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C. Bulutay Lecture 2Topics on Semiconductor Physics
� Electronic Bandstructure: General Info
In This Lecture:
C. Bulutay Lecture 2Topics on Semiconductor Physics
Electronic Bandstructure
all-electron
(FPLAPW)
valence electronPAW
Acronyms
FPLAPW: Full-potential linearized
augmented plane wave
PAW: Projector augmented wave
LMTO: Linearized muffin tin orbital
EPM: Empirical pseudopotential
method
ETB: Empirical tigh-binding
ab-initio semi-empirical
planewave pseudopotential
(FP) LMTO EPM ETB k·p
C. Bulutay Lecture 2Topics on Semiconductor Physics
Perfect Crystal Hamiltonian (cgs units)
2 2 2 22' ''
, ' , ' ,''
1 1
2 2 2 2
j i i i i
j j j i i i i jj i i ij j j i
p P e Z Z e ZeH
m M R Rr r r R= + + + −
−− −∑ ∑ ∑ ∑ ∑� � �� � �
over all e’s e-nucleiover all nuclei
� 1st Approximation: core vs. valence e’s
Still Eq. Above Applies with:
core e’s+nucleus
e’s
ion core
valence e’s
semicore
C. Bulutay Lecture 2Topics on Semiconductor Physics
� ions are much heavier (> 1000 times) than e’s
So,
for e’s: ions are essentially stationary (at eql. lattice sites {Rj0})
� 2nd Approximation: Born-Oppenheimer or adiabatic approx.
for ions: only a time-averaged adiabatic electronic potential is seen
In other words,
using the adiabatic approximation, we separate the (in principle
non-separable) perfect crystal Hamiltonian
C. Bulutay Lecture 2Topics on Semiconductor Physics
Under adiabatic approximation…
( ) ( ) ( )0, ,ion i e j i e ion j i
H H R H r R H r Rδ−= + +� � �� �
e-phonon interaction (resistance, superconductivity…)phonon spectrum electronic
band structure
Ref: Yu-Cardona
C. Bulutay Lecture 2Topics on Semiconductor Physics
Electronic Hamiltonian
2 22'
, ' ,' 0
1
2 2
j ie
j j j i jj j j j i
p e ZeH
m r r r R= + −
− −∑ ∑ ∑ �� � �
over all valence e’s >1023 cm-3
� 3rd Approximation: Mean-field Approximation
2
1 ( ); ( ) ( )2
e
pH V r V r R V r
m= + + =
�� � �
Density Functional Theory
VH+Vx+Vc
A direct lattice vector
C. Bulutay Lecture 2Topics on Semiconductor Physics
Translational Symmetry & Brillouin Zones
( ) ( ); : Translational Operator
Introduce a function, ( ) ( ), where ( ) ( ) with
( ) ( ) ( )
R R
ikx
k k k k
ikR
R k k k
T f x f x R T
x e u x u x nR u x n
T x x R e x
ψ
ψ ψ ψ
≡ +
≡ + = ∈
= + =
ℤ
eigenvalues of T
[ ]1
1
Since , 0,
Eigenfunctions of can be expressed also as
e R
e
H T
H
=
eigenfunctions of
2 2: is only defined modulo , i.e., and are equivalent
RT
nNB k k k
R R
π π+
eigenvalues of TR
C. Bulutay Lecture 2Topics on Semiconductor Physics
ax
Real space
Direct Lattice
Momentum space
Reciprocal lattice
k
One-dimensional Lattice
k=-π/a k=π/a
1st BZ
[Real Space] Primitive lattice vector: a
[Mom. Space] Primitive lattice vector: 2π/a
C. Bulutay Lecture 2Topics on Semiconductor Physics
Three-dimensional Lattice
1 2 3
1 1 2 2 3 3
1 2 3
[Real space] , ,
A general direct lattice vec
Prim
tor (DLV): ;
[Mom. space] , ,
itive lattice vectors:
Primitive lattice vectors:
i
j k
a a a
R n a n a n a n
b b b
a a
= + + ∈
×
� � �
� � � �ℤ
� � �
� �� Volume of the real
1
where, 2(
j k
i
a ab
aπ
×≡
×
� ��
� �2 3
1 1 2 2 3 3
)
A general reciprocal lattice vector (RLV): ;
: 2
i
i j ij
a a
G n b n b n b n
NB b a πδ
⋅
= + + ∈
⋅ =
�
� � ��ℤ
� �
Volume of the real space primitive cell
1st Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice
C. Bulutay Lecture 2Topics on Semiconductor Physics
Direct vs. Reciprocal Lattices
1ˆ
ˆ
a a x
a a y
=
=
�
�
1
2ˆ
2
b xa
π
π
=�
�
� SC
Direct Lattice Reciprocal Lattice
also a SC lattice!2
3
ˆ
ˆ
a a y
a a z
=
=
�
� 2
3
2ˆ
2ˆ
b ya
b za
π
π
=
=
�
�
also a SC lattice!Primitive translation
vectors
1st BZ of SC lattice is again a cube
C. Bulutay Lecture 2Topics on Semiconductor Physics
� BCCDirect Lattice Reciprocal Lattice
forms an fcc lattice!
1
2
3
ˆ ˆˆ( )2
ˆ ˆˆ ( )2
ˆ ˆ ˆ ( )2
aa y z x
aa z x y
aa x y z
= + −
= + −
= + −
�
�
�
1
2
3
2ˆ ˆ( )
2ˆˆ ( )
2ˆ ˆ ( )
b y za
b z xa
b x ya
π
π
π
= +
= +
= +
�
�
�
a is the side of the conventional cube
1st BZ of bcc lattice
a is the side of the conventional cube
Ref: Kittel
C. Bulutay Lecture 2Topics on Semiconductor Physics
� FCCDirect Lattice Reciprocal Lattice
forms an bcc lattice!
a is the side of the conventional cube
1
2
3
ˆ ˆ( )2
ˆˆ ( )2
ˆ ˆ ( )2
aa y z
aa z x
aa x y
= +
= +
= +
�
�
�
1
2
3
2ˆ ˆˆ( )
2ˆ ˆˆ ( )
2ˆ ˆ ˆ ( )
b y z xa
b z x ya
b x y za
π
π
π
= + −
= + −
= + −
�
�
�
1st BZ of fcc lattice
a is the side of the conventional cube
Ref: Kittel
C. Bulutay Lecture 2Topics on Semiconductor Physics
fcc 1st BZ Cardboard Model
Truncated Octahedron
Ref: Yu-Cardona
Ref: Wikipedia
C. Bulutay Lecture 2Topics on Semiconductor Physics
More on Symmetry
� The star of a k-point
All have the same energy eigenvalues
Symmetry of
the direct latticeSymmetry of the
reciprocal lattice
� Wavefunctions can be expressed in a form such that they have definite
transformation properties under symmetry operations of the crystal
Selection rules: certain matrix elements of certain operators vanish
identically…
� Formal analysis is remedied by the use of Group theory
C. Bulutay Lecture 2Topics on Semiconductor Physics
Symmetry Points & Plotting the Band Structure
Diamond BZ
0
5
EPM Bandstructure of Si
-15
-10
-5
KLWX ΓΓ
En
erg
y (
eV
)
C. Bulutay Lecture 2Topics on Semiconductor Physics
Bloch Functions vs Wannier Functions
�Felix Bloch provided the important
theorem that the solution of the
Schroedinger equation for a periodic
potential must be of the special form:
Cell-periodic functions Orthonormality:* ( ) ( )
nnnk n k kkr r drψ ψ δ δ′′ ′ ′
=∫ � � ��� � �
( ) ( ), with ( ) ( ),ik r
nk nk nk nkr e u r u r u r Rψ ⋅= = +
� �
� � � �
�� � � �
Transformation relations
Wannier functions
all space
( ) ( ) nnnk n k kk
r r drψ ψ δ δ′′ ′ ′=∫
is the crystal momentum (more on this later)
1( ; ) ( ),
1( ) ( ; )
i
i
i
nk nk nk nk
ik R
n i nkk
ik R
n inkR
k
a r R e rN
r e a r RN
ψ
ψ
− ⋅
⋅
=
=
∑
∑
� �
�
�
� �
�
�
�ℏ
�� �
�� �
C. Bulutay Lecture 2Topics on Semiconductor Physics
Bloch vs. Wannier Functions
xBloch functions are extended
x
Wannier fn’s are localized around lattice sites Ri
Wannier form is useful in describing impurities, excitons…
But note that the Wannier functions are not unique!
C. Bulutay Lecture 2Topics on Semiconductor Physics
Crystal Momentum
( ) ( )ik r
nk nkr T e rψ ψ⋅+ =
� �
� �
�� �
determines the phase factor by which a BF is multiplied under a translation in real space
k�
k�
labels different eigenstates together with the band index n
�k�
is determined up to a reciprocal lattice vector; this arbitrariness can be removed by restricting it to 1st BZ
A typical conservation law in a xtal: k q k G′+ = +� � ��
Any arbitrariness in labelling the BFs can be absorbed in these additive RLVs w/o changing the physics of the process
Physically, the lattice
supplies necessary
recoil momentum so
that linear momentum is exactly conserved
C. Bulutay Lecture 2Topics on Semiconductor Physics
Be ware of the Complex Bandstructure
( )ik r
nke u r
⋅� �
��
“Real” bandstructure of Si
What if we allow k to become complex?
Ref: Chang-Schulman PRB 1982