solid state physics i - snuphysics.snu.ac.kr/~ssphy/2001-2/ssp-lecture-0.pdf · basic physical...

Solid State Physics I

JAEJUN YUSEOUL NATIONAL UNIVERSITY

[email protected]

c2001 JAEJUN YU

1

PrefaceThe purpose of this lecture is to help students to understand various physical phe-nomena in condensed matter systems based on the fundamental physical principles.A primary objective of the lecture as a part of the two-semester course is to discussbasic physical concepts and related examples in solid-state systems including crystallattice and defects, phonons, electrons in energy band, metals and semiconductors, andmagnetism. I will try to illustrate physics with several (simplified) models, which mayrequire some backgrounds in quantum mechanics, electrodynamics, as well as classicalmechanics of the undergraduate level.

Since this is a try-out version of the e-Lecture, I will try to give out guidelines forstudents through the web e-Lecture sever so that we can save some lecturing time inthe classroom. Since more time will be allocated for discussions and question-and-answers, active discussions during the class among students are highly desired.

SNU Lecture: Solid State Physics I (2001-1) Chap. 0 Overview Jaejun Yu http://phya.snu.ac.kr/ssphy/ 2

List of Topics (Solid State Physics I)

0. Overview

1. Crystal structure and x-ray diffraction

2. Diffraction in perfect and imperfect crystals

3. Lattice dynamics in one dimension

4. Lattice dynamics and heat capacity

5. Dynamics of free electron gas

6. Electron energy bands and states

7. Semiconductors

8. Ising model and ferromagnetism

9. Antiferromagnetism


Contents0 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

0.1 Macroscopic vs. microscopic objects . . . . . . . . . . . . . . . . . . . . . . 60.2 Most condensed matter systems are quantum mechanical by

nature! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110.3 How do we approach the condensed matter system in order to

understand the physics of a black box? . . . . . . . . . . . . . . . . . . 120.4 Particle zoo in the condensed matter systems . . . . . . . . . . . . . . . . 150.5 Energy scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Solutions to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19


0. OverviewWhat is the condensed matter physics?

Suppose that we understand all about of the fundamental lawsof physics and know all the elementary particles constituting thematter, what can we say about the physical properties of con-densed matter systems?

Question 0.1. How do we understand the physical properties of single par-ticle systems in a classical or quantum sense? In other words, what do wemeasure or observe?

O

x

p

?


0.1. Macroscopic vs. microscopic objects

Observables for the macro object consisting ofmore than 1020 particles:

specific heat cv entropy, temperature bulk modulus or compressibility pressure P polarization P, magnetization M electro-magnetic field E, B reflectivity, color, conductivity, ... etc.


Question 0.2. For an example, what physical property of the cube (ordisk) makes this happen, i.e., a cube float over a magnet against grav-itation?

(Hint: This is the phenomenon called magnetic levitation, which is mainly attributed

to the Meisner effect of superconductors.)


Observables for the micro or nano object oforder 102 particles: conductance G / electron tunneling current I force F magnetic Flux charge density distribution, electron cloud (bonding), ... etc.


Question 0.3. What do we really see in these images?What physical quantities do they represent?

an AFM image an STM image a LEED image

An example of the SEM System:


Question 0.4. MRI Image: What do we really see in this image?What physical quantity does it represent?

(Hint: This is an image of our brain probed by using the technique of the nuclear magnetic

resonance.)


0.2. Most condensed matter systems are quantum mechanical bynature!

~p d

Unfortunately, however, there is no quantum mechanical solution available exceptfor free (i.e., non-interacting) particle systems.

An example of exactly solvable models:

H = ~(a+a+ 12

)

a+a|n = n|n

a|O = 0


0.3. How do we approach the condensed matter system in orderto understand the physics of a black box?

Experimental Methods:By disturbing the black box by phtons, phonons, electrons, and/or neutrons, try toget a hint on elementary excitations in the system.

Theoretical Methods:By guessing possible elementary excitations and working out the QM equation ofmotions, see if we can predict the physics of the system. If wrong, correct the modelfor the elementary excitations for the better description.

How to disturb the black box?


Question 0.5. Measuring the dc resistance is a way of disturbing the sys-tem? What really happens inside the black box when we apply a biasvoltage?

Current-Voltage curve of a YBCO high Tc superconductor


Question 0.6. What happens when we shine a light on the matter?

Light scattering experiment: (Photoemission / IR spectroscopy)


0.4. Particle zoo in the condensed matter systems

elementary excitations in solids

quasi-particles: electrons, holes, polarons, excitons, Cooper pairs, ...

collective excitations: phonons, magnons, zero-sound, plasmons, ...


0.5. Energy scale

The condensed matter system is merely a collection of atoms, where each atom consistsof electrons and a nucleus (me mN ).

From the uncertainty principle xp ~,

p =

2meE

3mkBT

Thus, practically, the size of atom the size of electrons.

atomic unit: ~ = e2 = me = 1

x aB =~2

me2= 0.529177A (Bohr radius)

EB = 1

2

e2

aB= me

4

2~2= 13.6058eV = 1Ry

In atomic unit, aB = 1, EB = 1/2, c = 1/ 137,kB 3 106 , ...( = e2/~c: fine structure constant)


energy of an electron in a box of size L:

x L p ~L

Eo =p2

2m 1

2L2

speed of electrons in a box:

ve 1

L c

L

electrons in solid

Fermions: Pauli exclusion principle Degenerate Electrons: lowest possible excitations near the Fermi

energy


Question 0.7. Estimate the scale in atomic unit:

1. ground state energy of an electron in a cube of length 1 nm.

2. energy and momentum of a photon with = 104 A.

3. energy of an electron in a magnetic field of 1 T.

4. thermal wave length of an electron and a neutron in a thermal bath of300 K.

5. Fermi energy of a degenerate electron (neutron) gas with its density1022 cm3 respectively.


Solutions to Exercises

Question 0.1. In classical dynamics, the state of a single particle is deter-mined by the observables {x(t),p(t)}. It can be extended to the systemwith many particles where the state of the system is described by the set ofobservables {xi(t),pi(t)|i = 1, 2, ..., N}. However, when N 1025, it ispractically impossible to trace the orbits of all, even the part of, the parti-cles. Here it is the point where the statistical physics comes into playing arole. Instead of following each individual particles, we measure a quantityby an (ensemble or time) average of the given quantities adopted in classi-cal dynamics. In addition, now we have to deal with new observables suchas entropy, temperature, ....

The same thing applies for the case of quantum systems. The only dif-ference is the dynamics state of the quantum system is determined by astate vector |(x, t) for a single particle and |(x1,x2, ...,xN , t).


J


Question 0.2.

To be in euilibrium, the magnetic force on the cube must compensatethe one by the gravitaton:

~Fg = +m~g, ~F = (~ ~B), U = ~ ~B

~ = (4)1 ~B v ~Fm =v

4B2

~Fg + ~Fm = 0 : +m~g v

4B2 = 0

= 4mv~g = B2


What about the stability in the lateral direction:

* hint : pinning of vortex

Main Part

EM force in balance with m~g pinning of vortex magnetic field


J


Question 0.3.

AFM (Atomic Force Microscopy)

In case of charge q, we can measure the force acting on the tip:~F = qU(~r)

Either the variation of ~F with the constant height h = h(~r) or thevariation of h(~r) at the constant ~F can be scanned over the surface.


STM (Sanning Tunneling Microscopy)

Tunneling rate G I

T ed, =

2m(U E)T exp[d

2m(U E)]


where the potential barrier height U as well as the barrier thickness dare a function of h(~r).

LEED (Low Energy Electron Diffraction)

Why low energy electron?

e }p 1

Ewhere E =

p2

2me, i.e., p

2E

if E 110 , then e & 2(interatomic spacing)interference(wave nature of electrons)


But, the actual measurement is done by counting the number ofelectrons in d~k!! or the current through that direction.

J


Question 0.4. NMR(Nuclear Magnetic Resonance)

Resonanceabsorptionimaginary part of the magnetic susceptibility ()L(): inductancemeasuring the phase shift V ()/I()

J


Question 0.5.

~J = ~E: is what to be measured.

vd =eEm drifting velocity(average)

In Drude model(above figure): = ne2

m

J


Question 0.6. * photon-electron scattering:

1

2m(~p e

c~A)2 =

~p2

2m emc

~A ~p+ ....

* light scattering : () + i()


J


Question 0.7.

1. l = 1nm 20 a.u.

E0 =}2

2m(

1

l2+

1

l2+

1

l2) 3

2l2 3

16 103 2 104

2. E = }, p = }k

E = = kc =2

c = 2

2 104 137 500

104 5 102

p = k =2

3 104

* wavelength of an electron with E = 5 102.

p =

2E =

103 3 102


e =2

p 2

3 102 2 102

size of quantum well vs. photon wave length

3.

EM = ge}

2mcB = B B

where B is Bohr magneton

EM B 1T 2 106a.u.

4. kB 3 106, kB(300K) 103 = E, mn 2 103

pe 3 102 2 102

pn

2 2 103 103 2 n 2

p 3


5. Fermi Energy : EF =p2F2m

N = (L

2)3 k

EF,electron k2F2 1

20 101

EF,neutron k2F2m 1

20 1

2 103 104

J


OverviewMacroscopic vs. microscopic objectsMost condensed matter systems are quantum mechanical by nature!How do we approach the condensed matter system in order to understand the physics of a red ``black box''? Particle zoo in the condensed matter systemsEnergy scale

Solutions to Exercises

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