1

PHYS 745: Solid State Physics

Syllabus, viewgraph sets, and other course related materials:

http://academic.brooklyn.cuny.edu/physics/tung/GC745S15

BUSINESS ITEMS

Self Introductions

Lecture Plan

Textbooks, homework, exams, grades, rules of the classroom, etc.

About Your Instructor

• B.S. National Taiwan University (physics) ; Ph.D. University of Pennsylvania (physics)

• Worked in basic research for 21 years at Bell Labs (AT&T, Lucent, Agere,...)

• Joined Brooklyn College in 2002.

RESEARCH AREAS

Electronic material processing and properties.

Interface electronic properties.

rtung@brooklyn.cuny.edu718-951-5807Office Hours: (GC) Thursday before lecture

(BC) Tuesday 10:30-12:30

2

Solid State Physics (Condensed Matter Physics)

A very wide range of topics, ranging from crystallography, chemistry, quantum mechanics, thermodynamics, low-temp physics, materials science, fabrication science, engineering science, etc., ranging in nature from fundamental phenomena to practical applications. This field is still growing.

What is solid state physics?

If we have to summarize, condensed matter physics is a study of the behavior and properties of electrons in condensed materials.

Course On Solid State Physics?

What specific topics should a graduate-level introductory course on solid state physics cover?

What kind of background knowledge is most useful for research scientists in various fields of condensed matter physics?

Fundamental principles and general techniques for problems in condensed matter physics.

Basic principles and in-depth knowledge in specific areas of current interest.

… and a general knowledge of some of the important CURRENT happenings in condensed matter physics. An emphasis on new ideas, emerging areas, and practical stuff.

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Overall Plan

BASIC CONCEPTS (Follow Ashcroft and Mermin)

Crystal lattice, reciprocal lattice, x-ray diffraction, symmetry groups, structure determination, Drude theory of metals, Sommerfeld theory of metals, electron levels in periodic potential, nearly free electrons, tight binding method, semiclassical model of electron dynamics, interacting electrons, surface effect, classical and quantum theories of the harmonic crystal, dielectric properties of insulators, semiconductors, p-n junction, etc.

In addition, overviews on several topics of current and practical interest in condensed-matter and materials physics will be selected, based on students’ interest, and discussed, toward the end of the semester. Possibilities include hetero-junctions, quantum wells; epitaxial growth, nanofabrication; mesoscopic transport; ULSI devices, processing; ferroelectrics, non-volatile memories; spintronics, magnetic materials; solar cells; solid state ionics, fuel cells, etc.

Student input needed

Tentative Lecture Schedule

1/29, Drude Theory of Metal 2/5, Sommerfeld Theory

2/12, no class 2/19, Crystal Lattice & Reciprocal Lattice

2/26, X-ray & Crystal Classif. 3/5, Bloch Thrm & Weak Periodic Pot.

3/12, Tight Binding & Other Methods

3/19, Semiclassical Electron Dynamics

3/26, Conduction in Metals 4/2, Electron Interaction, HF, DFT

4/9, no class 4/16, Harmonic Crystal

4/23, Semiconductors 4/30, Inhomo. Semiconductors

5/7, Special Topic 5/14, Special Topic

4

Homework, Exam, Grade, & Classroom Rules

• Grade will be based largely on homework. Homework needs to be turned in on time. Late submission will not be accepted, no exceptions. In case a student cannot come to class on the due date of a homework set, he (or she) should e-mail (or fax) me with his work no later than 10:00 of that day.

• Recitation classes may be held upon request from majority of students, in which case all are required to attend.

• Year-end final exam will be conducted, the main purpose of which is to verify that the students have done their homework on their own.

• Students are required to attend classes on time. They are also encouraged to ask questions and participate in discussions in class.

• Attendance will be taken and may be used to raise students’ grades in borderline cases.

Bohr radius

Chapter 1, Drude Theory of Metals

3

41 3sr

nN

V

3/1

4

3

nrs

AZn m /1002.6 23

esue 101080.4 )1300( Cesu

Nuclei are immobile. Some loosely bound electrons (Z per atom) are mobile.

massatomicA:

densitymassm :

cmema 82120 10529.0

typically, 62/ 0 ars

Early View Of Electrons In Matter:

5

Drude Model of Conduction in Metals

A few years after electron was discovered, Drude constructed theory on electrical and thermal conduction from kinetic theory of gases.

DRUDE’S ASSUMPTIONS1. Weakly bound valence electrons move freely and independently between collisions. 2. Electrons only collide with positive ions (instantaneous collisions, very short-range forces).3. Collisions between the electrons and the immobile positive ions is characterized by a relaxation time (collision time, mean free time), i.e. the probability of collision per unit time t is (t /. is independent of electron energy, position, velocity.4. Electrons achieve thermal equilibrium through collisions: immediately after each collision, an electron emerge with a speed (random direction) characteristic of the local prevailing temperature.

What are the probabilities that an electron will go time t without suffering a collision?

dtNdN }/exp{)0()( tNtN

DC Conductivity

Experimentally observed Ohm’s law

To estimate the average velocity of electrons (in a constant electric field), first figure out, on the average, how long has it been since any electron, picked at random, had its last collision.

jE

ALR /

venEj

mne /2

0

/)(

tedt

tm

Eevavg

sa

r

ne

m

ne

m s

cm

14

3

022

1022.0

IRV

1

6

mean free path

Relaxation-Time From DC Conductivity

0v

sa

r

ne

m s

cm

14

3

02

1022.0

Using measured metal resistivities, was estimated to be 10-14 to 10-15 sec at room temperature. Using classical equipartition of energy ½ mv2 = 3/2 kT, the mean free path was estimated to be 0.1 – 1nm, agreeing with interatomic spacing.

However, apparent agreement is fortuitous.

Drude model still useful for analysis of phenomena not directly governed by relaxation-time. Examples: static magnetic field and homogeneous ac E field.

Time-Dependent Field (Spatially Homogeneous)

)(tfp

dt

pd

m

pnej

p: average momentum

f: (average) external force

dttfdt

tpdt

tpdttp )(1)()()(

Momentum gain for electrons unscattered in dt. Contributions for scattered electrons: ~ (dt)2

akin to frictional damping

MAIN RESULT OF THE MODEL

7

From Drude’s model:

Magnetic field H in z direction, current (wire) in x direction. Accumulation of charge induces an electric field to offset the Lorentz force.

Hall Effect & Magnetoresistance

p

Hmc

pEe

dt

pd

)(

Hall CoefficientHj

ER

x

yH

Magnetoresistance

Cyclotron Frequency

cm

Hec

x

x

j

EH )(

c (109 Hz) = 2.80 * H (kilogauss)

Hall Effect & Magnetoresistance

xxc

y jnec

HjE

0

/0

/0

yxcy

xycx

ppeE

ppeE

Look for steady-state solution. No net current in y direction. Set py=0

cenRH

1

cx

y

E

Etan Hall Angle: angle

between E and j

both independent of H

The dimensionless product c is a measure of the effect of magnetic field on the electronic orbits in metals.

2)(

ne

mH

8

Closed Cyclotron Orbits In Crossed E Field

)(1 HvEc

e

dt

vd

yzyyxzxzyz

yyyyxyxyyy

xyyyxxxxyx

eEvvc

eHv

eEvvc

eHv

eEvvc

eHv

)(

)(

)(

Assume magnetic field to lie in the z-direction and the electric field to lie in the y-direction

Nearly free electrons:

0

z

yxy

yx

vm

eEv

cm

eHv

vcm

eHv

xy

yx

vmc

eHv

vmc

eHv

yy

yxx

vvH

cEvv

change of variables

Compare With Experimental Results

Actual experimental Hall coefficient: positive or negative; depends on magnetic field, temperature, sample preparation.

RH is constant only for pure sample at low T, high H.

cenRH

1

aluminum

Something’s wrong with Drude’s model!

9

AC Conductivity of a Metal

p

Hmc

pEe

dt

pd

)(

])(Re[)( tieptp

])(Re[)( tieEtE

ii

mne

11

/)( 0

2

220)1(

1

220)2(

1

applied field

m

pnej

)()()( Ej

Steady state solution: everything has e-it time dependence.

ac conductivity

)()(

)(

)()(

)(

2

Em

nejji

Eep

pi

but also

Electromagnetic Waves In Metal

Applicable to incident EM wave. Long-wavelength (>> MFP). Still ignore magnetic force (factor of v/c smaller).

t

H

cE

1 4 1j E

Hc c t

0 HE

Ei

cE

)

41(

2

22

dielectric conductive bound el. free el.

E

c

iE

cc

i

c

HiEE

4)(2

Steady state: everything has e-it time dependence.

Ec

E

)(2

22

i4

1)(

10

plasma frequency

Propagation Of EM Waves

when >> 1, is purely imaginary

i

1

)( 0

2

2

1)(

p

m

nep

22 4

i4

1)( 22

022

0

1

/4

1

41)(

i

cm

sp a

r

1

106.12/3

0

2

Origin of Complex AC Conductivity

Complex ac conductivity indicates a difference in phase between the current and the electric field.

Electric field accelerates electrons (between collisions). However, the current is related to the instantaneous velocities of the electrons (acceleration integrated in time).

Two extreme cases:1. -1 >>

2. -1 <<

11

Charge Density Oscillation

2

2

1)(

p

04

1)( i

)()(4)( i

)()( ij

tj

p

)(4)( E

With a dielectric constant like this, can a metal support oscillations in the charge density?

There is a solution if

)(4 i or,

Charge density can oscillate at the plasma frequency!

eVarsp 1.47/ 2/30

Plasma Oscillation

Metal is transparent above plasma frequency, typically 1016 Hz. Light with wavelength shorter than ~ 30 nm can propagate inside metal.

nmarsp22/3

0 1026.0)/(

dNnedNm 24

m

nep

22 4 Metal

wf (eV)

Ag 9.01Au 9.03Cu 10.83Al 14.98Be 18.51Cr 10.75Ni 15.92Pd 9.72Pt 9.59Ti 7.29W 13.22

12

Thermal Conductivity of Metal

In Drude’s picture, the electrons arriving from the hotter end of the metal will have higher kinetic energy than electrons arriving from the cold side. After collision, electrons are assumed to have equilibrated local thermal energy, i.e. the velocity distribution of electron after the collision does not depend on the incoming velocity. The flow of heat can be estimated to be

Therefore,

from kinetic theory of gases

)(}])[(])[({2

1 2

dx

dT

dT

dnvvxTvxTnvjq

vv cvlcv3

1

3

1 2

Te

k

ne

mvcB

v 2

2

2

2

33

1

Tjq

TcL0 Wiedemann-Franz law, empirical.

cL a constant (Lorenz number).

2/3 Bv nkc Tkmv B232

21

Wiedemann-Franz law

Te

k

ne

mvcB

v 2

2

2

2

33

1

28 /1011.1 KwattT

fortuitous agreement! cv depends on T

Drude result:

13

Thermopower

thermoelectric field (Seebeck effect) TQE

2

2v

dx

d)]()([21 vxvvxvvQ in one-dimension

TdT

dvvQ

2

6

3-D

mEevE / Since and , we get0 EQ vv

ne

cmv

dT

d

eQ

323

1 2

Drude used cv = 3nkB/2 to obtain , which is two orders of magnitude larger than experiment.

Some material even show thermopower in direction opposite to that predicted by the Drude picture.

e

kQ B

2

Summary of Chapter 1

Drude’s model borrows concepts from kinetic theory of gases. The concept of relaxation time (collision with nuclei) leads to predictions in dc conductivity, ac conductivity, uniform magnetic field, thermal conductivity, etc.

Poor agreement with experimental results indicates problems/over-simplifications of the crude Drude picture of carrier movement. These problems, as will be discussed later, turn out to arise from

(1) Electrons obey Fermi-Dirac statistics: the concept of Fermi surface.(2) Electrons are not “free” between collisions: the effect of periodic lattice.(3) Electrons are not independent: screening, spin interaction.(4) Positive ions are not immobile: vibrations and phonons.(5) Crystals are imperfect: surfaces, interfaces, defects.

14

Free Electron Gas & Sommerfeld Theory

Maxwell-Boltzmann: number of electrons per volumewith velocity v

kinetic gas model of conduction electrons

Bv nkc 23Tnku B2

3

Pauli’s exclusion principle leads to Fermi-Dirac distribution.

Sommerfeld theory(1) Thermodynamic properties of free electron gas(2) Transport properties of free electron gas

)2/(

2/32

2)( Tkmv

BB

BeTk

mnvf

Chapter 2

nedyyneTk

mdvvnvfvd yTkmv

BB

B

0

2/1)2/(

2/3

0

2 22

4)(2

TnkedyyTnkeTk

mdvmvvnu B

yB

Tkmv

B

B

23

0

2/32/1)2/(

2/3

0

2212 2

2)(4

2

]/)exp[(1

1

2

)/()(

2213

3

TkCmv

mvf

BD

Periodic Boundary Condition (Born-von Karman)

particles in a box

Schrodinger eq.

PBC

solutions:

)()(2 2

2

2

2

2

22

rErzyxm

)()ˆ()ˆ()ˆ( rzLryLrxLr zyx

rki

keVr

2/1)(

provided k satisfies

(nx, ny, nz: integers),

2,

2,

2

z

zz

y

yy

x

xx L

nk

L

nk

L

nk

VLLL zyx

m

kk

2)(

22

ipop

kkp

)(

15

dependence

Summations and Density Of States (DOS)

kdV

kdkgkdkg

38)()(

Density of allowed states per volume, g, in k-space is homogeneous. For large V, replace summations with integrals.

spin

)(4

)(),()(3

,

kFkdV

kFskgkdkFsk

number of allowed states within a volume d3k

m

kk

2)(

22

Free electron density of states at energy :

0,0

0,2

)(222

mm

g

F is any quantity we wish to sum.

Frequently, quantities of interest depends only on the energy of the state 2/1

Construct zero-temperature ground state by filling levels with lowest energies, two to a particular k level. All states with k<kF (inside the Fermi surface) are occupied.

FERMI LEVEL

Fermi Surface and Fermi Level (T=0K)

nV

NkF

3

4

4

1 3

3

34

1)()(

kgkg

2

3

3Fk

n

3/1

4

3

nrs ss

F rrk

92.1)4/9( 3/1

scm

arm

kv

s

FF /10

/

20.4 8

0

eVRy 6.131

20

22

)/(

1.50

2 ar

eV

m

k

s

FF

RynaF3/22

03/43/23 FBF Tk

F

FF

nmkg

2

3)(

22

24

3)(

FF

ng

16

Properties of Electron Gas At T=0K

Total Energy

Fk

k

kFkdV

kF03

)(4

)(2

m

k

m

kdkk

V

E Fkk

k

F

2

5222

0

23 102

)4(4

1

FBFF Tk

m

k

N

E

5

3

5

3

10

3 22

Kar

Ts

F4

20

10)/(

2.58

Pressure

Bulk Modulus

FnV

EP

V

PVB

3

2

9

10

3

5

V

E

V

N

V

EP F

N 3

2

5

3

3/2~ VE 3/5~ VP

spin

Very different from Drude result! (P=B=0 at T=0)

Fkkd

VN

031

4

Helmholtz free energy F (A) (=U-TS)

Fermi-Dirac Distribution

In equilibrium, the probability of an energy E being “observed” in an N-electron system

: all possible N-electron states

)(

,)( N

N

FE

E

E

N ee

eEP

Probability that a one-electron level i is occupied in an N-electron system

)(

, )(1)(

i

NNN EPif

NN FN

E eZe

,

)(

, )()(

i

NNN EPif

Probability that the one-electron level i is unoccupied in an N-electron system

)(

)(

)(

,,

)(1

)(

11

i

E

Ni

E

N

iNi

N eZ

ee

Z

)(,11

1

)(

)( )()(

)(1

)(1 ,1

iNN

N

N

i

E

N

EPZ

Zee

Z

e iN

i

chemical potential

NN FF 1)(

)(1)(1 1][

1][ 1 ifeife NN

FF iNNi

)(1

1)(

ieif N

This step assumes that electrons do not interact!

summed over all containing occupied i

17

Sum over occupied states(per volume)

special case: T 0

Electron Gas At Finite Temperature

iTk

iN

BieifN

/)(1

1)(

chemical potential depends on T and N

FT

0

lim

)(,0

)(,1lim

0 k

kf

skT

)()( fgdu

)()()())(()(

40 3

fgdorkfk

kd

0,0

0,2

32

)(

2/1

222

FF

nmm

g

)()( fgdn

Sommerfeld Expansion At Finite Temperatures

)()( fgdu

)()( fgdn

Only f depends on temperature. Typically, T << TF. Finite temperature integrals differ from T=0 integrals only because of contributions near F (a few kBT within F).

FTTdTfH

?),()(

dxxHK )()(

)()()()(

fKdfHd

n

n

n

n K

nK

)(

!

)()(

0

12

12

1

2 )(

)!2(

)()()()(

n

n

n

n H

n

fdHdfHd

)1)(1( )()(

ee

f even function about

12

12

1

2 )()()(

n

n

nn

nB

HaTkHd

expand about

numbers

12

12

1

2)(

)!2(

1

)1)(1( n

n

n

n

xx

H

n

x

ee

dx

18

free electron gas

Non-Interacting Fermion Properties: T << TF

)()(

360

7)()(

6)()()( 4

42

2

HTkHTkHdfHd BB

)()( fgdu

)()( fgdn

)()(6

)( 22

gTkgdn B

)()( FF g

In general, decreases with T, reverting to Maxwell-Boltzmann statistics at very high T.

)}()({)(

6)( 2

2

ggTkgdu B

24

3)(

FF

ng

)()(6

22

0 FB gTkuu

free electron gas

)(6

)()( 2

F

FBF g

gTk

2

2

12

)(1

F

BF

Tk

)(3

22

FB

n

gTk

T

uc

BF

B nkTk

c

2

2

FF

ng

2

3)(

interpretation of the specific heat expression

a1 a2

positive

Heat Capacity

Linear dependence only observed at very low T for metals. Lattice vibration contribution dominates at high T. Acoustic phonons give a contribution proportional to T3.

3TATc

For free electronsF

B

T

kn

2

2

A plot of cv/T against T2 distinguishes the linear and the cubic terms.

3nkB

Insulator metal

19

Low Temperature Specific Heat

For free-electron-like metals with Z electrons per atom, the electronic contribution to the per-mole heat capacity is

214

2

0

2

10169.02

Kmolcal

a

rZ

T

RZn s

F

Sommerfeld Theory of Conduction

)2/(

2/32

2)( Tkmv

BB

BeTk

mnvf

]/)exp[(1

1

22)(

221

3

Tkmv

mvf

B

equilibrium velocity distribution

There is a concern about specifying a velocity distribution and a spatial distribution of electrons at the same time, because of the uncertainty principle. This is not a problem whenever the dimension of the sample is large compared to the wavelength and the wavelength is large compared to the mean free path.

There is also the concern that electron dynamics based on the independent electron picture, which allows Fermi-Dirac statistics to be used in the first place, may not hold because of the exclusion principle. For example, two electrons may evolve into the same state according to classical equations of motion. Fortunately, the evolution of states of non-interacting electrons preserves volume in phase space (Liouville’s theorem). Therefore, between collisions, electrons behave as if they were classical and independent. A wide range of metallic behavior is well described by classical mechanics.

20

unchanged

Sommerfeld Theory of Conduction

cenRH

1mne /2

i

1

)( 0

Within the constant relaxation time model, only those predictions of Drude’s model that require the knowledge of electronic velocity distribution are affected by switching to Sommerfeld’s free electron gas theory.

FN

E 5

3

Fvv ||nm

arv

cm

sF 2.9

)/( 20

mTkB /3

Te

k

ne

mvcB

v 2

2

2

2

33

1

BF

B nkTk

c

2

2

22Fvv

Te

kB

22

3

F

BB Tk

e

kQ

6

2

BF

B nkTk

c

2

2e

k

ne

cQ Bv

23

Wiedmann-Franz

Drude Sommerfeld

Chapter 2 Summary

Pauli’s exclusion principle requires that independent electrons obey Fermi-Dirac statistics. The consequences of this requirement on the thermodynamic properties of conductors and the Drude’s ideal gas model of electron dynamics are worked out by Sommerfeld.

Some of the serious problems with Drude’s model are removed, e.g. low temperature specific heat and thermopower. Others are left unexplained, e.g. Hall coefficient and magnetoresistance.

21

Chapter 3 Summary

Inadequacies in free electron transport coefficients

1. Hall coefficient

2. Magnetoresistance

3. Thermoelectric field

4. Wiedemann-Franz law

5. T Dependence of DC Conductivity

6. Anisotropy of DC Conductivity

7. AC Conductivity

Chapter 3 Summary

Inadequacies In Static Thermodynamic Predictions

1. Linear term in specific heat

2. Cubic term in specific heat

3. Compressibility of metal

Fundamental Mysteries

1. What determines the number of conduction electrons?

2. Why are some elements nonmetal?

22

1/29, Drude Theory of Metal 2/5, Sommerfeld Theory

2/12, no class 2/19, Crystal Lattice & Reciprocal Lattice

2/26, X-ray & Crystal Classif. 3/5, Bloch Thrm & Weak Periodic Pot.

3/12, Tight Binding & Other Methods 3/19, Semiclassical Electron Dynamics

3/26, Conduction in Metals 4/2, Electron Interaction, HF, DFT

4/9, no class 4/16, Harmonic Crystal

4/23, Semiconductors 4/30, Inhomo. Semiconductors

5/7, Special Topic 5/14, Special Topic

Topics for Lectures in May

Giant Magnetoresistance (GMR) and Magnetoresistive Random Access Memory (MRAM)

Ferroelectric RAM and Phase-Change RAM.

Solid State Ionics, Solvation, Electron Transfer?

Surface States, 2D Electron Gas?

Quantum Hall Effect?

Homework Set #1

Chapter 1: Problems 2 - 3.

Chapter 2: Problem 1.