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MSCI406

HOMEWORK 05

8-1:   Impurity orbits.   Indium antimonide has   E g   = 0.23eV ; dielectric constant     = 18;electron effective mass  me = 0.015m. Calculate (a) the donor ionization energy; (b) theradius of the ground state orbit. (c) At what minimum donor concentration will appre-

ciable overlap effects between the orbits of adjacent impurity atoms occur? This overlaptends to produce an impurity band-a band of energy levels which permit conductivitypresumably by a hopping mechanism in which electrons move from one impurity siteto a neighboring ionized impurity site.

a.   E d = 13.6eV  ×  m∗

m ×   1

2  6.3× 10−4eV 

b.   r =  aH  × ×   mm∗  6× 10−6cmc. Overlap will be significant at a concentration  N  =   14π

3  r3 ≈ 1015atoms/cm3.

8-2:   Ionization of donors.   In a particular semiconductor there are  1013donors/cm3 with an

ioniation energy E d  of  1meV  and an effective mass  0.01m. (a) Estimate the concentrationof conduction electrons at  4K . (b) What is the value of the Hall coefficient? Aussumeno acceptor atoms are present and that  E g  kBT .

a. From Eq. (53),   n    (n0N d)1/2e−E d/2kBT , in an approximation not too good for the presentexample,   n0  ≡   2

m∗kBT 2π

 

  2

3/2 ≈   4 ×  1013cm−3; and   E d2kBT 

    1.45,   e−1.45   0.23, therefore,   n  0.46× 1013electrons/cm3.

b.   RH  = −   1nec  −1.3× 10−14 in CGS units.

9-1:  Brillouin zones of rectangular lattice.   Make a plot of the first two Brillouin zones of a

primitive rectangular two-dimensional lattice with axes  a,  b = 3a.

9-2:   Brillouin zone, rectangular lattice.  A two-dimensional metal has one atom of valencyone in a simple rectangular primitive cell   a = 2Å;   b = 4Å. (a) Draw the frist Brillouinzone. Give its dimensions, in  cm−1. (b) Calculate the radius of the free electron Fermisphere, in  cm−1. (c) Draw this sphere to scale on a drawing of the first Brillouin zone.

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Make another sketch to show the first few periods of the free electron band in theperiodic zone scheme, for both the first and secound energy bands. Assume there is asmall energy gap at the zone boundary.

a.   πb   = 0.78× 108cm−1 and   πa   = 1.57× 108cm−1

b.

N   = 2× πk2

F 

(2π/L)2

n   =   N/L2 = k2F /2π = 1

8 × 1016els/cm2

kF    =√ 

2πn  = 0.8× 108cm−1

c.

9-4:  Brillouin zones of two-dimensional divalent metal.   A two-dimensional metal in the formof a square lattice has two conduction electrons per atom. In the almost free electronapproximation, sketch carefully the electron and hole energy surfaces. For the electronschoose a zone scheme such that the Fermi surface is shown as closed.

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Since the diagonal =   π√ 2

a  . Then, 2πk2F 

a2

4π2  = 2 →   kf  =   2

√ π

a  , so that,   π

a  < kF   <

  π√ 2

2  .

9-7:   De Haas-van Alphen period of potassium.   (a) Calculate the period   ∆(1/B)   expectedfor potassium on the free electron model. (b) What is the area in real space of theextremal orbit, for   B   = 10kG   = 1T ? The same period applies to oscillations in theelectrical resistivity, known as the Shubnikow-de Haas effect.

a. ∆   1H 

 =  2πe

 

cS , where  S  = πk2

F , with  kF   = 0.75× 108

cm−1

from Table 6.1, for potassium. Thus,∆

 1

H 

  2

137k2F e  0.55× 10−8G−1.

b.   ωcR =  vF , therefore,  R =  V F mceB

  = 

  kF ceB  0.5× 10−3cm, and πR2 0.7× 10−6cm2.