11. (a) Show that the angle φ between two planes in a cubic system with Miller indices (h1 k1 l1) and (h2 k2 l2) is given by
2/12 2
2 2
2 2
2/12 1
2 1
2 1
212121
)lk(h)lk(h
llkkhhcos ++++
++ =φ
(b) Show that the interplanar spacing dhkl between planes hkl in a cubic crystal with lattice parameter a is given by
2/1 )lkh(
ad 222hkl
++ =
2. Explain the following terminologies. (a) plasmon (b) exciton (c) polariton (d) polaron (e) magnon 3. Brillouin zone, rectangular lattice. A two-dimensional metal has one atom of valency one in a simple rectangular primitive cell a=2Å; b=4 Å. (a) Draw the first Brillouin zone. Give its dimensions, in cm-1. (b) Calculate the radius of the free electron Fermi sphere, in cm-1. (c) Draw this sphere to scale on a drawing of the first Brillouin zone. Make another sketch to show the first few periods of the free electron band in the periodic zone scheme, for both the first and second energy bands. Assume there is a small energy gap at the zone boundary.
4. (a) In a linear harmonic chain with only nearest-neighbor interaction, the normal-mode dispersion
relation has the form ω(k)=ω0sin(ka/2), where the constant ω0 is the maximum frequency (assume when k is on the zone boundary). Show that the density of normal modes in this case is given by
. a
0 ω−ωπ =ω
The singularity at ω=ω0 is a van Hove singularity. (b) In three dimensions the van Hove singularities are infinities not in the normal mode density itself,
but in its derivative. Show that the normal modes in the neighborhood of a maximum of ω(k), for example, lead to a term in the normal-mode density that varies as (ω0- ω)1/2.
+
-
Nd
dn
-Na
-dp
(a) If the dielectric constant of the semiconductor is ε, derive the potential φ(x) due to this charge distribution..
(b) If the potential difference between p side and n side is φ,
What is the width of the depletion region dn + dp?
(c) Assume the doping concentration on p side and n side are 1018cm-3, ε=16, φ=0.67V, calculate the
depletion width. (d) If a forward voltage 0.3V is applied, calculate the depletion width.
6. Diamond structure structure factor(a)(100) (b) (111) (c) (200) (d) (220) (e) (222) (f) (331) (g) (440)
7. Find the atomic scattering factor for solid hydrogen, given the charge density
0a a/r23 0
2 a100aa e)a/e()r(e)r( −π=Ψ=ρ where )r( a100Ψ is the normalized s-state function and
a0=0.53Å is the Bohr radius. 8. 0.862 g/cm3Atomic mass: 39.0983 1.40×1022 cm-3 a. KFermi wavevector, Fermi velocity, Fermi energy in eV, and Fermi temperature. b. Kthermal massDebye temperature 9. (a) Explain the meaning of Bravais lattice, primitive unit cell, and Wigner-Seitz cell. (b) Prove that the Wigner-Seitz cell for any two-dimensional Bravais lattice is either a hexagon or
a rectangle. (c) Show that the ratio of the length of the diagonals of each parallelogram face of the Wigner-
Seitz cell for fcc lattice is 1:2 . (d) Show that every edge of the polyhedron bounding the Wigner-Seitz cell of bcc lattice is
4/2 times the length of the conventional cubic cell.
10. (a)Show that the structure factor for a monatomic hexagonal close-packed crystal structure can
take on any of the six values 1+exp 3
inπ , n=1,...,6, as K rang through the points of the simple
hexagona1 reciprocal lattice. (b) Show that all reciprocal lattice points have non-vanishing structure factor in the plane
perpendicular to the c-axis containing K=0. (c) Show that points of zero structure factor are found in alternate planes in the family of
reciprocal lattice planes perpendicular to the c-axis. (d) Show that in such a plane the point that is displaced from K=0 by a vector parallel to the
c-axis has zero structure factor. (e) Show that the removal of all points of zero structure factor from such a plane reduces the
triangular network of reciprocal lattice points to a honeycomb array. 11. Nearly Free Electron Fermi Surface Near a Single Bragg Plane. To investigate the nearly free electron band structure near a Bragg plane, it is convenient to measure the wave vector k with respect to the point G2
1 on the Bragg plane (Brillouin zone boundary). If we
write qGk += 2 1 , and resolve q into its components parallel (q//) and perpendicular ( ) to G, then
the electron energy can be written as
⊥q
2/1
22 //
2
2/
22
G m h
Gλ , and UG is the Fourier component of U(r) at G.
It is also convenient to measure the Fermi energy Fε with respect to the lowest value assumed by the equation above in the Bragg plane, writing:
Δ+−= GG UF 2/λε
so that when Δ<0, no Fermi surface intersects the Bragg plane. (a) Show that when 0<Δ<2|UG|, the Fermi surface lies entirey in the lower band and intersects the Bragg plane in a circle of radius
2
Δ =
mρ .
(b) Show that if Δ>|UG|, the Fermi surface lies in both bands, cutting the Bragg plane in two circles of radii ρ1 and ρ2, and that the difference in the areas of the two circles is
( ) GUm 2
2 1
2 2
4 h
πρρπ =− .
12. Show that the velocity of an electron moving in a periodic potential in one dimension,
, vanishes at the zone boundaries. ∑ −δ= j
0 )jax(V)x(V
13. Verify that in a crystal with an fcc monatomic Bravais lattice, the free electron Fermi sphere for
valence 1 reaches (16/3π2)1/6=0.903 of the way from the origin to the zone face, in the [111] direction.
14. Consider a linear chain in which alternate ions have mass M1 and M2, and only nearest neighbors
interact (a) Show that the dispersion relation for the normal modes is
)kacosMM2MMMM( MM
K 21
2 2
2 121
2 ++±+=ω
where K is the spring constant between M1 and M2, a is the lattice constant. (b) Discuss the form of the dispersion relation and the nature of the normal modes when M1 >> M2. (c) What is the dispersion relation when M1 ≈ M2. 15. Consider a monatomic linear chain that the ions interact all other ions, then the potential energy is
given by
(a) Show that the dispersion relation is
, M
2
m∑ >
=ω
(b) Show that the long-wavelength limit of the dispersion relation is
,k)M/Km(a 2/1
0m m
2∑ >
=ω
(c) Show that if Km=1/mp (1 < p < 3), so that the sum does not converge, then in the long-wavelength limit
2/)1p(k −∝ω . (hint : replace the sum by an integral in the limit of small k.) (d) Show that in the special case p=3,
.klnk∝ω
16. Find the dielectric constant of a solid in which the local field has the form )12(
PEE r0
eff +εε +=
r rr
assuming N atoms per unit volume, each of polarizability α.
17. Does the Fermi surface of an alkaline earth metal, e.g. Be, Mg, extend to the 2nd Brillouin zone? Why? 18. The crystal NaCl has a static dielectric constant εr(0)=5.6 and optical index of refraction n=1.5. (a) What is the reason for the difference between εr(0) and n2? (b) Calculate the percentage contribution of the ionic polarizability. (c) Plot the dielectric constant versus the frequency, in the frequency range 0.1ωt to 10ωt, ωt=3.05×
1013rad/s is the optical phonon of NaCl. 19. The right figure shows the de Haas-van Alphen oscillations in Ag. The magnetic field is along a <111> direction as shown in the lower figure. How can you determine the area ratio of the two extremal orbits on the Fermi surface? What is the ratio? Why? 20. (a) Determine the ratio of the lattice constants c and a for a hexagonal close packed crystal
structure and compare this with the values of c/a found for the following elements, all of which crystallize in the hcp structure: He (c/a=1.633), Mg (1.623), Ti (1.586), Zn (1.861). What might explain the deviation from the ideal value?
(b) Supposing the atoms to be rigid spheres, what fraction of space is filled by atoms in the primitive cubic, fcc, hcp, bcc, and diamond lattice.
21. A two-dimension electron gas (2DEG) has an areal density of 2×1011 cm-2. How large is the magnetic field perpendicular to the 2DEG needed to have the Landau filling factor to be 1. Assume that the spins are already split by the Zeemann effect. 22. Landau levels. The vector potential of a uniform magnetic field zB is in the Landau gauge. The Hamiltonian of a free electron without spin is
xA ˆBy−=
. 2 1
h (in SI units)
We will look for an eigenfunction of the wave equation εψψ =H in the form )](exp[)( zkxkiy zx += χψ .
(a) Show that )(yχ satisfies the equation
( ) 0 2 1
m c zhh ,
where meBc /=ω and . (b) Show that this is the wave equation of a harmonic oscillator with frequency
eBky x /0 h=
2 1 h h ++= ωε .
(c) Calculate the degeneracy per unit area of each Landau level. 23. (a) Prove that the thermal conductivity of a metal due to the conducting electrons is
m TnkK B
el 3
22 τπ =
(b) Explain “Wiedemann-Franz law” and “Lorentz nunmber”. 24. 0.1nm 25. (a) Prove the Larmor theorem, i.e., that a classical dipole μ in a magnetic field B precesses
around the field with a frequency equal to the Larmor frequency ωL=eB/2m. (b) Evaluate the Larmor frequency, in hertz, for the orbital moment of the electron in a field
B=1T. (c) What is the precession frequency for a spin dipole moment in the same field?
26. A system of spins (j=s=1/2) is placed in a magnetic field H=5×104amp-m. Calculate the following:
(a) The fraction of spins parallel to the field at room temperature (T=300K). (b) The average component of the dipole moment along the field at this temperature. (c) Calculate the field for which <μz>=(1/2)μB, where <μz> is the average of dipole moment
along the magnetic field. (d) Repeat parts (a) and (b) at the very low temperature of 1K.
27. CTT2 2AT T C
+= γ
1234 θBkNA ′= θDebye Temperature
28. Rms thermal dilation of crystal cell. (a) Estimate for 300K the root mean square thermal dilation ΔV/V for a primitive cell of sodium. Take the bulk modulus as 7×1010 erg cm-3. Note that the Debye temperature 158 K is less than 300K, so that the thermal energy is of the order of kBT. (b) Use this result to estimate the root mean sqsuare thermal fluctuation Δa/a of the lattice parameter.[hint: The
B
potential energy associated with the dilation is 32 2 1 )/( aVVB Δ per unit cell.]
29. Umklapp Processs Te 21 θ
τ −
∝ θ Debye
Temperature 30. (a) Show that the angle φ between two planes in a tetragonal system with Miller indices (h1 k1 l1)
and (h2 k2 l2) is given by
2/1 2
2 2
=φ
(b) Show that the interplanar spacing dhkl between planes hkl in a tetragonal crystal with lattice parameter a, c is given by
2
2
2
22
+ +
=
31. Show that the unit cell volume of hexagonal crystal is 0.866a2c and it is abcsinβfor monoclinic
crystal where a, b, c are cell constants, βis the angle between a and c axes.
32. Empty lattice approximation periodic potential
reduced zone scheme
a[11] 5
k=0 33. Empty lattice approximation periodic potential
reduced zone scheme
a[11] 5
k=0
34.(a) Explain the difference between Rayleigh scattering, Raman scattering and Brillouin scattering. (b)A monochromatic light beam, λ0=6328Å, from water at room temperatureleads to a Brillouin
sideband whose shift from the central line is Δν=4.3×109Hz at scattering angle of 90o. Knowing that the refractive index of water is 1.33, what is the velocity of sound in this substance at room temperature?
35.(a) Using the continuity equation and Poisson’s equation, show that an excess localized charge in a
semiconductor decays in time according to the equation , where τ−ρΔ=ρΔ e)t( 0 σ ε
=τ is the
dielectric relaxation time and is the initial excess density. 0ρΔ
(b) Calculate τ for GaAs at low field for a carrier concentration of 1021m-3, μe=8500cm2/volt-s, and εr=10.9.
36.The indices and observed θvalues of Zn crystal made with Cu Kα1(λ=1.54Å) radiation are
line hkl θ 1 002 18.8o
2 100 20.2o
3 101 22.3o
4 102 27.9o
Calculate theθvalues if the radiation X-ray is Co Kα1(λ=1.79Å).
37. The following questions are concerning to the SEM experiment.
(a) Describe the origin and energy range of secondary electron, backscatted electron, Auger electron
and X-ray quanta.
(b) Explain E-beam lithography, EDX, Scanning Electron Acoustic Microscopy and their applications.
38.αCo has hcp crystal structure with lattice spacing of a=2.51Å and c=4.07Å, β-Co has fcc crystal structure with lattice spacing of a=3.55Å . Assume that the wavelength λof the X-ray radiation is 1.54Å. plase calculate and compare the position of first five X-ray powder diffraction peaks for each pattern.
39. Brillouin zones of two-dimension divalent metal. A two-dimensional metal in the form of a square
lattice has two conduction electrons per atom. In the almost free electron approximation, sketch carefully the electron and hole energy surfaces. For the electrons choose a zone scheme such that the Fermi surface is shown as closed.
40. Open orbits. An open orbit in a monovalent tetragonal metal connects opposite faces of the
boundary of a Brillouin zone. The faces are separated by G= 2×108 cm-1. A magnetic field B=103 gauss= 10-1 tesla is normal to the plane of the open orbit. (a) What is the order of magnitude of the period of the motion in k space? Take v~108 cm/sec. (b) Describe in real space the motion of an electron on this orbit in the presence of the magnetic field.
41. Alkali metals, like Li, Na, etc., have bcc lattice structures. What is the ratio of the Fermi
wavevector (kF) to the shortest distance between the Brillouin zone center Γ to the zone boundary
(N)? Assume the Fermi surface is not distorted by the lattice potential. Explain why the transport properties of alkali metals can be explained by Sommerfeld free Fermi gas theory?
42. Debye Model? 3 Debye Model Density of States Model
Debye Temperature θ acoustic phonon v Debye Temperature θ v N/V
T>>θ Blat NkC 3≈ ; T<<θ
34
dx e
x x
43.Using the periodic boundary condition, find the average energy per particle for a free electron gas
system of N electrons confined in (a) 3D, (b) 2D, and (c) 1D space of size Ld, where d gives the dimensionality of the space.
44.Atoms are arranged in a 1D chain with lattice spacing a. Each atom is represented by the potential
V(x)=aV0δ(x), Determine the energy gaps between the bands, assuming that the nearly free electron approximation applies.
45.Given a sodium sample of width 5mm, thickness 1mm. Now a current of 100mA is applied through
the sample while it is placed in a magnetic field of 0.1T (the magnetic field is perpendicular to the direction of current), find the Hall voltage. Given that the sodium is of body center structure with lattice constant a=4.28Å.
46.Given that the Fermi energy of silver at absolute zero is 5.5eV, at what temperature would the
electronic molar specific heat be equal to that of the lattice is silver? (the Debye temperature for silver is 210K).
47. Given a 1D diatomic chain of atoms with average inter-atomic spacing a=3.0Å. If the highest
frequency for the optical and acoustical branch of lattice wave are given as 4×1013rad/s and 4×
1013rad/s respectively, find the propagation speed of acoustic wave in the long wavelength limit. 48.Given the Einstein temperature and Debye temperature for diamond are ΘE=1320K and ΘD
=1860K respectively, Using Einstein model and Debye model calculate the specific heat of diamond at temperature T=2000K and T=0.2K.
49.We have a 2-D hexagonal lattice of lattice spacing a=3Å and one electron per unit cell. If the
electrons are considered free within the 2D plane, what is the Fermi energy εF. 50. Explain what ”heavy hole”, ”light hole” and ”split-off hole” are. 51.The crystal structure of KCL and KBr is fcc, please discuss and compare the
x-ray reflection patterns of their powders.
52.Electrons of mass m are confined to one dimension. A weak periodic potential
described by the Fourier series V ( ) ⋅⋅⋅⋅⋅+++= xVxVVx
aa ππ 4cos2cos is 210
ill the nearly free-electron approximation
at
applied. (a)Under what conditions w work?Assuming that the condition is satisfied,sketch the three lowest energy bands in the first Brillouin zone. Number the energy bands (starting from one
ak π= the lowest band). (b)Calculate (to first-order) the energy gap at
(between the first and second band) and 0=k (between the second and third
3. Using the tight binding method, considering only the s orbital at each atomic site, show that for a
band).
5 fcc structure with a lattice constant a, the electron energy can be written as:
)coscoscoscoscoscos(4)( 111111 akakakakakaka 222222 yxzxzy ++−−= γαε k ,
neighbor. What is the
where , and ρm is the position of the nearest
width of the band? Calculate the inverse effective mass tensor near the zone center.
αα −=−−= ∫∫ )()(,)()( ** rρrrr smsss HdVHdV
54. Using the tight binding method, considering only the s orbital at each atomic site, show that for a simple cubic structure with a lattice constant a, the electron energy can be written as:
)coscos(cos2)( akakak ++−−= zyxγαε k ,
where , and ρm is the position of the nearest
neighbor. width of the band? Calculate the inverse effective mass tensor near the zone center.
5. Please plot the electron density as a function of 1000/T for a silicon sample with 1016 cm-3 n-type
of the
6.The dimensionality of a system can be reduced by confining the electrons in
αα −=−−= ∫∫ )()(,)()( ** rρrrr smsss HdVHdV
5 doping atoms. Please estimate the temperature for the electron density of the system becoming saturated from intrinsic, and from saturated to frozen-out regime. Assume the ionization energy donor is 5 meV. 5 certain directions. Consider an electron gas in an exterial potential : V = 0 for
2 dz < and V = V for 0 2
dz > . What is the density of states as a function of
e for ?∞→V (Disuss what happens at low and high energies.) Assume nergy
d = 100 . Up to what temperatures can we consider the electrons to be
of
7.(a)Please prove that the density of orbital of a free electron in two dimensions is
0
o
Α
two-dimensional? If we can produce a potential of 100 me V and reach a temperature of 20 mK, what is the range of thickness feasible for the study such a two –dimensional electron gas? 5
independent of energy: ( ) 2hπ mEg = , per unit of area of specimen. (b)Show
that the chemical potential of a F o dimensions is given by ermi gas in tw
( ) −= 1exp
2nnTkT hl πμ , for n electrons per unit area. TmkB B
8.Given M = for the mass of the Sun, estimate the number of electrons in
ons f an
5 2× g3310 the Sun. In a white dwarf star this number of electrons may be ionized and contained in a sphere of radius 2 910× cm; find the Fermi energy of the electr in electrons volts.(b)The energy o electron in the relativistic limit 2mc>>ε is related to the wavevector as kcpc h=≅ε .Show that the Fermi energy in this
limit is ( ) 3 1
/VNcF h≈ε , roughly. (c) If the above number of electrons were
contained within a pulsar of radius 10 km, show that the Fermi energy would be eV. This value explains why pulsars are believed to be composed largely 810≈ of neutrons rayher than of protons and electrons, for the energy release in the reaction is only 0.8 eV,which is not large enough to enable −+→ epn 610×
many electrons to form a Fermi sea. The neutron decay proceeds only until the electron concentration builds up enough to create a Fermi level of eV, at 6108.0 × which point the neutron,proton, and electron concentrations are in equilibrium. 59.Consider the free electron energy bands of an fcc crystal lattice in the approximation of an empty lattice, but in the reduced zone scheme in which all ,k
r s are transformed to lie in the first Brillouin zone. Plot roughly in the [1 1 1]
direction the energies of all bands up to six times the lowest band energy at the
zone boundary at ( )( 2 1
2 1
r .Let this be the unit of energy. This problem
shows why band edges need not necessarily be at the zone center. Several of the degeneracies (band crossings) will be removed when account is taken of the crystal potential.
60.(a)Show that for the diamond structure the Fourier component of the GU r
crystal potential seen by an electro is equal to zero for AG rr
2= ,where A r
is a basis vector in the reciprocal lattice referred to the conventional cubic cell.(b) Show that in
the usual first-order approximation to the solutions of the wave equation in a periodic lattice the energy gap vanishes at the zone boundary plane normal to the end of the vector A
r .
61.Consider a square lattice in two dimensions with the crystal potential ( ) ( ) ( )a
y a
xUyxU ππ 22 coscos4, −= . Apply the central equation to find approximately the energy gap at the corner point ( )aa
ππ , of the Brillouin zone It will suffice to solve a determinatal equation. 22× 62.For the drift velocity theory, show that the static current density can be written in
matrix form as ( ) ( )
. In the high
magnetic field limit of 1>>τωc , show that xyyx Bne σσ −== . In this limit
0=xxσ , to order τω c
1 . The quantity yxσ is called the Hall conductivity.
63.Show that the conductivity at frequency ω is ( ) ( ) ( )



+ +
0 = .
64.A metal with a concentration n of free electrons of charge –e is in a static magnetic field zBˆ . The electric current density in the xy plane is related to the
electric field by yxyxxxx EEj σσ += ; yyyxyxy EEj σσ += . Assume that the
frequency cωω >> and τω 1>> , where meBc ≡ω andτ is the collision time. (a)Solve the drift velocity equation to find the components of the
magnetoconductivity tensor: ; πωωσσ 4/2 pyyxx i==
, where . (b)Note from a Maxwell 22 4/ πωωωσσ pcyxxy =−= mnep /4 22 πω ≡
equation that the dielectric function tensor of the medium is related to the
conductivity tensor as σωπε rrr )/4(1 i+= . Consider an electromagnetic wave
with wavevector zkk ˆ= r
. Show that the dispersion relation for this wave in the
medium is . At a given frequency there two modes of ωωωωω /22222 pcpkc ±−=
propagation with different wavevectors and different velocities. The two modes correspond to circularly polarized waves. Because a linearly polarized wave can be decomposed into two circularly polarized waves, it follows that the plane of polarization of a linearly polarized wave will be rotate by the magnetic field. 65.Assuming concentrations n, p; relaxation time eτ , hτ ; and masses m ,m , show e h
that the Hall coefficient in the drift velocity approximation is ( )2
21 nbp nbp
where h
eb μ μ= is the mobility ratio. In the derivation neglect terms of order
.(Hint:In the presence of longitudinal electric field, find the transverse 2Β electric field such that the transverse current vanishes.)
66.Consider the energy surface ( )
222 2 , where is the tm
transverse mass parameter and is the longitudinal mass parameter. A surface lm
on which ( )kE r
is constant will be a spheroid. Use the equation of motion
Bve dt kd rr r
h ×−= with Ev k rh
r ∇= −1 , to show that ( ) 2
1
=ω when the
static magnetic field B lies in the xy plane. 67.Consider a conductor with a concentration n of electrons of effective mass and relaxation timeem eτ ; and a concentration p of holes of effective mass hm and relaxation hτ . Treat the limit of very strong magnetic fields, 1>>τωc . (a)
Show in this limit that ( ) Bepnyx /−=σ . (b) Show that the Hall field is given
by, with τωcQ ≡ , ( ) x he
y E Q p
( )




+−+


+=
B eσ . If n = p, . If n2−∝ Bσ ≠ p,σ
saturates in strong fields; that is, it approaches a limit independent of B as B → . ∞
68. Explain why the projection of the trace of the motion on the x-y plane for an electron moving in a uniform magnetic field B in the z direction has the same shape as the trace in the k-space but with 90-degree rotation? What is the ratio of the area between the closed shape in real space and that in k-space?
69. Explain what is (a) zone folding (b) Bloch oscillation (c) type-II band alignment (d) modulation
doping (e) reduced zone scheme (f) exchange energy (g) self-consistent band calculation (h) local density approximation (i) crystal momentum.
70. Ionization of donors. In a particular semiconductor there are 1013 donors/cm3 with an ionization
energy Ed of 1meV and an effective mass 0.01m. (a) Estimate the concentration of conduction electrons at 4 K. (b) What is the value of the Hall coefficient? Assume no acceptor atoms are present and that Eg>>kBT.
71. Impurity orbits. Indium antimonide has Eg=0.23 eV; dielectric constant ε=18; electron effective
mass me=0.015m. Calculate (a) the donor ionization energy; (b) the radius of the ground state orbit. (c) At what minimum donor concentration will appreciable overlap effects between the orbits of adjacent impurity atoms occur? This overlap tends to produce an impurity band- a band of energy levels which permit conductivity presumably by a hopping mechanism in which electrons move from one impurity site to a neighboring ionized impurity site.
72. (a) Calculate and plot the dielectric constant as a function of ω for a neutral plasma with free electron density n and an uniform fixed positive charged background. Assume the dielectric constant
for the background is 1. The mass of the electron is m. (b) Derive the dispersion relation of an electromagnetic wave in the plasma described above. 73. Explain why the alkali metals become transparent in the ultraviolet region. 74. Explain “Mott metal-insulator transition”. 75.(a) Use less than 10 words to define“phonon.
(b) What is the“residual resistance.
(c) Explain the difference between the “perfect conductorand “superconductor.
(d) Describe the condition in which Van Hove singularities occur.
(e) Briefly describe“Bloch Theorem.
(f) What are“Landau levels?
(g) Describe the“normal processand “Umklapp process.
x
y
primitive translation vectors primitive cell basis
structure factor 77. (a) Write down the coupled equations of E and P for a polariton in a simple cubic crystal, where P is the volume density of the dipole moment induced by the electric field E. Assume the effective charge of the dipole moment in the crystal is q* and the relative displacement between the positive ion and the negative ion is u. The reduced mass of the oscillating ions is M. The high and low frequency limits of the dielectric constant are ( )∞ε and ( )0ε , respectively. The TO phonon of the crystal is
Tω .
2 /*)()0( T
L , where Lω is the zero of )(ωε .
(d) Show that . This is the so-called LST relation. )()0( 22 ∞= εωεω LT
(e) Plot the phase velocity versus ω and show that no electromagnetic wave with frequency between
Tω and Lω can propagate in the crystal. (f) Plot qualitatively the dispersion relation near the coupling region. 78. A 2-D single atom basis square lattice film is constructed with lattice constant
with orientation such that its fundamental translation vectors are o
Aa 2= iaa ˆ 1 = r
and jaa ˆ=2 r , where and unit vectors along positive x and y directions i j
respectively. An X-ray of wavelength 6.1=λ is incident upon the film from o
A
a direction of ),( θ which is defined identical to the conventional spherical coordiate. (a) If is fixed at , find the value of 040 θ that will result in Bragg reflection. In terms of angle find also directions of the reflected wave. (b) For all possible values of incident angle ]2,0[ π ∈ find the smallest value for θ that could produce Bragg reflection. 79.A weak periodic potential in the form of )2cos()( 0 xkVxV F= is created for a 1-D electron system ( is the Fermi wavenumber). CalculateFk ( )kE for the lowest energy band. 80.Show that as applied tight-binding method to 1-D atomic chain, if only nearest neighbor interaction is taken into account, the electronic s-band can be written as , ( ) )(sin4 2
min akJEkE π+= where gives the lowest energy value of the band, is the overlap integral. minE J Find also the band width, the electronic effective mass near the bottom and the top of the band. 81.Consider electrons on a 2-D square lattice in the tight-binding approximation,
. (a) With one electron per site in this crystal,draw )]cos()cos(2[0 yx kakaEE −=
the Fermi surface. Is this a metal or an insulator? (b) With two electrons per site, draw the Fermi surface. Is this a metal or an insulator? 82.If the magnetic field is applied in the z-direction, the cyclotron effective mass is defined as
2 1
M m
Calculate the cyclotron frequency for electron at the Fermi surface in a nearly
empty tight-binding band, described by
, )]cos()cos()cos([ 321 zyx kcEkbEkaEE ++−=
And show that the result indeed corrrsponds to the cyclotron frequency of free electrons of mass . *m 83.Calculate the leading term in the temperature dependence of the chemical potential for one-, two-, and three-dimensional free electron gases
**Sommerfeld Expansion:
84. Sommerfeld expansion a. chemical potential μ T(T<<TF) b. [hint:U(T)]
85.At temperature , only electrons within the range of near the Fermi KT 0≠ TkB
surface get excited to higher energy levels and each excited electron has kinetic energy of . Based on this model, calculate the average contribution to the TkB)2/3( heat capacity of each electron.
86. Explain what “Peierls instability” is. 87. Proof that for a point charge q in a metal with electron density n0, the effective potential due to this charge becomes
( ) ( )rk r
,
where ks is a function of n0. Please also derive the form of ks. 88.For a 1-D monatomic chain of N identical atoms of mass m, show that the number of vibration mode per unit frequency is
2 1
π ω
m Ng ,
where β is the restoring force conatant. 89.Providing that the Fermi energy for sodium is . If the conductivity at eVEF 2.30 = zero temperature is measured as , find the electronic 117 )(101.2 −⋅Ω× cm
relaxation time τ for sodium at zero temperature. 90.It is known that the equi-energy surfaces for silicon conduction band are ellipsoids with 6 minima in the conduction band edge. If the electronic effective mass along the long and short axis of the ellipsoid are and respectively, show that if the lm tm
electronic mobility can be expressed as then the “conductivity */ cmeτμ =
effective mass is given by *cm


+=
* .
91.Surface subbands in electric quantum limit. Consider the contact plane between an insulator and a semiconductor, as in a metal-oxide-semiconductor transistor or MOSFET. With a strong electric field applied across the - interface the poten- 2SiO Si tial energy of a conduction electron may be approximated by eExxV =)( for x positive
and by for∞=)(xV x negative, where the origin of x is at the interface. The wave-
function is 0 for x negative and may be separated as )(),,( xuzyx =ψ exp , )]([ zkyki zy +
where satisfies the differential equation )(xu uuxVdxudm =∈+− )(/)2/( 222h
With the model potential for the exact eigenfunctions are Airy functions, but we can find a fairly good ground state energy from the variational trial function
)(xV
x exp . (a) Show that)( ax− ( ) aeEam 2/32/ 22 +=∈ h . (b) Show that the energy is a minimum when
3 1)2/3( 2heEma = . (c) Show that 3
2 3
eEmh=∈ . In the exact solution for the
ground state energy the factor 2.26 is replaced by 1.78. As E is increased the extent of the wavefunction in the x direction is decreased. The
function defines a surface conduction channel on the semiconductor side of the interface. The various eigenvaluse of define what are called electric subbands. Besause the eigenfunctions are real functions of
)(xu )(xu
x the state do not carry current in the x direction, but they do carry a surface channel current in the plane. zy,
The dependence of the channel on the electric field E in the x direction makes the device
a field effect transistor.
92. Plasma frequency and electrical conductivity. An organic conductor has recently been found by
optical studies to have s15108.1 ×=pω -1 for the plasma frequency, and τ=2.83×10-15 s for the
electron relaxation time at room temperature. (a) Calculate the electrical conductivity from these data. The carrier mass is not known and is not needed here. Take 1)( =∞ε . Convert the result to units (Ω cm)-1. (b) From the crystal and chemical structure, the conduction electron concentration is
4.7×1021 cm-3. Calculate the electron effective mass m*. 93. Derive the static Thomas-Fermi dielectric function ε(0,K) for a metal with conduction electron
density n0. 94. Causality and the response function. The Kramers-Kronig (KK) relations are consistent with the principle that an effect cannot precede its cause. Consider a delta-function force applied at time t=0:
ω π
−== 2 1)()( ,
whence πω 2/1=F . (a) Show by direct integration or by use of the KK relations that the oscillator response function
( ) 122 0)( −
−−= ωρωωωα i
gives zero displacement, under the above force. For t < 0 the contour integral may be completed by a semicircle in the upper half-plane. (b) Evaluate x(t) for t > 0. Note that α(ω) has
poles at
95. Dissipation sum rule. By comparison of α'(ω) from
( )∑ ∑ +−
+− =
−− =
ssP ω
α π
ωα in the limit ∞→ω , show that the following sum rule for the oscillator
strengths must hold:
.
96. Reflection at normal incidence. Consider an electromagnetic wave in vacuum, with field components of the form
)()inc( tkxi y AeE ω−= .
Let the wave be incident upon a medium of dielectric constant and permeability the same as vacuum that fills the half-space x > 0. Show that the reflectivity coefficient r(ω) as defined by E(refl)=r(ω)E(inc) is given by
1 1)(
= iKn iKnr ω .
Where , with n and K real. Show further that the reflectance is 2/1ε=+ iKn
22
22
97. Hagen-Rubens relation for infrared reflectivity of metals. The complex refractive index n+iK of a metal for 1<<ωτ is given by
0
02 1)( ωε σε iiKn +=+≡ ,
where σ0 is the conductivity for static fields. We assume here that intraband currents are dominant; interband transitions are neglected. Using the results of last problem for the reflection coefficient at normal incidence, show that
2/1
0
081
σ .
98. Explain “Kramer-Kronig relations”. Why are they useful in analyzing optical measurement data? 99.Surface plasmons. Consider a semi-infinite plasma on the positive side of the plane . A solution of Laplace , s equation in the plasma is
, whence ; .
zi ekxkAE −= cos kz xi ekxkAE −= sin
(a) Show that in the vacuum forkzekxAzx cos),(0 = 0<z satisfies the boundary
condition that the tangential component ofΕ be continuous at the boundary; that is find (b) Note that 0xE ;)( ii ED ω=∈ 00 ED = . Show that the boundary condition that the
normal component of D be continuous at the boundary requires that 1)( −=∈ ω , whence from (10) we have the Stern-Ferrell result:
2 2 12
ps ωω =
for the frequency sω of a surface plasma oscillation. 100.Interface plasmons. We consider the plane interface 0=z between metal 1 at
and metal 2 at . Metal 1has bulk plasmom frequency 0>z 0<z 1pω ; metal 2 has
2pω . The dielectric constants in both metals are those of free electron gases. Show
that surface plasmons associated with the interface have the frequency
2 1)]([ 2
2 2 12
1 pp ωωω += .