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Lundstrom ECE-606 S13
Notes for ECE-606: Spring 2013
L3: Quantum Mechanics
Professor Mark Lundstrom Electrical and Computer Engineering
Purdue University, West Lafayette, IN USA [email protected]
1 1/14/13
Lundstrom ECE-606 S13 2
tetrahedral bonding in the diamond lattice
4 nearest neighbors
J. S. Bhosale, Ph.D. defense, 1/11/13
! = 109.5!
Lundstrom ECE-606 S13 3
Anisotropic etching of Si
P. Campbell, P. M.A. Green, “Light trapping properties of pyramidally textured surfaces,” J. Appl. Phys. 62, 243-249 (1987).
Lundstrom ECE-606 S13 4
Anisotropic etching of Si
370 ! 400 µm
24.5% at 1 sun Martin Green Group UNSW – Zhao, et al, 1998
Lundstrom ECE-606 S13 5
motivation for quantum mechanics 1) Black body radiation (Planck)
2) Photoelectric effect (Einstein)
3) Atomic spectra (Bohr)
4) Wave-particle duality (de Broglie)
E = hf = !!
E = hf = !!
p = !k = h !
Lundstrom ECE-606 S13 6
Schroedinger wave equation KE + PE = E
! x,t( ) = ei kx"#t( )
KE = p2
2m0
!i!!x
" x,t( ) = !k" x,t( )
p! x,t( ) = !k! x,t( )
p ! !
i""x
p p! x,t( )"# $% = !k( )2
! x,t( )
!!2 "2
"x2 # x,t( ) = !k( )2# x,t( )
p2! x,t( ) = "!2 #2
#x2 ! x,t( )
p2
2m0
! x,t( ) = "!2
2m0
#2
#x2 ! x,t( )
= !2k 2
2m0
! x,t( )= KE! x,t( )
Lundstrom ECE-606 S13 7
Schroedinger wave equation
KE( )! x,t( ) + PE! x,t( ) = E! x,t( )
! x,t( ) = ei kx"#t( )
PE =U x( )
! !
i""t
# x,t( ) = !$ # x,t( )
!!2
2m0
"2
"x2 # x,t( ) +U x,t( )# x,t( ) = E# x,t( )
E = ! !
i""t
E! x,t( ) = !" ! x,t( )
!!2
2m0
"2
"x2 # x,t( ) +U x,t( )# x,t( ) = ! !i"# x,t( )
"t
Lundstrom ECE-606 S13 8
Time-independent Schroedinger wave equation
! x,t( ) =" x( )# t( )
!!2
2m0
"2
"x2 # x,t( ) +U x,t( )# x,t( ) = ! !i"# x,t( )
"t
!!2
2m0
"2# x( )"x2 $ t( ) +U x,t( )# x( )$ t( ) = ! !
i"$ t( )"t
# x( )
!!2
2m0
"2# x( )"x2
1# x( ) +U x( ) = ! !
i"$ t( )"t
1$ t( )
“separation of variables”
Lundstrom ECE-606 S13 9
Time independent Schroedinger wave equation
!!2
2m0
"2# x( )"x2
1# x( ) +U x( ) = c
! !
i"# t( )"t
1# t( ) = c
! !
i"# t( )"t
= c# t( )
! x,t( ) =" x( )# t( ) =" x( )e$ i%t
c = !! = E ! !
i"# t( )"t
= !$# t( ) = c# t( )
!!2
2m0
d 2" x( )dx2 +U x( )" x( ) = E" x( )
! = E!
! t( ) = e" i#t
Lundstrom ECE-606 S13 10
Time independent Schroedinger wave equation
!2" x( )!x2 +
2m0
!E #U x( )$% &'" x( ) = 0
!!2
2m0
d 2" x( )dx2
1" x( ) +U x( )
E >U x( )
k 2 =
2m0
!E !U x( )"# $%
! x( ) = Ae± ikx
E <U x( )
!2" x( )!x2 #$ 2" x( ) = 0
! 2 =
2m0
!U x( )" E#$ %&
! x( ) = Ae±"x
Lundstrom ECE-606 S13 11
solutions of the SE
x
E U x( )
E1
! x( ) = Ae± ikx
! x( ) = Ae"#x
! x( ) = Ae+"x
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quantum effects
1) Plane waves
2) Quantum confinement
3) Quantum reflection and tunneling
4) Plane waves and quantum confinement
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Plane waves
d 2! x( )dx2 +
2m0
!E "U0#$ %&! x( ) = 0
d 2! x( )dx2 + k 2! x( ) = 0
! x( ) = Ae± ikx
! x( ) = 1
Leikx (normalized in 1D)
Lundstrom ECE-606 S13 14
particle in a box
x0 W
U = 0
d 2! (x)dx2
+ k2! = 0
! (x) = sin knx
!n =!2kn
2
2m* =!2n2" 2
2m*W 2
!1
!2EF
light mass narrow width ! high subband energy
k2 = 2m
*E!2
!U x( )
Lundstrom ECE-606 S13 15
carrier densities
x0 W
U = 0
!1
!2EF
W0
classical
n(x)
x
quantum mechanical
n(x)!" * x( )" x( )
!U x( )
Lundstrom ECE-606 S13 16
triangular quantum well
!
x
U(x) = qExx
x
! (x)
x =2Ei
3qE
! (x) : Airy functions
! 1
!i =3hqEx
4 2m*i + 3 4( )"
#$
%
&'
2 /3
i = 1,2,3,...
!U x( )
Lundstrom ECE-606 S13 17
quantum effects
1) Plane waves
2) Quantum confinement
3) Quantum reflection and tunneling
4) Plane waves and quantum confinement