Questions and Answers of Solid State Physics Essential Concepts

(a) Prove the assumption made above, that if the probability of a collision per unit time is dt/τ, then the average time since the last collision is τ. This follows a similar procedure as Exercise
Verify the statement above, that a hydrostatic stress leads to a hydrostatic strain, using the cubic (but not necessarily isotropic) compliance tensor deduced in part (a) of the prior exercise. Is it
Show that the Debye–Waller factor e−W vanishes in the case of a one- or two-dimensional system, but is finite in three dimensions. This reflects the fact, discussed in Sections 10.5, 10.7, and
The electro-optic matrix for lithium niobate, LiNbO3, has the form given in Table 3.3, with four independent constants. Write down the index ellipsoid matrix η̃ for an electric field along the
Verify the equations (3.8.24) through (3.8.27) for a plane wave reflecting from a free surface in an isotropic medium, using the boundary conditions (3.8.13) and (3.8.14). A program like Mathematica
(a) Determine the compliance tensor S in terms of the elements of the cubic, or isotropic, elastic constant matrix C given in Table 3.1.(b) If a uniaxial stress is applied along the [110] axis, what
Show analytically that if the two masses are equal and the two spring constants are equal, then the dispersion relation becomes the simple relation for a linear monatomic chain,where a is the spacing
Using the Fresnel equations, determine Brewster’s angle, the angle of incidence at which no p-polarized light can be reflected (r∥ → 0). Before polaroid plastic polarizers were invented,
Determine the angle of the extraordinary ray, relative to the normal, for light incident normal to the surface of a uniaxial crystal with ne = 1.45, no = 1.65, with the crystal cut with the c-axis at
Write down the Christoffel wave equation for a wave in a cubic crystal propagating along the [111] direction. For the choice C11 = 1, C12 = C44 = 0.5, and ρ = 1, use a program like Mathematica to
Determine the principal polarizations for a wave traveling in the direction (x̂ + ŷ)/√2 in a crystal with dielectric tensor (3.5.23). A program like Mathematica can easily diagonalize a 3 × 3
A uniaxial crystal with ne = 1.45, no = 1.65, is cut with the c-axis at 45° relative to the normal. If light is incident on the surface at 45° (90° relative to the caxis), determine the angles,
Earthquakes can also generate surface acoustic waves. Compute the Rayleigh wave speed for a SAW in rock that has vl = 10 km/s and vt = 5 km/s.
A uniaxial crystal is cut with the c-axis parallel to the surface, as shown in Figure 3.22, with ne = 1.7 and no = 1.77. For light incident normal to the surface, determine the polarizations of the
Use (4.10.5) to find the rms displacement of the atoms in a crystal of silicon at T = 300 K, for a phonon speed of v = 5×105 cm/s and density of 2.3 g/cm3. For the unit cell size of silicon 5.43 Å,
Prove that the commutation relation (4.1.10) follows from the definitions (4.1.9) and the commutation relation (4.1.4). [x, p] = iħ. (4.1.4)
Show that the spatial field operators ψ(r) and ψ†(r) defined in (4.5.1) have commutation relations similar to those of the momentum state operators, that is,You will need the identity [(r), ¹
Prove the commutation relation (4.2.17) using the definitions of ak and a†k and the commutation relation [xn, pn′ ] = ih̄δn,n′. The identity (4.2.9) is useful. [ak, ay] = dk k', (4.2.17)
Verify that the superposition of Fock states given in (4.4.5) for the coherent state has the eigenvalue given in (4.4.1). aklak) = aklak), (4.4.1)
Show that the commutation relation for the electron spatial field operators isusing the normalization relation for the Bloch cell functions {¥₂G), ¥ † (Gr')} = 8&r — 7'′)§n,n', - n (4.6.6)
Prove the statement given in this section that the time evolution operator is unitary, under the assumption that the norm of the wave function 〈ψt|ψt〉 = 1 at all times. S(t, to) = e(i/h)
Show that (4.2.13) is equal to (4.2.14) by substituting into (4.2.14) the definitions of ak and a†k. You will need to use the definitions of xk and pk in terms of xn and pn, and the
Convert (4.9.10) into an equation for the radiant energy per unit wavelength for a blackbody radiation source. From this equation, determine the peak emission wavelength, that is, the wavelength at
Prove the relations (4.8.11) using the known commutation relation [a, a†] = 1 for bosons and the anticommutation relation {a, a†} = 1 for fermions. [Ñk, ak] = -akk,k' [Nk, at] =a8kk, (4.8.11)
Prove, using the result of the previous exercise, that the fermion many particle state (4.6.5) is normalized ifData from in previous exercise [ d³r þ;* (7)þi(r) = Sij.
The state (4.4.5) is a Poisson distribution of photons. Assuming that only one k-state is occupied, calculate the uncertainty defined byShow it is equal to the expected value for a Poisson
In theoretical work that eventually led to the laser, Einstein considered a set of N two-level atoms, as shown in Figure 4.8, in a box with Nk photons, wherethe energy h̄ωk corresponds to the
Calculate the phonon specific heat, in units of J/K-kg, of a solid with speed of sound v = 4 × 105 cm/s, and density ρ = 5 g/cm3, at a temperature of 77 K (much less than the Debye temperature).
Write down the quantum Boltzmann equation for the evolution of a population of fermionic particles interacting with a population of phonons through an interaction term of the formwhere a†k3 is the
Fill in the missing steps from (4.9.30) to (4.9.31). First, write a Sommerfeld expansion to second order in T for the total electron number N. The lowest-order terms of both U and N can then be
Show that (4.8.21) satisfies the detailed balance condition (4.8.20). A critical step in the proof is to invoke energy conservation, Ek + Ek1 = Ek2 + Ek3. f(Ek3)f(Ek₂)(1±ƒ(Ek₁ ))(1 ±ƒ(Ek))
Prove that the relation (4.4.11) is true. You will need the quantum mechanical relation [A,f(B)] = [A, B] ƒ' (B). (4.4.15)
Find the rate of transitions from initial state |i〉 with Np phonons in state p(vector) and Nq in state q(vector), and none in any other states, to the final many-body state |f〉 with Np − 1
Calculate the temperature at which the photon energy density in a solid will exceed the phonon energy density in the same solid, if the solid has a Debye temperature of 300 K, phonon velocity of 4 ×
Prove the identity (4.2.9) by computing the sum for finite N and then taking the limit N → ∞. δη,η = Σακα(n-1), k (4.2.9)
Show explicitly using formula (4.4.10) that Dk(αk)|0〉 = |αk〉. Dk(z) = e¯¹²¹²/²e²ake-z*ak = ek²1²/2e-zakeza. (4.4.10)
If there is a heat source within the medium, the heat diffusion equation (5.7.2) is altered towhere G is a generation term which can depend on x(vector) and t. Solve this equation in one dimension
In a solid with density 1 g/cm3, a sound wave has frequency 1 MHz, velocity 5 × 105 cm/s, and intensity 1 W/cm2. What is the root-mean squared displacement of the atoms?
What is the amplitude in V/cm of a laser beam with diameter 2 mm, with 1020 photons in a cavity of length 1 m? Assume each photon is in the visible range with energy h̄ω ≃ 2 eV. What is its
Show that the interaction Hamiltonian (5.1.10) is Hermitian. - Σοκα και HD = h 2pVwk ilab -b *k'nky +kn,kı - akton). a-b (5.1.10)
Estimate the Seebeck coefficient for a metal with a Fermi level of 0.5 eV, at T = 300 K. To do this, you will need to approximate D(E) as a peaked function at E = EF, according to the discussion at
Show that the interaction Hamiltonian (5.2.8) is Hermitian, even if 〈n′|p(vector)|n〉 is complex. The fact that both n and n′are summed over is important. He-phot m X h ΣΣΣ, Ξενω
For an electronic system with constant scattering time τ = 10−11 s and effective electron mass one-tenth the vacuum electron mass, determine the magnetic field at which magnetoresistance effects
Prove that the force law (3.4.23) follows from the definition of the energy density (5.12.1) and the energy conservation law Joij ax; Σ pui, j (3.4.23)
(a) Show that a typical drift velocity is of the order of 105 cm/s for electrons in a solid subject to an electric field of 5 V over a distance of 1 cm, for τ around 10−11 s as calculated.(b) Plot
If we assume that scattering of phonons leads to breakup of the coherent properties of sound, then the attenuation of sound in a solid can be computed using the above formulas. Calculate the
Show that the scattering rate formula (5.4.8) follows from the form of the Hamiltonian (5.4.6). Hanharm 1 = fav /zyce³. (5.4.6)
Determine the interaction Hamiltonian for Coulomb scattering of electrons confined to move in only two dimensions. Use the screened Coulomb interaction (5.5.5). For the actual two-dimensional
Determine the average electron–phonon scattering time for a metal with electrons in a Fermi–Dirac distribution with μ = 0.5 eV above the ground state, and T = 300 K, instead of a
Show that the electric field term dominates over the diffusion term in (5.8.10) in a typical situation. Compute D using ¯v given by the root-mean-squared speed of electrons in a Maxwell–Boltzmann
Compare the screening length for a three-dimensional electron gas at room temperature, in the Maxwell–Boltzmann limit with density n = 1018 cm−3, with that of a two-dimensional electron gas with
Show that the spontaneous-emission lifetime is a few hundred femtoseconds for a conduction electron initially in a state with kinetic energy 30 meV falling into an empty valence-band state. Assume
Fill in the missing mathematical steps from (5.4.12) to (5.4.14). Ә P = — — (U - TS) av || awk Nk av En- k (5.4.14)
For the piezoelectric tensor for a cubic crystal given in Table 3.4, show that the piezoelectric interaction Hamiltonian couples electrons to transverse phonons in the [110] direction but not
Estimate the melting temperature of a solid with lattice constant 6 Å, optical phonon energy 30 meV, and effective mass of the unit cell given by the mass of a silicon atom. Does your answer seem
Show that for a three-dimensional isotropic electron gas, (5.5.16) becomes (5.5.19) when f(E) ∝ e−E/kBT, that is, when the electrons have a Maxwell– Boltzmann distribution. K 2 = af - ²/² ( =
In a cubic crystal, a triply degenerate band with p-symmetry has the following deformation Hamiltonian, known as the Pikus–Bir Hamiltonian (derived in Section 6.11.2):operating on the basis of the
Derive the Wiedemann–Franz law (5.8.21) using the definitions of the electrical conductivity, the thermal conductivity, and the diffusion constant given in the previous sections, along with the
Formula (5.2.14) gives an absorption rate. Convert this to an absorption length, that is, show that for a constant fluence of photons hitting a surface at x = 0, the probability of photon absorption
If the Rayleigh scattering of light in the atmosphere is known to be dominated by scattering from a particulate with radius 50 nm and index of refraction n = 2, estimate the particulate density if
Show that not only σxx , but also ρxx vanishes in the limit τ → ∞, where ρxx is the diagonal resistivity defined by õ = Pxx Pxy 0 - Pxy Pxx 0 0 :) = 0 | =õ–¹ po (5.10.10)
For a typical resistance R = 100 Ω in a resistor of length l = 2 mm and cross-sectional area A = 1 mm2, where R = l/Aσ , calculate the average scattering time τ for an electron, for a free carrier
Prove that (5.6.15) is a solution to the diffusion equation (5.6.14) with the initial condition n(x(vector), 0) = δ(x(vector)). Show that it conserves the total number of particles.What is the full
Use (5.5.32) to show that in the limit of low density, the total scattering time (dephasing time) for an electron gas in vacuum due to Coulomb scattering at room temperature is independent of density
Show that in vacuum, an electron cannot directly recombine with a hole in the Dirac sea, that is, a positron, by emission of a single photon, because this would violate energy and momentum
Prove that substituting (5.2.29) into (5.2.25) gives (5.2.8). See Exercise 5.2.1.Data from Exercise 5.2.1Show that the interaction Hamiltonian (5.2.8) is Hermitian, even if 〈n′|p(vector)|n〉 is
Show, following a procedure similar to that of Section 5.5, that if the realspace interaction potential between two particles is proportional to a δ-function, δ(r(vector)1 − r(vector)2), then the
Show explicitly that the fluctuation–dissipation theorem applies to the case of the two-level oscillator driven by an electric field, discussed in Section 7.4, using the susceptibility defined in
Show that the result (9.9.12) is the same as expected for a one-dimensional Planck distribution, following the logic of Section 4.9.2 using a one-dimensional density of states. — R|I(ƒ)/² = kBT
Prove that (9.6.17) follows from putting the definition of ρ(x(vector), t) into (9.6.16). C₂ (²₁1) = = = f d³x V d³x' (p(x, t)px' + x,0)), (9.6.16)
(a) In Chapter 5, we did not calculate any interaction Hamiltonian terms that would couple a photon directly to a phonon. However, a second-order process is possible in which an electronic transition
(a) Find the basis state of the Γ−2 irreducible representation for this molecule as a linear superposition of the original atomic states. (b) Working with an explicit representation for the
Show that a symmetry operator R must be unitary, that is, its adjoint is equal to its inverse; for a matrix representation this means that its inverse is equal to the complex conjugate of the
Verify the characters in Table 6.3 for the irreducible representation Γ5 which has basis functions which are the three components of a vector in real space along x, y, and z, by explicitly
(a) In cubic symmetry, there are six C4 rotations (90° rotations about the x, y, and z axes), eight C3 rotations (120° rotations about the [111] directions, or cube corners), and three C2 rotations
Verify the values of P,Q, R, and S in (6.11.10) by evaluating four of the terms in (6.11.9) above, using the coupling coefficients of Table 6.9. R = 1-m 6 1 S = n(kxkz -
Use a program like Mathematica to solve (7.5.10) for ω(k). Setting c = 1 and ϵ(∞) = 1, plot both branches of ω(k) for various values of the splitting ω2L−ω2T , and show that in the limit of
Verify the multiplication rules given in Table 6.8, Γ6 Ο Γ5 = Γ7 Θ Γ8, (6.6.3)
Prove the statement above, that U = eiϕσy satisfies the requirement (6.9.5). You will need to use the anticommutation rule {σi, σj} = 2δij and σ∗y = −σy. S₁T = TS₁ SUKU
Determine how the Γ4 states in a crystal with Td symmetry will be split if uniaxial stress is applied along the [111] direction. To do this requires access to a set of symmetry tables, which can be
Use a program like Mathematica to diagonalize the 4 × 4 block for the Γ8 valence band states Hamiltonian (6.11.9) and verify the form of the solution (6.11.11). Determine A,B, and C in terms of
Determine all the dipole-allowed optical transitions between states in a crystal with Td symmetry, for electronic bands with Γ6, Γ7, and Γ8 symmetry.
Verify, using the coupling coefficients for the Γ6 ⊗ Γ5 product states in Table 6.9, that the interaction (6.6.5) acting on the product states gives the same energy shift for both Γ7 states, and
In Figure 6.4, the states at the critical points X and L, in the [100] and [111] directions, respectively, correspond to different symmetries from the Td symmetry of the crystal as a whole. Determine
For an optical transition in a crystal with Td symmetry between a valence band with Γ8 symmetry and a conduction band with Γ6 symmetry, which states are coupled to each other by left-circular and
For couplings between spins, we can construct four independent Hermitian coupling matrices,Show explicitly, by calculating the characters for the C2 and C4 rotations, that one of these matrices can
In many semiconductors, a biexciton can be formed as a bound state of two excitons, analogous to a hydrogen molecule formed from two hydrogen atoms. For the Γ6 conduction band and Γ7 and Γ8
Determine the polarization-angle dependence of the matrix element for light absorption by each of the orthoexciton states in the crystal Cu2O, for light directed along the [111] axis.
Determine the polarization dependence of the dipole and quadrupole optical transitions to paraexciton and orthoexciton states in Cu2O if the direction of the propagation of the light is along [110],
Suppose that instead of a harmonic oscillator, we have a metal in which the electrons move directly in response to the electric field, that is, there is a damping constant but no spring constant, so
The optical response of a semiconductor can be treated as a set of twolevel oscillators, with the number of oscillators at a given frequency equal to the joint density of states at that frequency.
The change of basis function for the orthoexcitons from xz, yz in Oh symmetry to Lx, Ly in D4h was allowable because these functions transform the same under all D4h symmetry operations. The
The dielectric function is subject to various sum rules. For example, show that the Kramers–Kronig relations implyfor a set of single oscillators, no matter what their frequency distribution is.
(a) Show that the wave equation (7.6.4) allows sum and difference frequency generation in the case of two input waves with different frequencies. Show that the amplitude of the sum wave with
Use a program like Mathematica to solve equations (7.5.13)–(7.5.15) for ω(k). Setting c = 1, ±(∞)/ϵ0 = 2, and Eg−Ryex = 1, plot both branches of ω(k) for various values of the coupling
Section 8.1 of Chapter 8 will show that we can write an imaginary part of the energy of an electron state, corresponding to h̄/τ, where τ is the time to scatter out of that state. Put in Ec −
For a solid with index of refraction 1.6 and speed of sound 5 × 105 cm/s, determine the sound frequency needed to deflect a light beam of λ = 600 nm by an angle of 30°.
Using the result of the previous exercise, calculate the skin depthλ = 1/kI for the penetration of an electromagnetic wave into a metal surface, as a function of frequency and the material
Show explicitly that the diffraction grating made by two beams interfering in a medium will give diffracted output of a third beam consistent with momentum conservation of the photons. In other
Surprisingly, near a resonance, not only the phase velocity, but also the group velocitycan exceed the speed of light. This still does not violate the theory of relativity, because in the region near

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