Notes for ECE-606: Spring 2013 Energy Band Essentials Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA [email protected]1 1/29/13 Lundstrom ECE-606 S13 2 Steve Chu’s advice “Learning science and thinking about science or reading a paper in science is not about learning what a person did. You have to do that, but to really absorb it, you have to turn it around and cast it in a form as if you invented it yourself. … How do you do that? … try to internalize it in such a way that it really becomes intuitive.
Transcript
Lundstrom ECE-606 S13
Notes for ECE-606: Spring 2013
Energy Band Essentials
Professor Mark Lundstrom
Electrical and Computer Engineering Purdue University, West Lafayette, IN USA
“Learning science and thinking about science or reading a paper in science is not about learning what a person did. You have to do that, but to really absorb it, you have to turn it around and cast it in a form as if you invented it yourself. … How do you do that? … try to internalize it in such a way that it really becomes intuitive.
Lundstrom ECE-606 S13 3
free electron
E
U = 0
x→ +∞−∞← x
ψ x, y, z( )∝ e+ ikxxe± iky ye± ikzz
ψ x, y, z( )∝ e− ikxxe± iky ye± ikzz
Lundstrom ECE-606 S13 4
electron in a very large box E
x =Wx = 0
U =U0U =U0
ψ x, y, z( )∝ sin knx( )e± iky ye± ikzz
U = 0
εn =
2n2π 2
2m*W 2
W >>> a
εn <<< kBT q
L >>> a
L <<<W
Lundstrom ECE-606 S13 5
electron in a very large box E
x =W
U =U0U =U0
x = 0
ψ x, y, z( )∝ e± ikxxe± iky ye± ikzz
U = 0
Lundstrom ECE-606 S13 6
electron in a very large box E
U =U0U =U0
L >>> a
L <<<W
ψ x( )∝ e± ikxx → u x( )e± ikxx
u x( )
x = 0 x =W
u x( ) = u x + a( )
Lundstrom ECE-606 S13 7
bulk semiconductor, x-direction
( ) ( ) xik xkx u x eψ =
, xx k0 xL
( ) ( )0 1x xik LxL eψ ψ= → =
2 1,2,3,...x xk L j jπ= =
2x
x
k jLπ=
2
xLπ
dk
( )# of states 22
xk
x
dk N dkLπ
= × =
= = density of states in -spacexkLN kπ
xk aπ=
Lundstrom ECE-606 S13 8
counting states
( ) ( ) xik xkx u x eψ =
, xx k
0 xL
( ) ( )0 1x xik LxL eψ ψ= → =
2 1,2,3,...x xk L j jπ= =
2x
x
k jLπ=
2
xLπ
dkx
x AL N a=
2 2x
x A
jk jL a Nπ π= =
2xA
j xik x iN a
θ π⎛ ⎞
= = ⎜ ⎟⎝ ⎠
max Aj N= max2kaπ=
0 xkaπ+aπ−
Lundstrom ECE-606 S13 9
density of states in 1D
x
yz
N = f0
k∑ Ek( )
( )01
L k xj BZ
n f E dkπ
=∑ ∫( ) -10
1 cmx
L kkx
n f EL
= ∑
sum over subbands
=22kL LNπ π
⎛ ⎞× =⎜ ⎟⎝ ⎠Lx
nL =
1Lx
f0 Ek( ) Nk dkxBZ∫
Lundstrom ECE-606 S13 10
density of states in 2D
x
y
z
A
t
( )0 kk
N f E=∑r
nS =
j∑ 1
2π 2 f0 Ek( ) dkx dkyBZ∫
nS =
1A
f0k∑ Ek( ) cm-2
2 2=24 2kA ANπ π
⎛ ⎞× =⎜ ⎟⎝ ⎠
Lundstrom ECE-606 S13 11
density of states in k-space
Nk =2 ×
L2π
⎛⎝⎜
⎞⎠⎟=
Lπ
Nk =2 ×
A4π 2
⎛⎝⎜
⎞⎠⎟=
A2π 2
Nk =2 ×
Ω8π 2
⎛⎝⎜
⎞⎠⎟=
Ω4π 3
1D:
2D:
3D:
dk
x ydk dk
x y zdk dk dk
12
recap 1) Within any finite region with N atoms, we have N discrete k-
states. 2) When the region is large, the allowed k’s are spaced VERY
closely, so we can integrate rather than sum (using the density of states in k-space).
3) We know the allowed k-states, but to get the allowed
energies, we must solve the wave equation for the given crystal potential, u(x).
4) But the unique solutions are all contained in a Brillouin zone
(0 to 2pi/a, or –pi/a to +pi/a, or in a more complicated volume of k-space in 3D. Lundstrom ECE-606 S13
Lundstrom ECE-606 S13 13
solving the wave equation
−2
2m0
∇2 + u r( ) + q2ψ j
* ′r( )ψ jr( )
r − ′rd ′τ∫
j≠ i∑
⎡
⎣⎢⎢
⎤
⎦⎥⎥ψ ir( ) = ε iψ i
r( )
ψr( ) = u r( )ei
k ir
u r( ) = u r +
R( )
1) Pick a k 2) Solve for the eigen energies
E
kx−π a +π ak1
(Hartree approximation)
Lundstrom ECE-606 S13 14
bandstructure
(“Bandstructure Lab” at www.nanoHUB.org)
silicon GaAs
15
model bandstructures
silicon
100< >111< >
L X1.12eV
hh
lh *00.16lhm m=
*00.49lhhm m=
*00.92m m=l
*00.19tm m=
kr
E k( ) = EV −
2k 2
2mp*
E
GaAs
100< >111< >
L X1.42eV
hh
lh
*00.063m m=
E
kr
Lundstrom ECE-606 S13
16
GaAs model bandstructure GaAs
100< >111< >
L X1.42eV
hh
lh
*00.063m m=
E
k
E k( ) = EC +
2k 2
2mn*
k 2 = kx
2 + ky2 + kz
2
E k( )− EC =
2 kx2 + ky
2 + kz2( )
2mn*
The constant energy surface for the conduction band is a sphere in k-space. Lundstrom ECE-606 S13
17
Si model bandstructure
E k( ) = EC +
2 kx2 + ky
2( )2mt
* +2kz
2
2m*
The constant energy surface is an ellipsoid in k-space.
