The Oxford Solid State Basics, Solution Manual

199 Pages·2015·2.436 MB·English

by  Steven H. Simon

The Oxford Solid State Basics
Solutions to Exercises
Steven H. Simon
Oxford University
.
CLARENDON PRESS OXFORD
2015

iii
These are the solutions to exercises from the Book The Oxford Solid
StateBasics byStevenH.Simon,publishedbyOxfordUniversityPress,
2013 edition. Please do everyone a favor and do not circulate these
solutions. Donotpostthesesolutionsonyourwebsite. Donotputthem
onRussianwebsites. Donotcopythemandhandthemouttostudents.
Whilethereisnowayformetoenforcethesereasonablerules,beassured
thatI,beingaprofessoratHogwarts,aminpossessionofpowerfulhexes
which I have used to protect the secrecy of these solutions. Those who
attemptto circulatethesesolutionsunlawfully willactivatethe hex and
will suffer thirty years of bad luck, including spiders crawling into your
underwear.
Some of these solutions have been tested through use in severalyears
of courses. Other solutions have not been completely tested. Errors
or ambiguities that are discovered in the exercises will be listed on my
web page. If you think you have found errors in the problems or the
solutions please do let me know,and I will make sure to fix them in the
next version. Doing so will undoubtedly improve your Karma. ,
Steven H Simon
Oxford, United Kingdom
January 2014

Contents
1 About Condensed Matter Physics 1
2 SpecificHeatofSolids: Boltzmann, Einstein,and Debye 3
3 Electrons in Metals: Drude Theory 15
4 More Electrons in Metals: Sommerfeld (Free Electron)
Theory 21
5 The Periodic Table 35
6 What Holds Solids Together: Chemical Bonding 39
7 Types of Matter 47
8 One-Dimensional Model of Compressibility, Sound, and
Thermal Expansion 49
9 Vibrations of a One-Dimensional Monatomic Chain 55
10 Vibrations of a One-Dimensional Diatomic Chain 71
11 Tight Binding Chain (Interlude and Preview) 81
12 Crystal Structure 95
13 Reciprocal Lattice, Brillouin Zone, Waves in Crystals 99
14 Wave Scattering by Crystals 111
15 Electrons in a Periodic Potential 125
16 Insulator, Semiconductor, or Metal 135
17 Semiconductor Physics 139
18 Semiconductor Devices 149
19 Magnetic Properties of Atoms: Para- and
Dia-Magnetism 159
vi Contents
20 Spontaneous Magnetic Order: Ferro-, Antiferro-, and
Ferri-Magnetism 167
21 Domains and Hysteresis 175
22 Mean Field Theory 179
23 Magnetism from Interactions: The Hubbard Model 191
About Condensed Matter
1
Physics
There are no exercises for chapter 1.

Specific Heat of Solids:
Boltzmann, Einstein, and
2
Debye
(2.1) Einstein Solid ity should be 3Nk = 3R, in agreement with the law of
B
(a) Classical Einstein (or “Boltzmann”) Solid: Dulong and Petit.
Consider a three dimensional simple harmonic oscilla-
(b) Quantum Einstein Solid:
tor with mass m and spring constant k (i.e., the mass
NowconsiderthesameHamiltonianquantummechan-
is attracted to the origin with the same spring constant
ically.
in all three directions). The Hamiltonian is given in the
(cid:3) Calculate thequantum partition function
usual way by
p2 k
H = + x2 Z = e−βEj
2m 2
(cid:3) Calculate theclassical partition function Xj
dp
Z = dxe−βH(p,x) where thesum overj is a sum over all eigenstates.
(2π~)3 (cid:3) Explain therelationship with Bose statistics.
Z Z
Note: inthisproblempandxarethreedimensionalvec- (cid:3) Find an expression for the heat capacity.
tors. (cid:3) Show that the high temperature limit agrees with
(cid:3) Usingthepartitionfunction,calculatetheheatca- thelaw of Dulong of Petit.
pacity 3k . (cid:3) Sketchtheheat capacity as a function of tempera-
B
(cid:3) Concludethatifyoucanconsiderasolidtoconsist ture.
of N atoms all in harmonic wells, then the heat capac- (Seealso exercise 2.7 for more on thesame topic)
(a)
p2 k
H = + x2
2m 2
dp
Z = dxe−βH(p,x)
(2π~)3
Z Z
Since,
∞
dye−ay2 = π/a
Z−∞
p
in three dimensions, we get
3
Z = 1/(2π~) π/(β/2m) π/(βk/2)) =(~ωβ)−3
h p p i
with ω = k/m. From the partition function
p U = (1/Z)∂Z/∂β =3/β =3k T
B
−
4 Specific Heat of Solids: Boltzmann, Einstein, and Debye
Thus the heat capacity ∂U/∂T is 3k .
B
(b) Quantum mechanically, for a 1d harmonic oscillator, we have
eigenenergies
E =~ω(n+1/2)
n
with ω = k/m. The partition function is then
p Z = e−β~ω(n+1/2)
1d
n≥0
X
= e−β~ω/21/(1 e−β~ω)
−
= 1/[2sinh(β~ω/2)]
The expectation of energy is then
E = (1/Z)∂Z/∂β =(~ω/2)coth(β~ω/2)
1 h i − 1
= ~ω(n (β~ω)+ )
B
2
0.75
where n is the boson occupation factor
B
0.5 n (x)=1/(ex 1)
B
−
(hence again the relationship with free bosons). The high temperature
0.25
limitgivesn (x) 1/(x+x2/2)=1/x 1/2sothat E k T. More
B B
→ − h i→
generally, we obtain
0
0 1 2
eβ~ω
C =k (β~ω)2
Fig. 2.1Heatcapacity intheEinstein B (eβ~ω 1)2
−
model (per atom) in one dimension.
UnitsarekbonverticalaxisandkbT/ω In 3D,
onhorizontal. Inthreedimensions,the
heat capacity per atom is three times E =~ω[(n +1/2)+(n +1/2)+(n +1/2)]
aslarge. n1,n2,n3 1 2 3
and
Z3d = e−βEn1,n2,n3 =[Z1d]3
n1,n2,n3≥0
X
and correspondingly
1
E =3~ω(n (β~ω)+ )
B
h i 2
So the high temperature limit is E 3k T and the heat capacity
B
h i →
C =∂ E /∂T =3k . More generally we obtain
B
h i
eβ~ω
C =3k (β~ω)2
B (eβ~ω 1)2
−
Plotted this looks like Fig. 2.1.