r) sin(2kF r) F K (r) = −g ( F ) − (π )3 (2kF r)3 (2kF r)4 We see that the response oscillates in space that is a consequence of the Fermi degeneracy (Friedel oscil lations). These oscillations are important for specific effects − but if we are interested in the distances much greater than kF 1 the oscillations are smeared and we return to the picture of the spheres of atomic scale. So one can use the Thomas-Fermi approximation to get estimates....
Conductivity Tensor. Calculations.
Simplified version for isotropic case In a magnetic field the Boltzmann equation reads ( E p 1 f − f0 f+ = 0. v r) − e + [v × H] c τtr We look for a solution as f = f0 + (v · G) , We have − e ∂ 1 [v × H] + c ∂ p τtr |G| ∝ E . (6.48)...
The solutions are: a0 = C0 , t1 a1 = [I (C0 ) − e (v(t2 )E)] dt2 + C1 , . . . 0 t1 ak = [I (ak−1 ) − e (v(t2 )E)] dt2 + Ck . . .
0...
One can see that the result is strongly dependent on the scattering mechanism and oscillates in the case of the Fermi statistics....
Here n, and φ are Fourier components of the concentration and the potential, respectively. To relate the potential φ to the external field one has to employ Poisson equation iqD = 4π en , We get 0q 2 φ + iqDe = 4π en . D = −i 0qφ + De ....
−p kT nd expand it up to the first order in the energy difference, . p− p Wp p = Wpp 1 + kT Expanding also the distribution function we get finally
exp...
¤ ¥ ¤ ¥ ¤ ¥ ¤ ¥ ¤ ¥ ¤ ¥ ¤ ¥ ¤ ¥ ¤ ¥ ¤ ¥ ¤ ¥ ¡ ¡ ¡ ¡ ¡ ¡...
where the collision integral is determined by the scattering processes. The most important of them are • Phonon-phonon processes. These processes are rather complicated in comparison with the electron-electron ones because the number of phonons does not conserve. Consequently, along with the scattering processes (2 → 2) there are processes (2 → 1) and (1 → 2). Scattering processes could be normal (N) or Umklapp ones. Their − − frequency and temperature dependencies are different (τN 1 ∝ T ω , τU 1 ∝ exp(Θ/T )). • Scattering by static defects. Usually it is the Rayleigh scattering (scattering by imperfections with the size less than the wave length, τ −1 ∝ ω 4 ). • Scattering phonons by electrons. All these processes make the phonon physics rather complicated. We are not going to discuss it in detail. Rather we restrict ourselves with few comments. Probably most important phenomenon is phonon contribution to thermal conductivity. Indeed, phonon flux transfers the energy and this contribution in many cases is the...
Substituting the estimate for ϕm we get 2e2 (σr )ik (ω , H ) ∼ (2π )3 d ϕ δ ni nk . −w K (ϕ) 1−e...
We see that the result depends on the character of extreme (max√ um or minimum). The im resonant contribution is maximal at ∆ ∼ (ω τ )−1 : Imax ∼ (1/k ) ω τ For k ∼ 1 it means √ Imax ∼ ωc τ . Consequently, at the maximum the ratio Zr /Z acquires the extra large factor √ ωc τ , and √ Zm rδ r ∼ . Z ax c This ratio could be in principle large but usually is rather small. The procedure employed is valid at ωc τ 1. In the opposite limiting case the oscillatory part of the impedance is exponentially small. In real metals there are many interesting manifestations of the cyclotron resonance corresponding to different properties of FS. A typical experimental picture is shown in Fig. 7.8....
we return to the static case. In the general case we can write 1 −i(qv − ω ) + τ −1 1 −i(cos ϑ − ω /q v ) + (q v τ )−1 = = i(qv − ω ) + τ −1 (qv − ω )2 + τ −2 q v (cos ϑ − ω /q v )2 + (q v τ )−2...
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