computational solid state physics 計算物性学特論 第4回 4. electronic structure of...
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Computational Solid State Physics
計算物性学特論 第4回
4. Electronic structure of crystals
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Single electron Schroedinger equation
)()(
)(2
22
rr
r
H
Vm
H
nmmn
nnn
Sd
a
rrr
rr
)()(*
)()(
Expansion by base functions Φn
: overlap integral
m: electron mass
V(r): potential energy
h: Planck constant
2h
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mnnm
nmn
nnmn
n
nnnm
nnnm
nnn
nnn
HdH
dadHa
dadaH
aaH
r
rr
rr
*
**
**
n
nmnn
nmn aSaH
:matrix element of Hamiltonian
:algebraic equation
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2
1
a
a
a
2221
1211
HH
HH
H
2221
1211
SS
SS
S
aa SH
mnnm HdH r *
mnnm Sdrrr )()(*
:expression of algebraic equation by matrixes
and vectors
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aa SH
nmnmS
0||det
IH
H
aa
nmnmI
: ortho-normalized bases
: unit matrix
eigenvalue equation
condition of existence of inverse matrix of
secular equation
)( IH
Solution (1)
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Solution (2)
nmnmS
aa HS 1
0||det 1 IHS
aa SH
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Potential energy in crystals
mln
VV
cbaR
rRr
)()(
a,b,c: primitive vectors of the crystaln.l.m: integers
G
G rGr ivV exp)( G: reciprocal lattice vectors
:periodic potential
Fourier transform of the periodic potential energy
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Primitive reciprocal lattice vectors
baG
acG
cbG
c
b
a
2
2
2
0
2
cGbG
aG
aa
a
cba
3)2( cbaG GGGV
: volume of a unit cell
Volume of 1st Brilloluin zone
Properties of primitive reciprocal lattice vectors
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Bloch’s theorem for wavefunctions in crystal
)()exp()( rRkRr kk ii i
)(exp()( rr)ukr kk ii i
)()( rRr ikik uu mln cbaR
(1)
(2)
k is wave vectors in the 1st Brillouin zone.
Equations (1) and (2) are equivalent.
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Plane wave expansion of Bloch functions
)(]exp[)( rrkr kk ii ui
G
Gk,k rGr ]exp[)( icu ii
G
Gk, rGkr )(exp)( iciik
)()( rRr ikik uu
G : reciprocal lattice vectors
mln cbaR
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Normalized plane wave basis set
)()exp()(
)()(*
])(exp[1
)(
''
rRkRr
rrr
rGkr
kGkG
GGkGkG
kG
i
d
iV
kG :satisfies the Bloch’s theorem
V : volume of crystal
mln cbaR
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Schroedinger equation for single electron in crystals
'
02
2
)(
)(2
)(
GGvH
vm
H
GG'
GG
k
Gkk
2
1
G
G
a
a
a
0|)()(|det
IkkH
H
aa
: Bragg reflection
G
G rGr ivV exp)( : potential energy in crystal
: secular equation to obtain the energy eigenvalue at k.
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Energy band structure of metals
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Zincblende structure
a
bc
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Brillouin zone for the zincblende lattice
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Empirical pseudopotential method
Energy band of Si, Ge and Sn
Empirical pseudopotential method
Si Ge Sn
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Tight-binding approximation
nlmiii
ijii
j
mlnmlniN
c
)()](exp[1
)(
)()(
cbarcbakr
rr
k
kk
)()exp()( rakar kk ii i
)( ii mln cbar i-th atomic wavefunction at (n,l,m)-lattice sites
Linear Combination of Atomic Orbits (LCAO)
satisfies the Bloch theorem.)(rik
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1-dimensional lattice (1)
a
'
'
,
'
)()exp(
)(])'()(exp[1
)()(*)'exp(1
)()(*
)()exp(1
)(
kk
lkk
mmn
nm crystal
ss
crystal
kk
sk
lSikal
mnSmkkimnikaN
dmnamikikanN
d
nikanN
rararrrr
arr
S(n-m)
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rrr
rrr
rr
d
dHk
kH
kk
kk
kk
)()(*
)()(*)(
)()()(
)exp()exp(
)()(*)exp(
)'()(*)]'(exp[1
)()(*
101000
'
ikaHikaHH
dHnikan
dnHnnnikaN
dH
n
nnkk
rrar
rararrrr
:Schroedinger equation
1-dimensional lattice (2)
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1-dimensional lattice (3)
)cos(2)( 0 katk
ε0=H00: site energy
t=H10=H-10: transfer energy
ka
ε(k)/-t
ak
a
t < 0
Energy dispersion relation
1st Brillouin zone
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Valence orbits for III-V compounds
4 bonds
;4/]111[
;4/]111[
;4/]111[
;4/]111[
4
3
2
1
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Matrix elements of Hamiltonian between atomic orbits
ssV
spV
ssV
ssV
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Matrix element of Hamiltonian between atomic orbit Bloch functions
ksksss acac HkH ||)(
nlm
cnlmsnlmksi
Nc )()exp(
1)( τRrRkr
mlnnlm cbaR
nlm
anlmsnlmksi
Na )()exp(
1)( τRrRkr
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)()( 0 kgVkH ssss ac
Calculation of Hamiltonian matrix element
i
dikssss
iac eVkH )(
][ 4/)(4/)(4/)(4/)( akkkiakkkiakkkiakkkiss
zyxzyxzyxzyx eeeeV
4321
4321
4321
4321
)(
)(
)(
)(
3
2
1
0
dikdikdikdik
dikdikdikdik
dikdikdikdik
dikdikdikdik
eeeekg
eeeekg
eeeekg
eeeekg
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Matrix element between atomic orbits
ppppxy
ppppxx
spsp
ssss
VVE
VVE
VE
VE
3/13/1
3/23/1
3/ 2/1
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Hamiltonian matrix for the zincblende structure
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1-fold
Bottom of conduction band: s-orbit
Top of valence band: p-orbit
0)0()0()0(
4)0(
321
0
ggg
g
Energy at Gamma point (k=0)
3-fold
;422
;422
2
2
2
2
xx
ap
cp
ap
cp
ss
ax
cs
as
cs
EE
EE
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Energy band of Germanium
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Energy band of GaAs, ZnSe, InSb, CdTe
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Spin-orbit splitting at band edge
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Efficiency and color of LED
Periodic table
B C N
Al Si P
Ga Ge As
In Sn Sb
PL energy is determined by the energy gap of direct gap semiconductors.
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Bond picture (1): sp3 hybridization
]|||[|2
1|
]|||[|2
1|
]|||[|2
1|
]|||[|2
1|
4
3
2
1
zyx
zyx
zyx
zyx
pppsh
pppsh
pppsh
pppsh
ijjihh |
[111]
[-1-1-1]
[-11-1]
[-1-11]
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Bond picture (2)
32
23
VV
VVH
3V
Hamiltonian for two hybridized orbits
2V
22
23 VV
: hybridized orbit energy
: transfer energy
bonding and anti-bonding states
Successive transformations of linear Combinations of atomic orbitals, beginning with atomic s and p orbitals and proceeding to Sp3 hybrids, to bond orbitals, and finally to band states. The band states represent exact solution of the LCAO problem.
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Problems 4
Calculate the free electron dispersion relation within the 1st Brillouin zone for diamond structure.
Calculate the energy dispersion relation for a graphen sheet, using a tight-binding approximation.
Calculate the dispersion relation for a graphen sheet, using pane wave bases.