Introduction К Современной Физике Твердого тела, Introduction To Modern Solid State Physics(477s).pdf:

Introduction К Современной Физике Твердого тела

Introduction To Modern Solid State Physics(477s).pdf

Size 4.6Mb
Date Dec 9, 2005

Cites:

Then we can return to Cartesian co-ordinates by the transform k ξi = αik xk Finally we get V (r) = b Vb eibr ....

Figure 1.13: The Ewald construction. (RL) and then an incident vector k, k = 2π /λX starting at the RL point. Using the tip as a center we draw a sphere. The scattered vector k is determined as in Fig. 1.13, the intensity being proportional to SG ....

The two names are due i) to the fact that these forces has the same nature as the forces in real gases which determine their diﬀerence with the ideal ones, and ii) because they are...

Figure 2.5: Linear diatomic chain. elastic constants to be C1,2 we come to the following equations of motion: m1 un = −C1 (un − vn ) − C2 (un − vn−1 ) , ¨ m2 vn = −C1 (vn − un ) − C2 (vn − un−1 ) . ¨...

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)...

Consequently we come to the Schr¨dinger equation o − q 2 ∂2 12 ˆ (P , Q) = ˆ H − ωj (q)Q2 (q) j 2 2 ∂ Qj (q) 2 j...

that is negative. Very often such a quasiparticle is called a hole (see later) and we deﬁne its mass as the electron mass with opposite sign. So, in a simple cubic crystal the hole mass, m∗ = |m∗ |, near the band top is the same as the electron mass near its bottom. p n In the general case one can expand the energy as in lth band near an extremum as ∂2 0 1α ε(k) εl (k) = (kα − kα0 )(kβ − kβ 0 ) (3.36) 2 ,β ∂ k α ∂ k β and introduce the inverse eﬀective mass tensor ∂2 0 ε(k) −1 (m )αβ = ∂ kα ∂ kβ This 2nd-order tensor can be transformed to its principal axes....

Also remains the motion along z-axis with the velocity vz = ∂ ε/∂ z . Now we discuss one very useful trick to calculate the properties of electrons in a magnetic ﬁeld. Namely, let us introduce the time of the motion along the orbit as d c p t1 = eH v⊥...

4.2. STATISTICS OF ELECTRONS IN SOLIDS is the Heaviside unit step function. In this approximation we easily get ζ0 =
2 2kF 2 = (3π 2 n)2/3 . 2m 2m...

The concentration nT for the room temperature and the free electron mass is 2.44 · 1019 cm−3 , it scales as (mn mp )3/4 T 3/2 ....

Now the magnetic susceptibility can be calculated for any limiting case. For example, the temperature-dependent part for a strongly degenerate gas (see Problem 4.5) 1 k 2. 2 π BT (4.22) χ = µ2 g ( F ) − B 12 F We see very important features: at low temperatures the main part of magnetic susceptibility is temperature-independent. Comparing the free electron susceptibility (4.22) with the Langevin one (4.21) we see that for electrons the role of characteristic energy plays the Fermi energy F . For the Boltzmann gas we return to the formula (4.21)....

eH (4.26) mc is the cyclotron frequency. Note that ωc = 2ωL , where the Larmor frequency ωL was introduced earlier. The corresponding magnetic moment is ωc = µ=
2 erc v⊥ mv⊥ /2 = . 2c H...

Figure 4.4: Landau levels as functions of H . magnetic ﬁelds its dependence on magnetic ﬁeld is very weak. We see that if magnetic ﬁeld is small many levels are ﬁlled. Let us start with some value of magnetic ﬁeld and follow the upper ﬁlled level N . As the ﬁeld increases, the slopes of the ”fan” also increase and at a given treshold value HN for which εN (HN ) = F . As the ﬁeld increases the electrons are transferred from the N -th Landau level to the other ones. Then, for the ﬁeld HN −1 determined from the equation εN −1 (HN −1 ) = F the (N − 1) becomes empty. We get 1≈ mc c F me HN ≈ , so∆ . e N 1 H cc F We see that electron numbers at given levels oscillate with magnetic ﬁeld and one can expect the oscillating behavior of all the thermodynamic functions. To illustrate the oscillations the functions x N 1 ˜ Z (x) = −N − 2 as well as the diﬀerence between its classical limit (2/3)x3/2 are plotted in Fig. 4.5....

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...

is the unit vector directed along p. In this case d w c 4 Ω I (f ) = f (ε) W (θ) os(p , f ) − cos(p, f ) π here Ω is the solid angle in the p -space, θ ≡ p, p is the angle between p and p , while (Check!) ni |v (θ)|2 W (θ) = π (ε) . (6.9) g Then we can transform the integral as follows. Chose the polar axis z along the vector p. Now let us rewrite the equation p · f = pz fz + p⊥ · f⊥...

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....

We observe if the screening is neglected rs → ∞ than the the transport relaxation time τtr → 0, and the transport relaxation rate diverges (long-range potential!). The function Φ(η ) slowly depends on the energy, so τtr ∝ ε3/2 ....

To get a similar formula for absorption one should make a similar substitution. The result can be obtained from that above by the replacement N ↔ N + 1, f ↔ 1 − f , a1 f1 (1 − f1 ) (N + 1 − f2 ) − a2 f2 (1 − f2 ) (N + f1 ) and then replace 1 ↔ 2 in the δ -functions to take into account the conservation law. Finally,
− kB T Fk,k−q = a1 f2 (1 − f2 ) (N + 1 − f1 ) − a2 f1 (1 − f1 ) (N + f2 ) → absorptiom...