ZHEJIANG UNIVERSITY Department of...
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ZHEJIANG UNIVERSITY Department of Physics Issued: 2014/04/17 Solid State Physics I Problem Set #6 Due: 2014/05/27 1. Bloch oscillation. Consider a one-dimensional energy band with the dispersion relation ε(k)= ε 0 + ε 1 cos 2 ka 2 (1) (a) Find a general expression for the effective mass in this band and evaluate it at k =0 and k = π/a. (b) Find an expression for the density of states (number of states/volume having energy between ε and ε + dε) for this band and evaluate it at k = π/a. (c) A dc electric field of magnitude E is applied in the positive k-direction. Cal- culate the dependence of k on time. Calculate the dependence of electrical current on time for a group of electrons of density n, having k =0 at t =0, and initial velocity v(0) at t =0. (d) What is the oscillation frequency of the current for E = 10, 000 Volts/cm and a =5 ˚ A? What would happen to the frequency of oscillation if a superlattice were made of this material with a period of 5a? Explain clearly with a sketch to illustrate your arguments. 2. Anisotropic DC conductivity. Silicon crystallizes in the diamond structure which has cubic symmetry. The lowest conduction band is at a Δ-point along ΓX of the fcc Brillouin zone. Six ellipsoidal (of revolution) constant energy surfaces are formed (A/M, Fig. 28.5). Take the constant energy surfaces to have the form: ε( k)= ¯ h 2 κ 2 l 2m l + ¯ h 2 (κ 2 t 1 + κ 2 t 2 ) 2m t (2) where m l is the effective component along the ΓX direction and m t is the effec- tive mass component perpendicular to this direction. Assume m l /m 0 =1.0 and m t /m 0 =0.2. (a) Assuming for the moment that there is only the single constant energy surface centered along [100]-direction (rather than 6 that are present in the reality), find the contribution to the electrical conductivity from this carrier pocket. [Hint: You need to generalize the DC conductivity to the anisotropic case. See A/M Problem 12.3.] 1
Transcript of ZHEJIANG UNIVERSITY Department of...
ZHEJIANG UNIVERSITYDepartment of Physics
Issued: 2014/04/17Solid State Physics I Problem Set #6 Due: 2014/05/27
1. Bloch oscillation. Consider a one-dimensional energy band with the dispersionrelation
ε(k) = ε0 + ε1 cos2
(
ka
2
)
(1)
(a) Find a general expression for the effective mass in this band and evaluate it atk = 0 andk = π/a.
(b) Find an expression for the density of states (number of states/volume havingenergy betweenε andε+ dε) for this band and evaluate it atk = π/a.
(c) A dc electric field of magnitudeE is applied in the positivek-direction. Cal-culate the dependence ofk on time. Calculate the dependence of electricalcurrent on time for a group of electrons of densityn, havingk = 0 at t = 0,and initial velocityv(0) at t = 0.
(d) What is the oscillation frequency of the current forE = 10, 000 Volts/cm anda = 5 A? What would happen to the frequency of oscillation if a superlatticewere made of this material with a period of5a? Explain clearly with a sketchto illustrate your arguments.
2. Anisotropic DC conductivity. Silicon crystallizes in the diamond structure whichhas cubic symmetry. The lowest conduction band is at a∆-point alongΓX of the fccBrillouin zone. Six ellipsoidal (of revolution) constant energy surfaces are formed(A/M, Fig. 28.5). Take the constant energy surfaces to have the form:
ε(~k) =h2κ2
l
2ml+
h2(κ2
t1+ κ2
t2)
2mt(2)
whereml is the effective component along theΓX direction andmt is the effec-tive mass component perpendicular to this direction. Assume ml/m0 = 1.0 andmt/m0 = 0.2.
(a) Assuming for the moment that there is only the single constant energy surfacecentered along [100]-direction (rather than 6 that are present in the reality),find the contribution to the electrical conductivity from this carrier pocket.[Hint: You need to generalize the DC conductivity to the anisotropic case.See A/M Problem 12.3.]
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(b) What is the direction of the current flow for the followingcases:
(i) The electric field~E ‖ [100]
(ii) The electric field~E ‖ [001]
(iii) The electric field~E ‖ [111]
(c) Find the contribution to the electrical conductivity from all 6 electron pocketsfor the same electric field directions as in part (b) and comment.
3. Cyclotron resonance for a spheroidal energy surface.
(a) [Kittel 8.4] Consider the energy surface
ε(~k) = h2
(
k2
x + k2
y
2mt+
k2
z
2ml
)
, (3)
wheremt is the transverse mass parameter andml is the longitudinal massparameter. A surface on whichε(~k) is constant will be a spheroid (or ellipsoidof revolution). Use the equation of motion
hd~k
dt= −
e
c~v × ~H (4)
with ~v = h−1∇kε, to show thatωc = eH/[(mtml)1/2c], when the static mag-
netic fieldH lies in thexy plane. This result agrees with
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m2c
=cos2 θ
m2t
+sin2 θ
mtml(5)
whenθ = π/2. The result is in CGS unit.
(b) Ashcroft / Mermin Chapter 28 Problem 2 (a) and (b).
4. Ashcroft / Mermin Chapter 12 Problem 4
5. Hall effect in compensated silicon.
(a) (Kittel 8.3) Assuming concentrationsn, p; relaxation timeτe, τh; and massesme, mh; show that the weak field limit (ωcτ ≪ 1), the Hall coefficient in thedrift velocity approximation is
RH =1
ec·p− nb2
(p+ nb)2(6)
in CGS unit, whereb = µe/µh is the mobility ratio.
(b) A silicon sample of resistivity10−3 Ω cm has zero Hall voltage at small mag-netic field strengths. Assumeµn = 1300 cm2/Vs andµp = 300 cm2/Vs anddetermine the carrier concentrations.
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6. Ashcroft / Mermin Chapter 12 Problem 6
7. Ashcroft / Mermin Chapter 13 Problem 2
8. De Haas-van Alphen period of ptassium
(a) Calculate the period(1/B) expected for ptassium on the free elctron model.
(b) What is the area in real space of the extremal orbit forB = 10 kG?
9. Open orbits
An open orbit in a monovalent tetragonal metal connects opposite faces on theboundaries of a BZ. The faces are separated byG = 2 × 108 cm−1. A magneticfield B = 103 gauss is normal to the plane of the orbit.
(a) What is the order of magnitude of the period of the motion in k-space? Takev ∼ 108 cm/s.
(b) Describe in real space the motion of an electron on this orbit in the presenceof a magnetic field.
10. Ashcroft / Mermin Chapter 14 Problem 1
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