Band Theory of Solids

固體的能帶理論

Schrodinger Equation

⇒ structure of hydrogen atom

⇒ properties of other atoms

⇒ periodic table

⇒ a system of atoms, crystals

⇒ semiconductors Chap. 24, 25

⇒ integrated circuits ( I.C. )

簡介 ( Introduction )

material σ ( Ω-m )-1

Cu 6 x 107

Al 3 x 107

Ge 2 x 10-2 1027 orders of magnitudeSi 4 x 10-4

glass ( SiO2 ) 2 x 10-11

polystyrene 1 x 10-20

一個好的理論必須能夠解釋

σ 這麼大的變化範圍 !

• problem of QMFE assume uniform P.E. inside the solid

• in actual cases P.E. inside the solid is a periodic function of position

Bloch 理論 ( Bloch’s Theorem )

- a general property of the wavefunctions in a periodic potential

• for a free e- with Ep = constant : χ( x ) = e±ikx

• a periodic potential with the period of d ( e.g. spacing of ions = d ) :

Ep( x ) = Ep( x+d )

• Bloch’s Theorem : for a particle moving in a periodic potential with

the period d χ( x ) = uk( x ) • e±ikx , uk( x ) = uk( x+d )

• χ*( x ) χ( x ) = uk*( x ) e-ikx uk( x ) e+ikx = uk*( x ) uk( x )

χ*( x+d ) χ( x+d ) = uk*( x+d ) uk( x+d )

= uk*( x ) uk( x ) = χ*( x ) χ( x )

the probabilities of finding the particle at the position of ( x ) and

at the position of ( x+d ) are the same

Kronig-Penny 模型 ( Kronig-Penny Model )

Ep( x ) = – ——— ——1 q⎪e⎪

4πε0 x

• well width : cwell spacing : bperiodicity :

d=b+c

• in region I :

Ep = 0 , χI : eigenfunction in region I

Kronig-Penney Model

– —— —— = E χIh22m

d2χIdx2

—— + γ2 χI = 0 , γ = 2mE / h2d2χIdx2

substitute χI = uI( x ) eikx

uI( x ) = Aei( γ-k )x + Be-i( γ+k )x

—— + 2ik —— + ( γ2-k2 ) uI = 0d2uIdx2

duIdx

Kronig-Penney Model• in region II :

Ep = Ep0

χII : eigenfunctionin region II

similarly, substitute χII = uII( x ) eikx uII = Ce( ε-ik ) x + De-( ε+ik )x

– —— ——— + EpoχII = E χIIh2 d2χII

2m dx2

——— + ε2χII = 0 , ε = ——————d2χIIdx2

2m( Epo – E )h2

• χ and dχ/dx must be continuous across a boundary

χI( c/2 ) = χII( c/2 ) , dχI( c/2 ) / dx = dχII( c/2 ) / dx

• periodicity requirement on u( x ) :

uI( -c/2 ) = uII( b+ c/2 ) , duI( -c/2 ) / dx = duII( b+ c/2 ) / dx

⇒ 4 linear algebraic equations for A, B, C, D⇒ acceptable solutions of A, B, C, D, and χ exist only if

P( sinγd / γd ) + cos γd = cos kd

P = ( mEpobd ) / h2 , γ = 2mE / h2

γ is a measure of the electron’s energy k is related to the electron’s momentum( k = 2π/λ and from de Broglie hypothesis : λ = h/p p = hk )

This equ. relates the e-’s energy to its wave vector k (and momentum)

• Dispersion Relationthe relation between a particle’s energy( E ) and its wave vector( k )

例 : for a free particle E = p2 / 2mp = h / λ ( de Broglie relation ) , λ = 2π / k p = h / λ = hk / 2π = hk⇒ E = h2k2 / 2m , E ∝ k2

例 : for an e- moving in a 1-dim array of potential wellsthe dispersion relation : P( sinγd / γd ) + cos γd = cos kd

γ = 2mE / h2

k

E

E = h2k2 / 2m

• Allowed and Forbidden Energy Bands

P( sinγd / γd ) + cos γd = cos kd

γ = 2mE / h2

this equation can be solved numerically : ( pick a value of E

obtain a corresponding k )

some values of E imaginary k physically unacceptable ⇒ these E’s are forbidden⇒ allowed and forbidden energy bands created

band discontinuities occur at k = ± nπ / d

• problems of Kronig-Penney model :(1) not much physical insight(2) does not give the # of energy states in a band

Tight-Binding Approximation

• infinite square potential well ( 1-dim. )

( - χ1 ) and ( - χ2 ) are also solutions of Schrodinger equ.

– —— ——— + Ep(x) (- χ) = E (- χ)h22m

d2(- χ)dx2

– —— —— + Ep(x) χ = E χh22m

d2χdx2

( - χ1 ) and χ1 same E1, and ( - χ1 )2 = χ12 (represent same wave function/quantum state)

( - χ2 ) and χ2 same E2, and ( - χ2 )2 = χ22 (represent same wave function/quantum state)

-χ1

-χ2

• finite potential well ( 1-dim )

• 2 finite potential wells ( 1-dim )

-χ2-χ1

-χB

-χC

2 kinds of combinations :

(1) χS = a ( χB + χC ) ( symmetric ) ( a : introduced for normalization )

(2) χA = a ( χB - χC ) ( antisymmetric )

for two wells far apart χS and χA are degenerate states with the same χ2 and energy

the same

the same

• when two wells get close enough :

χS is like the ground state for a well of width 2a

χA is like the 1st excited state for a well of width 2a

• when 2 wells get close enough degenerate states begin to break up into nondegenerate states

degeneracy of χS and χA begins to disappearE of χS < E of χA , but physically why ?

• consider the 1s state of 2 H-atoms :

2 kinds of combinations : ( 1 ) χS = a( χB + χC )

( 2 ) χA = a( χB – χC )

• an e- in χS state spend more time in between the 2 protonsstronger negative binding energy an e- in χS state has a lower energy than in χA state

• consider the 1s state of 6 H-atoms

When 2 atoms are brought together, two separate energy levels are formed from each level of the isolated atom.

What if six atoms are brought together ? Let’s start with the six individual 1s states ……

χsecond level = (χ1+χ2+χ3) - (χ4+χ5+χ6)

χfirst level = χ1+χ2+χ3+χ4+χ5+χ6

same wave function

• N atoms brought together ⇒ each level N discrete,

closely spaced energy levels⇒ a quasicontinuous band of

energy levels

例 : width of a band ~ a few eVif N = 1023

separation between adjacent levels ~ 10-23 eV

例 : Na : 1s2 2s2 2p6 3s1

• widths of the bands should not depend appreciably on N

• widths of the bands dependmainly on distance between adjacent atoms⇒ atoms closer to each other

greater bandwidth

• bandwidth of the low-lying levels < bandwidth of the higher energy

levels

導體 , 絕緣體 及 半導體 ( Conductors, Insulators, and Semiconductors )

2N

6N

2N

2N

⇑

# of e- states in the band= 2 • ( 2l + 1 ) • Nl : orbital quantum no.N : # of atoms

例 : Na crystal : 1s2 2s2 2p6 3s1

( 11 electrons/atom )

N atoms in the solidtotal of 11N e-

when ε appliede- gain energymove into empty, slightly higher energy statescurrent conduction

valence band : the highest band containing e-

conduction band : the band e- in which can conduct net current( in this case both are the 3s band )

( * in actual case, for Na, 3s band and 3p band overlap )

例 : Mg crystal : 1s2 2s2 2p6 3s2

( 12 electrons/atom )

N atoms in the solidtotal of 12N e-

for Mg, 3p band and 3s band overlap

when ε appliede- gain energymove into empty, slightly higher energy statescurrent conductionMg crystal is a conductor

例 : C crystal : 1s2 2s2 2p2

( 6 electrons/atom )

N atoms in the solidtotal of 6N e-

when ε applied ( @ T = 0 K or low T )e- has no higher energy level availableno electron conduction

conduction band is separated from valence band

band gap ( Eg ) : energy gap between conduction band and valence band

• C ( diamond ), Si, and Ge have similar band structures

• EgC (diamond) ~ 6 eVSi ~ 1.1 eVGe ~ 0.7 eV

6N

2N

4N

4N C : 2s 2pSi : 3s 3pGe : 4s 4pSn : 5s 5pPb : 6s 6p

interatomic distance ,

• at high temperatures⇒ some e- excited into conduction band free e-

⇒ create “holes” in the valence band effective free “+” charge• the probability of e- transition across the bang gap is very sensitive

to the magnitude of Eg• Eg determines whether a solid is an insulator or a semiconductor

• T ↑ free e- and holes ↑ conductivity ↑

Ge

Si

C ~ 6 eV insulator @ 300 K(diamond)Si ~ 1.1 eVGe ~ 0.7 eV semiconductor

等效質量 ( Effective Mass )

• when ε acts on a free e-

a = eε / m , m : mass of e-

• what if the e- is in a crystal under the influence of the lattice ion potentials ?

a = eε / m* , m* : effective mass of e-

m* = ?

• need to review group velocity first

• to describe a partially localized particle :mix a large number of sinusoidal traveling waves

Ψ( x,t ) = ∑ ∑ A( k,ω ) sin( kx – ωt )

Ψ( x,t ) = ∫ ∫ A( k,ω ) sin( kx – ωt ) dk dω

ω k ∞0

∞0

group velocity velocity of the particle

例 : Ψ1 = A sin( kx – ωt ) , Ψ2 = A sin[ ( k+Δk ) x – ( ω+Δω ) t ]assume Δk

The envelop travels at vgroup

vgroup = ——— = Δω / Δk ≈ dω / dk

λ = h / p λ = 2π / k υ = E / hυ = ω / 2π

vgroup = dω / dk = dE / dp = d( —— ) / d p = 2p / 2m = m vparticle / m = vparticle

⇒ vgroup = vparticle

k = p / h dk = dp / h

ω = E / h dω = dE / h

@ t=0

sin kx

cos Δkx/2x

λ = 2π/k , 2π/(Δk/2) >> 2π/k

Δω/2Δk/2

p22m

vgroup

vgroup

• vgroup = dE / dp• since in quantum mechanics, E is often expressed in terms of the

wave vector k : change dE/dp dE/dkp = h/λ , λ = 2π/k p = h k dp = h dk⇒ vgroup = ( 1/h ) dE/dk

• dE = dW ( work done on the particle )

= eε dx = eε ( dx/dt ) dt = eε vparticle dt = eε vg dt

dE/dt = eε vg• a = dvparticle/dt = dvg/dt = — — —— = — — ——

= ( eε / h ) dvg/dk = — —— eε eε = ———— a

⇒ m* = h2 / ( —— )

1 d dEh dt dk

1 d dEh dk dt

1 d2Eh2 dk2

h2

d2E/dk2

d2Edk2

例 : for a free electron E = h2k2 / 2m , m* = h2 / ( —— ) = m

例 : for an electron in a crystal (1) m* is not always equal to m(2) m* can be > m or even ∞(3) m* can be < m or even < 0

d2Edk2

• in classical free electron model : σ = q2Nτ / m , m : free e- mass

can be modified into :σ = q2Nτ / m* , m* : effective mass of e- in the crystal

for metals such as Cu, Na, Al, K : m* ≈ mfor metal such as Fe : m* ≈ 10 m Fe is not a very good conductor

• at the top of the allowed energy band, m* < 0

• a semi-classical view, when an external ε is applied :

a = - ⎪e⎪ε / m* ( e- with m* > 0 ) and ( e- with m* < 0 ) accelerate in opposite directions

( e- with m* > 0 ) and ( e- with m* < 0 ) drift in opposite directionsin a filled valence band, currents from ( e- with m* > 0 ) and ( e- with m* < 0 ) cancel each other, and cause no net current

eε = ———— a = m* a

m* < 0 e- accelerates in the direction opposite to classical e- ( m > 0 )e- drift ( vd ) in the direction opposite to classical e- ( m > 0 )

but why ?

h2

d2E/dk2

vd

vd

+ –ε

classical free e-

QM free e- with m* < 0

from Bragg reflection condition : when 2d sinθ = 2d = nλ = n 2π / k , i.e. k = nπ / d

electron wave reflected by the lattice ions(1) near the bottom of the band, k ~ 0

far from Bragg condition, no reflection

e- accelerated by ε ( m* > 0 ) (2) near the top of the band, k nπ / d

strong reflection opposite to ε acceleration

reflection overcome ε acceleration ( m* < 0 )(3) at a certain k in the middle of the band

reflection cancels ε acceleration no change in e- velocity m* approaches ∞

• a semi-classical view, when an external ε is applied : a = - ⎪e⎪ε / m* ( e- with m* > 0 ) and ( e- with m* < 0 ) accelerate in opposite directions

( e- with m* > 0 ) and ( e- with m* < 0 ) drift in opposite directionsin a filled valence band, currents from ( e- with m* > 0 ) and ( e- with m* < 0 ) cancel each other, and cause no net current

e- with m* < 0

e- with m* > 0

e- with m* < 0

e- with m* > 0

e- with m* < 0

e- with m* > 0

conductor :conduction/valence band

filled e- states

filled e- statesInsulator/semiconductor : @ 0 K , valence band

電洞 ( Holes )

• an empty state in the valence band

hole

• conduction by e- in the valence band= conduction by positive charge of

positive effective mass ( i.e. hole )• the number of +q, +m particles ( holes )

= the # of empty states in the valence band• for perfectly pure semiconductor :

# of free e- = # of holes

• under ε , when valence band is filled with e- Jfull = 0 = Jremaining + Ji

• under ε , when electron i is removed from the valence band : Jremaining = Jfull – Ji , Ji : contribution from the i electronJremaining = - JiJi = –⎪e⎪vi , vi : drift velocity of i electron would acquire from ε

Jremaining = ⎪e⎪vi = ⎪e⎪aiτi , ai = - ⎪e⎪ε / mi* , τi : relaxation time( mi* : effective mass of electron i , mi* < 0 )

ai = ⎪e⎪ε / ⎪mi*⎪ Jremaining = ⎪e⎪2ετi / ⎪mi*⎪

⇒ a positive charge with positive effective mass

electron i

霍爾效應的應用 : 判別移動電荷的正負性

移動電荷為負電荷

移動電荷為正電荷

VH 正負極性會不同

W

FE

FEFB

FB

FE

FE

FB

VH = ( 1/Nq ) IB/d = RH IB/d

RH ( m3/C )

Li -17 x 10-11

Na -25 x 10-11

Be 24 x 10-11

Zn 3 x 10-11

Cd 6 x 10-11