In 1926, Max Born postulated that x [] - Dongguk · 2011-12-28 · Energy Band - Band in Si crystal...
Transcript of In 1926, Max Born postulated that x [] - Dongguk · 2011-12-28 · Energy Band - Band in Si crystal...
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