Schrödinger's wave equationSchrödinger's wave equationg qg q

Wave function, - Meaning of the wave equation( , ) = ( ) ( )x t x tΦ ψ ⋅φ

In 1926, Max Born postulated that

[ ]2

2 2

( ) 2m E - V(x) ( ) 0x xx

∂ ψ+ ψ =

∂2( ) ( ) ( )*x x xψ = ψ ⋅ψ

E p

← E = KE + PE( ) ( ) ( )Probability of finding particle between

x ~ x x

x x xψ ψ ψ

≡+ Δ

2pParticle + V mv2m

Wave k

⇒

⇒ ω

)x( 2ψ

⇒ Time independent Schrödinger eq.

(Energy conservation)

Wave k⇒ ω3cmparticles/ of #

(Energy conservation)

V(x) : Potential energy

E( ) T t l f ti l + Δ

물리 전자/김삼동2-8

E(x) : Total energy of particles x x + Δx

SchrSchröödinger equationdinger equation

- Schrödinger's paradox

?

TNT

e-

물리 전자/김삼동5-9

One electron systemOne electron systemyy

Hydrogen tom By solving the Schrödinger eq., we have a total energy of the electron:zz

θ

P (r,θ, φ )-e

4o

H 2 2o

m e 13.6E eV, n =1, 2, 3, ...2(4πε hn) n

=− =−

, where n = principle quantum numbery

rNucleus

+e

⇒ Energy is quantizedx

φ

V (r)

r

- e 2

4πεore

V (r) = (SI unit)

물리 전자/김삼동2-10

+ e

One electron systemOne electron systemyy

- Quantum numbers

Th 4 diff t t b th t

- Degeneracy and Pauli Exclusion Principle

: There are 4 different quantum numbers; they are not independent, but are related by 1s 2s 2p

n=1

n = 1, 2, 3, : Principal quantum number

n=1 ℓ= 0n=2

ℓ= 0n=2

ℓ= 1

= n - 1, n - 2, n - 3, , 0 : Angular quantum number

m = 0, 1, 2, , , : Magnetic quantum number± ± ±

1

ℓ Spectroscopic symbol0 s (sharp)1 p (principal)1s = : Spin quantum number

2±

1 p (principal)2 d (diffuse)3 f4 gg. .. .

물리 전자/김삼동2-11

One electron systemOne electron systemyy

n = 1 : 1s ms = ±½ 2-fold degeneracyn = 1 n = 2

R1,0 R20

n = 2 :

ℓ= 0, 1

2 ±½ 2 f ld0

2 s

1,0 20

2s ms = ±½ 2-fold

2p 3-fold (mℓ= -1, 0, 1) × 2-fold (ms = ±½ ) = 6-fold

0 0.2 0.4 0.6 0.802 p

0 0.2 0.4

1s

0

R21

n = 3 :

ℓ= 0, 1, 2

3s m = ±½ 2-fold

r (nm)r (nm)

n = 2r2 |R2 0|23s ms = ±½ 2-fold

3p 3-fold (mℓ= -1, 0, 1) × 2-fold (ms = ±½ ) = 6-fold

4f 5-fold (mℓ= -2,-1, 0, 1, 2) × 2-fold (ms = ±½ )

n = 1r2 |R1,0|2

2s

| 2,0|

= 10-fold

M lti li it f ℓ 2 (2ℓ + 1) 0 02 0.4

1s

0 0

2 pr2 |R2,1|2

물리 전자/김삼동2-12

Multiplicity of ℓ = 2 (2ℓ + 1) 0 0.2 0.40 0.2 0.4 0.6 0.8

0

r (nm)r (nm)

Energy BandEnergy Bandgygy

Bond and Band - Band

E i t- Bond

• Spatial picture• Shared valence electrons form covalent bonding

• Energy picture• Shared valence electrons form electron filled

valence band and empty conduction band by QMEnergy band gap ; energy band that does not allowShared valence electrons form covalent bonding

• Bond breaking makes free moving conduction electrons that carries charge

• Need energy (or temperature) to break bond

• Energy band gap ; energy band that does not allow electrons

• Need energy as much as Eg to form conduction electronsgy ( p ) e ect o s

SiSiSiEnergy

Si Si Si

SiSiSi

Eg

conduction band

Si SiSi

Si Si

valence band

물리 전자/김삼동2-13

x

Energy BandEnergy Bandgygy

Formation of energy band Consider two racing cars

Energy lowering by solid formation. How? Why?

As atoms approach, the atomic wave function overlaps,

no interaction

v vwhich means that two electrons interact.

→ Larger overlapping between the valence band

orbitals.

require the same powersv v

The Pauli exclusion principle dictates not to have the

same quantum state v vq

→ splitting of energy levels leading car: more power, second car: less power

물리 전자/김삼동2-14

Energy BandEnergy Band

- Band in Si crystal

2 2 6 2 2

gygy

14 e-s → 1s2 2s2 2p63s2 3p2 (4 valence electrons)

Assume N atoms form Si solid → 3s band and 3p band interact (s-p perturbation) to→ 3s band and 3p band interact (s-p perturbation) to form sp3 hybridization

→ Conduction and valence bands are now neither s- Conduction and valence bands are now neither sorbital nor p-orbital, and has property of a mixture of them.

4N states

forbidden ( )

3p

conductionband

p like

s like

band (Eg)3s

atomvalence band

N atoms

p like

s like

물리 전자/김삼동2-15

solid(4N e-s)

4N states

Energy BandEnergy Bandgygy

- Band and bond model in Si crystals

at 0 oK

Electron breaks a covalent bond and becomes a free carrier

⇒ One electron has a transition from the valence b d t th d ti b d b ti i fband to the conduction band by generating a pair of negative and positive charges.

kT ~ 0.026 eVat RT

물리 전자/김삼동2-16

Energy BandEnergy Bandgygy

Charge carriers The current density due to motions of electrons is

NN

n ii=1

J = -e v∑

, where N = # of electrons/vol.

For the holes N

p ii=1

J = e v∑

Holes , where N = # of holes/vol.

물리 전자/김삼동2-17

Energy BandEnergy Bandgygy

Metal/Insulator/Semiconductors

- Metal : Partially filled band or overlapped CB & VB

- Semiconductor : Completely filled VB & Completely Metal InsulatorSemiconductorempty CB (at 0 K), Eg <3 eV,

- Insulator : Completely filled VB & Completely empty CB E

Metal InsulatorSemiconductor

(at 0 K), Eg >3 eV,

Electrons (or hole) can move only when there are empty

x

states nearby and current flows → Need to excite the e- in VB to CB in semiconductor & insulator or require overcoming the energy barrier Eg (by thermal optical etc)thermal, optical, etc)

Energy required to flow e- or current : M < SC < InsulatorCurrent flow or conductivity : M > SC > Insulator

물리 전자/김삼동2-18

Current flow or conductivity : M > SC > Insulator

Energy BandEnergy Bandgygy

E-k diagram h h 2p = mv = = = k2

π⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟λ λ⎝ ⎠⎝ ⎠: k = wave vector (number)

In fact, k is a vector.

2⎜ ⎟⎜ ⎟λ π λ⎝ ⎠⎝ ⎠

y

→direction // electron wave propagationmagnitude ~ electron momentumx

my

λ

2⎛ ⎞

= m = p v k

For a free particle, such as free electron,m

2y y cos x π⎛ ⎞= ⋅⎜ ⎟λ⎝ ⎠

2

2 22pE = k=

Free electron energy ∝ k2 : parabolic relationship

2k π=

λ“Wave number”

E k 2m 2m

물리 전자/김삼동2-19

Energy BandEnergy Bandgygy

Draw E vs. k

In general, E vs. k diagrams a function of the k-space direction in a crystal.

- Energy vs. Momentum (k-space)Spatial distribution cannot be described, but the band properties depending on the various crystal directions

: Quantum mechanically, the momentum of electrons and kinetic energy of electron is (near k=0)

can be described.

물리 전자/김삼동2-20

Energy BandEnergy Bandgygy

- E-k diagrams of GaAs and Si

물리 전자/김삼동2-21

Direct bandgap Indirect bandgap