advanced solid state physicsadvanced solid state...

by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )

Advanced Solid State PhysicsAdvanced Solid State Physics

Crystal binding & lattice vibrationCrystal binding & lattice vibrationA&M Ch. 19-25Kittel Ch 3-5Kittel Ch. 3-5

Classification of solids


• Previously, periodic ordering of ion cores → crystal latticeThe ion cores are perfectly localized with zero kinetic energy at the sites of a lattice.

• Here, physical properties based on the configuration of the valence electrons.

Cl ifi ti

Tip) Valence electrons → metals vs. insulators

• Classification

1.Molecular crystals : solid noble gases, such as Ne, Ar, Kr, Xe

Completely filled electronic shellsCompletely filled electronic shells.

All electrons remains very close to ion cores.

2 Ionic crystals : metallic + nonmetallic elements : NaCl LiF2. Ionic crystals : metallic + nonmetallic elements : NaCl, LiF

The electronic distribution near ion cores is not neutral, but charged.

Classification of solids (continued)


( )

3. Covalent crystals : C, Si, Ge

Partially filled band. → sharing electrons to form filled shells

The interstitial electrons are localized in certain preferred direction

: ‘Bonds’ between two atoms

4. Metallic crystals : Li, Na, K

Delocalized valence electrons

Molecular crystals


y• Solid noble gases : VIII elements

VIII = Ne, Ar, Krfcc structure

• atomic ionization of < 1%, but act as dipoles• Van der walls forces or fluctuating dipole force

A model of two harmonic oscillators for molecular crystal

22

22

21

210 2

1

2

1

2

1

2

1CxP

mCxP

mH +++= 2

0mwC =

y

2222 eeeeH −−+=

321

2

21211

2-

R

xxe

xRxRxxRRH

≈

−−

+−

−++=

xforxxx small ....1)1( 21 −+−=+ −

)(2

11 as xxx += )(

2

12 as xxx −= )(

2

11 as ppp += )(

2

12 as ppp −=

⎥⎤

⎢⎡

+++⎥⎤

⎢⎡

−+=+ 22

222

210 )

2(

11)

2(

11x

eCpx

eCpHH ⎥

⎦⎢⎣

+++⎥⎦

⎢⎣

++3310 )(

22)(

22 aass xR

Cpm

xR

Cpm

HH

⎥⎦

⎤⎢⎣

⎡⋅⋅⋅+−±≅⎥

⎦

⎤⎢⎣

⎡±=±

23

2

3

2

0

2

1

3

2

)2

(8

1)

2(

2

11/)

2(

CR

e

CR

ewm

R

eCw

6

2

3

22

8

1)2(

2

1

R

A

CR

ewwwwU ooas −=⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅−=−+= hhΔ Lowered energy

; Van der waals interaction (attractive)

Molecular crystals


y

VIII elements ; filled shells : 1s22s22p6 …

Pauli exclusion principle : repulsive interactionPauli exclusion principle : repulsive interactionTwo electrons can not have their quantum numbers equal

eV 98.78− εφ 4

V3859 eV38.59−

⎥⎤

⎢⎡ ⎞

⎜⎛−⎞

⎜⎛=

612

4)(RUσσε

For simplicity, choose a repulsive potential of ~1/R12

Lennard-Jones 6-12 potential

σR

⎥⎥⎦⎢

⎢⎣ ⎠

⎜⎝⎠

⎜⎝

= 4)(RR

RU εε ~ 0.01 eV

612 ⎤⎡dU σσAt equilibrium

R0)45.14)(6()13.12)(12(200

0 713=⎥

⎦

⎤⎢⎣

⎡−−===

=

=

RR

RRtot

RRN

dR

dUF

σσε εσ

NUR

6.8 ; 09.10 −==

Quantum correction for the kinetic energy is needed.

Ionic crystals


yimpenetrable charged spheres.electrostatic attraction between positively and negatively charged spheres.impenetrability by Pauli exclusion principle for the closed shell.

Alkali halides ( I-VII ) I = Li+, Na+, K+, Rb+, Cs and VII = F-, Cl-, Br-, I-)Cubic : Sodium chloride Cesium chloride Zincblende structuresCubic : Sodium chloride, Cesium chloride, Zincblende structures

II-VI ionic crystalsI = Be2+, Mg2+, Ca2+, Sr, Ba, and VI = O2-, S2-, Se, TeSodium chloride, Zincblende, Wurtzite structures

Ionic crystal examples


1) Na + 5.14ev → Na+ + e-

e- + Cl → Cl- + 3.61eVNa+ + Cl- → Na+Cl- + 7.9eV

2) Rb + 4.2 ev → Rb+ + e-e- + Br → Br- + 3 5 eVe- + Br → Br- + 3.5 eVRb+ + Br- → Rb+Br - + 4.2 eV

Electron affinities of negative ionsg

Narrow band width by small electronic overlap at the interstitial

Interionic Coulomb interaction in Ionic crystal


Electrostatic (or Madelung) energyAttractive interaction ← Coulomb interactionRepulsive interaction ← Pauli exclusion principle

+ - + -+ -

ijrepu s e te act o au e c us o p c p eUij : interaction between ith and jth atoms

jij

i UU '∑= Rprrqr

U ijijij

ijij ≡±⎟

⎠⎞

⎜⎝⎛−= exp

2

ρλR : nearest neighbor distancez : # of nearest

ijr

j rij⎠⎝ ρfor N molecules (2N ions)

⎥⎤

⎢⎡

⎥⎤

⎢⎡

∑−− q

Nq

NNUURR 22

' αλλ ρρ

z : # of nearest neighbor

Number of nearest neighbor atoms : z )(

⎥⎦

⎢⎣

−=⎥⎥⎦⎢

⎢⎣

∑−==R

qezN

RP

qezNNUU

ijj

itot λλ ρρ

ρR−

l ig

Madelung constant :

ijj P

)(' ±∑≡α

ρe∝ repulsive

1.7476Sodium Chloride

1.7627Cesium Chloride

Madelung constantCrystal structure

R

q2

attractiveα

−∝1.6381Zincblende R

From Ionic crystals to covalent crystals


y y

III-V ionic crystalsIII = Al3+, Ga3+, In3+, and V = P3-, As3-, SbZi bl d t tZincblende structuresemiconductors rather than insulators, partially ionic and partially covalent

Covalent crystals ( IV ) Co a e t c ysta s ( )IV = C, Si, Ge, Sn : SemiconductorDiamond structure, Tetrahedral bond

Share electrons, overlap of electron at the interstitial

Ionic crystalCovalent crystal

Perfect covalent Ge (IV) Covalent GaAs (III-V) Ionic CaSe (II-VI) Perfect ionic KCl (I-VII)


1H

2He

III+VIonic +covalent

II+VIionic

I+VIIionic

H He

3Li

1s22s

1

4Be

1s22s

2

5B

1s22s

2

6C

1s22s

2

2

7N

1s22s

2

8O

1s22s

2

9F

1s22s

2

10Ne

1s22s

2

IV

2p1 2p

22p

32p

42p

52p

6

11Na

12Mg

13Al

14Si

15P

16S

17Cl

18Ar

19K

20C

21S

22i

23V

24C

25M

26 27C

28Ni

29C

30 31G

32G

33A

34S

35 36K

3B 4B 5B 6B 7B 8B 9B 10B 11B 12B

19K

20Ca

21Sc

22Ti

23V

24Cr

25Mn

26Fe

27Co

28Ni

29Cu

30Zn

31Ga

32Ge

33As

34Se

35Br

36Kr

37Rb

38Sr

39Y

40Zr

41Nb

42Mo

43Tc

44Ru

45Rh

46Pd

47Ag

48Cd

49In

50Sn

51Sb

52Te

53I

54Xe

55Cs

56Ba

57La

72Hf

73Ta

74W

75Re

76Os

77Ir

78Pt

79Au

80Hg

81Tl

82Pb

83Bi

84Po

85At

86Rn

87Fr

88Ra

89AcFr Ra Ac

58Ce

59Pr

60Nd

61Pm

62Sm

63Eu

64Gd

65Tb

66Dy

67Ho

68Er

69Tm

70Yb

71Lu

Lanthanoids

90Th

91Pa

92U

93Np

94Pu

95Am

96Cm

97Bk

98Cf

99Es

100Fm

101Md

102No

103Lw

Actinoids

Covalent crystals


ySemiconductorsIV : 1s22s22p2 →1s22s12p3 for tetrahedral bonds Share electrons, overlap of electron at the interstitialS a e e ect o s, o e ap o e ect o at t e te st t a

spin dependent Coulomb energy ; exchange interaction

ψ SAχϕψ =

ASχϕψ =

Continuous crossover between the ionic and the covalent limits

Metallic crystals


yValence electrons are delocalized : free electron gasattractive interaction between the positive ions and the negative electron gas+ the kinetic energy of the electron gas

dkkkdm

kkdU

kk

kinetic

f

4 24

1 222

3== ∫ <

ππ

rhr•• Kinetic energy of electronsKinetic energy of electrons

Nucleus

Core electronsValence electrons Nucleus

Valence electrons

ion

atomeVrm

k

s

ff /

1.30

5

3

10

3 2

22

⎠⎞⎜

⎝⎛

=== εh

NucleusCore electrons

ion a0 ⎠⎜⎝

341r

V π== Vk

N fk f 3

34

23π

==2

0222 )( akek ff

f ==h

ε

0

2

2

0 A529.0==me

ah

atomeVU Coulomb /35.25

⎞⎛−=••Coulomb energy (for bcc)Coulomb energy (for bcc)

3 srNnπ

( )VN

L232 3

2ππ

022 amfε 10

0 A63.3 −

=s

f rak

rs : Wigner-Seitz sphere

ars

0⎟⎠⎞⎜

⎝⎛

atomeVUUU Coulombkinetic /35.251.30

⎞⎛−=+=

minimize

6.1 0 ==r

d

dU s

gy ( )gy ( )

atomeV

ar

ar

UUUss

/

0

2

0

⎟⎠⎞⎜

⎝⎛

⎟⎠⎞⎜

⎝⎛

+0adrs

In alkali metals, rs/a0=2~6

Shortcomings of the static lattice model


Shortcomings of the static lattice model

F il t l i ilib i ti f lidFailures to explain equilibrium properties of solid – Specific heat

– Thermal expansion

– Melting

Failures to explain transport properties– Temperature dependent resistivity : scattering, relaxation

– Superconductivity

– Thermal conductivity

– Transmission of solid

Failures to explain the interaction of various type of radiation– Inelastic scattering of light : Brillouin and Raman

– Scattering of X-rays and neutronsScattering of X rays and neutrons

Lattice vibration(phonon) should be considered to explain above properties.

Lattice vibration : normal modes of a 1D monatomic lattice



(n-2)a (n-1)a na (n+1)a (n+2)a (n+3)a

[ ]

[ ] ( )

2

1 2

]1[n

annahar

dU

uuKU +−= ∑ Harmonic potential energy

Periodic boundary condition[ ] (1) 2 ]1[]1[ anannana

na uuuKdu

dUuMF +− −−−=−== &&

)()0( ; )]1([)( Nauuanuau =+=y

nπknaeeuu iknaiwtikna 21 =∴== −

Traveling wave solution for eq.(1)Na

KM

uuuKuMikaika

naananna

2

]1[]1[2

)2(

)2( from(1) −+=−−

−+ω

nπkna eeuuna 210 =∴==

kaM

K

M

kaK

eeKM ikaika

21

2

sin2)cos1(2

)2(

=−

=

−+=−

ω

ω

Dispersion relation



kaK 1

sin2=ω


h l ω

kaM 2

sin2=ω

dk

dvvelocitygroup

kv city phase velo

g

ω

ω

=

=

:

:

dk

At a low frequency region (k=2π/λ→0, long wavelength limit)

kkM

Kaka

M

K νω ===2

2

12 velocitygroupv

dk

dkv city phase velo g :: ===

ωω

At a high frequency regionAt a high frequency region

)2( mM

K ωω ==02

1cos

2

1=

±=⎟⎠⎞

⎜⎝⎛==

πωων

kakaa

dk

dmg

π±=kan

n uu )1(−=At the zone boundary

Lattice vibration : normal modes of a 1D diatomic lattice


Harmonic potential energy

[ ] [ ]22

2

]1[22

2

21n

annan

nanahar

dUdU

uuG

uuK

U +−+−= ∑∑

Harmonic potential energy

[ ] [ ] [ ] [ ] (1) and ]1[12122

2]1[21211

1 annananana

naannananana

na uuGuuKdu

dUuMuuGuuK

du

dUuM +− −−−−=−=−−−−=−= &&&&

[ ] ( )Traveling wave solution for eq.(1)

iwtiknana

iwtiknana e u eu −− == 2211 and εε [ ] ( )

( ) [ ] (2) 0)(

0)( from(1)

22

1

212

=+−++

=+++− −

εωε

εεω

GKMGeK

GeKGKMika

ika

[ ]ika

ika

GeKGK

kaKGGKGeKGKM

++

++=+=+− −

1222

22222

1

cos2)(

ε

ωDispersion relation

Solve the determinant of eq.(2)

ikaGeK

GeKkaKGGK

MM

GK

++

=++±+

= m2

1222 and cos21

εεω


Lattice vibration : normal modes of a 1D diatomic lattice

cos21

222 kaKGGKMM

GK++±

+=ω

Dispersion relation

ika

ika

GeK

GeK

++

= m1 εε

MM

( ) 0 and

2 22 =+

= ωωM

GKIf k=0

If k /GK 2

d2 22

GeK +2ε

π±=ka

If k=π/aMM

and 22 == ωω

( ) 2εKG

Case 1, k<< π/a ( ) 2/1cos 2kaka −≈

( )

( )modeoptical 1 and )(

2

mode acoutic 1 and )(2

22

1

2

−=−+

=

=+

=

ε

εε

kaO M

GKω

kaGKM

KGω

p)(1εM

mode acoustic 1 and 2 2 ==

εε

M

Gω

Case 2, k= π/a

mode optical 1 and 2

1

2

1

−==εε

ε

M

Kω

M

1 12

; 1 and 1sin2

1

2

1

221 −=⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+==⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+=

εε

εε

G

KO

M

Kω

G

KOka

M

Gω

Case 3, k>>G

Case 4. k=G ?monoatomic?

Phonon modes


Phonon modes

•• Acoustic modeAcoustic modeAll the atoms go in the same direction.← Compressive wave or sound wave.

k

O ti l dO ti l dk

•• Optical modeOptical modeOpposite effect on the two different atoms ← Electromagnetic radiation

Not all frequencies can propagateOnly discrete bands are available.

For P atoms in the primitive cell.3 acoustic branches.

3P-3 optical branches

For example, KBr with two atoms, 1 LA (Longitudinal acoustic), 2 TA (Transverse acoustic)1 LA (Longitudinal acoustic), 2 TA (Transverse acoustic)1 LO (Longitudinal optical), 2 TO (Transverse optical)

Phonon energy and momentum phonon scattering


Phonon energy and momentum, phonon scattering

wave characteristics of lattice vibration → dispersion relationQuantization of lattice vibration (particle nature) → PhononQuantization of lattice vibration (particle nature) Phonon

Energy eigenvalue of lattice vibration (harmonic oscillator) by Quantum mechanics

ωh⎟⎠⎞

⎜⎝⎛ +=

2

1nE = ½ kinetic energy + ½ potential energy for time average

tkxuu ωcoscos0=

u2

1111 ⎞⎛ ∂The time average kinetic energy

Let’s quantize the mean square phonon amplitude

( )

( ) ωρ

ωωρωρρ

Vnu

nuVwtuVt

uVEkinetic

h

h

212

0

212

0222

02

4

2

1

8

1sin

4

1

2

1

+=∴

+===⎟⎠⎞

⎜⎝⎛∂∂

=

Kr

h

( )2

For phonon momentum

X-ray or neutron scattering by phonon

KGkkrrrr

±+=′Gkkrrr

+=′

y g y pElastic scattering vs. inelastic scattering

vectorlattice reciprocal ; Gv

Phonon heat capacity



VV T

UC ⎟

⎠⎞

⎜⎝⎛∂∂

=Heat capacity at constant volume

Troomat3NkC =From experiments

T lowat (metal) or )(insulator

T. roomat 33TCTC

NkC

VV

BV

∝∝

=From experiments,

⎞⎜⎛ ∂ ∫

U

Ratio of the number of harmonic oscillators in the (n+1) state to the number of in the n state

?)( ?,)( ?, )()( ====⎠⎞

⎜⎝⎛∂∂

= ∫ wDwnwwnwdwDUT

UC

VV h

( )Tk

NNB

Tkw

nn B

1exp1 ≡−=+ βwhere, h

Average excitation quantum number of an harmonic oscillator

Boltzmann distribution

)exp( e, wher)exp(

)exp()( −==

−

−=

∑∑

∑∑

wxx

sx

ws

wsswn

sS

s

S hh

h

ββ

β

1

Average excitation quantum number of an harmonic oscillator

111

)exp(

⎟⎠⎞

⎜⎝⎛−∑

∑∑

xxdxd

xxdxd

x

xws

S

s

SS

hβ1for

1

132 x x

xxxxS

s <−

=+++=∑ L

1)exp(11

1

−=

−=

−

⎠⎝==∑ wx

x

xdxx

dx

S

sS

hβ ; Planck distribution




{ } { })()()(expexp LzkLykLxkizkykxki yxyx +++++=++Density of states in 3-dimensions

{ } { })()()(expexp LzkLykLxkizkykxki zyxzyx +++++++L

L

N

LLkkk zyx

πππ±⋅⋅⋅±±=∴ ,

4,

2,0,,

L

3

3

82 ππVL

=⎟⎠⎞

⎜⎝⎛Density in k-space

4 33kL π⎞⎛ ⎞⎛ dk2

3

4

2

kLN

ππ

×⎠⎞

⎜⎝⎛= ⎟

⎠⎞

⎜⎝⎛==

dw

dkVkdw

dNwD 2

2

2)( π

Debye approximation : dispersion relation of Debye approximation : dispersion relation of acoustic phonon acoustic phonon c = w/kc = w/k

Ky

KD

KTKX

32

3

3

33

63

4

2 c

Vw

c

wLN

D

ππ

π=×⎟

⎠⎞

⎜⎝⎛=

Debye approxw( )1

KX

If phonons are filled up to wD only

Actual

Debye approx.w( )32323 66 VN

cwkV

Ncw DDD ππ === wd

⎞⎜⎛⎞

⎜⎛

∫∫2

3)()(3WD wVw

ddh

h

kkd

⎟⎠

⎜⎜⎝

⎟⎠

⎞⎜⎜⎝

⎛==

−∫∫

10 323 23)()(3

TkwD

B

D

e

w

c

VwdwwwnwdwDU

h

hh

π

Phonon heat capacity (by acoustic phonon?)


Phonon heat capacity (by acoustic phonon?)

d ⎞⎛

3

2

0

33

5

3

19 ⎟⎟

⎠

⎞⎜⎜⎝

⎛Θ

=−⎟⎟

⎠

⎞⎜⎜⎝

⎛Θ

= ∫D

B

x

xD

B

TTNk

e

xdx

TTNkU

D π xTkw

B≡h

w Θhh

Tkdxdw B=

kch

Θ

VV dT

dUC ⎟

⎠⎞

⎜⎝⎛=

⎠⎝⎠⎝

at low Temperatures

TTKwx D

B

Dd

Θ≡= h DB

D kk

=Θ

Ew

E wew E hh h ββ β +≈<< 11

∫∞

=−0

43

151

πxe

xdx

at low Temperatures

∞⇒⇒Tk

wx

B

DD

h

3443

5

12

53 ⎟⎟

⎠

⎞⎜⎜⎝

⎛==×⎟⎟

⎠

⎞⎜⎜⎝

⎛Θ

=D

BVD

B Θ

TNk

π

dT

dUC

TTNkU

π

33 EwN

NUh

h

On the other hand, Einstein model is On the other hand, Einstein model is

Consider N oscillators of the same frequency wE

( )21

3

3

3

=⎞⎜⎛=

==−

E

B

w

Tkw

EED

ewNk

dUC

ewnNU

h

h

h

h

β

β

BV NkC 3≈at high T

at low T 1>>Ewhβ Tk

w

B

E

eCh

−

∝( ) ( )213

−=

⎠⎜⎝

=EwEB

VV

ewNk

dTC

hh

ββ V

BeC ∝

Total (at low T)

Calorimetry for heat capacity measurment


Calorimetry for heat capacity measurment

tRiQTCQ V ∆=∆= 2 Measurement of the temperature variation for applied power

cond-mat/0303457

PRB, 64, 134426(2001)

Thermal conductivity by phonon scattering


Thermal conductivity by phonon scatteringQ: In insulators, how the energy is transferred from hot region to cold region?Energy can diffuse through phonon collisions in a solid.

τxvdx

dT

dx

dTT =⋅=∆ l T T+∆Tτ : collision time

Energy transmitted across unit area per unit time depends on temperature gradient

Tcnvj xu ><−= Δ

The net flux of energy : Energy transmitted per unit area per unit time

n : density, c : heat capacity per particle

ncCdx

dTCv

dx

dTncv

dx

dTcvn x =−=−=−= where,

3

1

3

1 222 τττ

lCvKdT

Kj1

=∴−=

Normal phonon collisionUmklapp phonon collision

lCvKdx

Kju 3 =∴−=

321 KKKvvv

=+ GKKKvvvv

+=+ 321

Phonon flux is independent of the lengthThermal resistivity = 0

Phonon flux is decreased as they moves to the rightThermal resistivity ≠ 0, Energy transfer ≠ 0

vectorlattice reciprocal: phonons:,, 321 GKKKrvvv

Raman scattering (by optical phonon?)


Raman scattering (by optical phonon?)Raman spectroscopy is associated with scattering of light by optical phonons in solids

Raman shift in wave numbers (cm-1) ν 11−=Raman shift in wave numbers (cm-1)

scatteredincient λλν −=

kr k ′

r

kr k ′

r

Kv

k

Kv

Ph i i (St k ) Ph b ti (A ti t k )Phonon emission (Stokes) Phonon absorption (Anti-stokes)

Raman spectrometer


Raman spectrometerApplications : structure determination, multicomponent qualitative analysis,

and quantitative analysis

PRB, 53, 3590(1996)

Inelastic X ray or Neutron scattering (by optical phonon?)


Inelastic X-ray or Neutron scattering (by optical phonon?)

GKkk ++=′v

wEE h±=′

tl ttii l

phonon :

veneutron wa scattered andincident ; ,

G

K

kk′

vectorlattice reciprocal : G

Inelastic X ray or Neutron scattering (by optical phonon?)


Inelastic X-ray or Neutron scattering (by optical phonon?)

cond-mat/0210700

advanced solid state physicsadvanced solid state...

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