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![Page 1: Lecture contents - University at Albanysoktyabr/NNSE508/NNSE508_EM-L9-bands-Bloch.pdfNNSE 508 EM Lecture #9 Nearly free electrons: 2D bands 10 Irrelevant to dimensionality, the following](https://reader033.vdocument.in/reader033/viewer/2022053019/5f22bd84bb208d539b598703/html5/thumbnails/1.jpg)
NNSE 508 EM Lecture #9
1
Lecture contents
• Bloch theorem
• k-vector
• Brillouin zone
• Almost free-electron model
• Bands
• Effective mass
• Holes
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NNSE 508 EM Lecture #9
2
Translational symmetry: Bloch theorem
)()( RrVrV 332211 amamamR
)()(2
2
rErrVm
p
If V(r) is a periodic function:
One-electron Schrödinger equation (each state can accommodate up to 2 electrons):
)()( ruer k
ikr
k
The solution is :
)()( Rruru kk
where uk (r) is a periodic function:
From:
• Linearity of the Schrödinger
equation
• Fourier theorem
22)()( Rrr kk
Quasi-wavevector k is analogous to a
wavevector for free electrons (V=const)
uk might be not a single valence electron
function but is close to linear combination of
valence electron wavefunctions
Important :
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NNSE 508 EM Lecture #9
3
• Introduced k-vector quantum number for
periodic potential (to enumerate states)
• Momentum is not conserved (not a quantum
number), however quasi-momentum is
conserved
• k-vector can be considered to lie in the first
Brillouin zone
• Solution with periodic boundary conditions
gives eigen-functions un,k for a given k which
forms orthogonal basis (compare with Fourier
expansion)
• n –values enumerate bands
• Electron occupying level with wavevector k in
the band n has velocity (compare to group
velocity)
Bloch theorem: consequences
)()( Rruru kk
)()()(1
2
22
ruErurVkim
kkk
)()( ruer k
ikr
k
)()(2
2
rErrVm
p
knkn Eru ,, ),(
)(1
)( kEkv nkn
22 2
2E k n
m a
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NNSE 508 EM Lecture #9
4 Reciprocal space (1D)
)()( ruer k
ikr
k Wavefunction of an electron in crystal :
1D reciprocal lattice vector :
m
kE
2
22
ma
b0
2
1D free electrons “band structure” is:
k’ k’-b
'
' ' ' 2,( ) ( ) ( ) ( )ik r ikr ibr ikr
k k k kr e u r e e u r e u r
periodic function
First Brillouin zone:
00 ak
a
2nd band
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NNSE 508 EM Lecture #9
5 Diamond or zinc-blende structures
• 4(Ga) + 4(As)=8 atoms in a cubic
unit cell
• 1+1 =2 atoms in a primitive unit cell
Brillouin zone (FCC):
Primitive unit cell (FCC):
0,1,12
1,0,12
1,1,02
03
02
01
aa
aa
aa
332211 bmbmbmb
...,,2 32
321
321
bb
aaa
aab
Primitive unit cell in a reciprical space
(1st Brillouin zone)
4
a
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NNSE 508 EM Lecture #9
6
Two wells: Illustration of Bloch Theorem
E
E211121 |||| VVE
212
1
0 L
x
)()(2
100 axx
a
)()(2
100 axx
)()( ruer k
ikr
k
12
1
0
0 )(2
1
m
ikmak maxe
How?
1
0
0)(
1
0
0)( )(
2
1)(
2
1
m
maxikikx
m
xxmaik maxeemaxe
equivalentarea
kandk
periodicxuk
2
!)(
ak
,0
0 0
0 0
1( ) ( ) , k=0
2
1( ) ( ) , k=
22
x x a
x x a
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NNSE 508 EM Lecture #9
7
Free electrons
)()()(2
)(ˆ2
rErrVm
rH
• Time-independent Schrödinger equation:
• Solution - plane wave
• Though free electron wave functions do not
depend on the structure of solid, they can be
written in the form of Bloch functions
• For any propagation vector k’ we can find
in the first Brillouin zone
• Then wave function (Bloch function) and
energy:
• For these wave functions we can plot the band
diagram, which become periodic with 2/a
with energy
0( )V r V const
'( ) ik rr e 2 2
0
'
2
kE V
m
'k k G
( ) ikr iGrk r e e
22
02
E V k Gm
periodic function
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NNSE 508 EM Lecture #9
8
Nearly free electrons: bandgap
2
( ) ( ) ( )2
V r r E rm
• Introduce weak periodic potential
• Let’s simplify the problem: 1D potential with
just one Fourier component:
• Electrons are waves : Bragg reflection occurs at
• In quantum mechanics degenerate states
can split when perturbation is applied:
• Wave functions corresponding to split states will
be linear combinations of :
or in a Fourier series
( ) ( )V r g V r
( ) iGrG
G
V r V e
22
2E k
m
1
2( ) cos
xV r V
a
, 1, 2...k n pa
ka
ka
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NNSE 508 EM Lecture #9
9
Nearly free electrons: Bandgap
• By first-order perturbation theory:
• Calculating the integral, find bandgap:
• Free electrons (plane waves) don’t interact with the
lattice much until wavevector becomes comparable with
1/a, then they are Bragg reflected and we have
interference between a plane wave and its oppositely
directed counterpart.
• These superpositions are standing waves with the
same kinetic energy, but total energy is different
1
2| cos |
xE V
a
2 211
0
2 2cos cos sin
L
g
V x x xE E E dx V
L a a a
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NNSE 508 EM Lecture #9
10 Nearly free electrons: 2D bands
Irrelevant to dimensionality, the following properties are valid:
• Within the first zone lie all points of allowed reduced wave vector
• “One-zone” and “many zone” descriptions are alternatives
• All the zones has the same “volume”
• The zone boundaries are the points of energy discontinuity
E-k curves for three different
directions for parabolic band
From Cusack 1963
The first three Brillouin zones of a
simple square lattice
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NNSE 508 EM Lecture #9
11 Nearly free electrons: 3D bands
First Brillouin zones for various 3D structures
From Cusack 1963
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NNSE 508 EM Lecture #9
12 Nearly free electrons: 3D bands
Electron bands in fcc Al compared to free electron bands (dashed lines)
First Brillouin zone for fcc structure Free electron bands of fcc structure
From Hummel, 2000
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NNSE 508 EM Lecture #9
13 Band structure for several fcc semiconductors
With diamond structure
From Burns, 1985
With zinc-blende structure
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NNSE 508 EM Lecture #9
14 Band-structures of Si and Ge
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NNSE 508 EM Lecture #9
Most essential bands in diamond/ZB
semiconductors
15
From www.ioffe.ru
GaAs
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NNSE 508 EM Lecture #9
16 Free electrons and crystal electrons
Free electrons
m
kdr
m
iv
*
Kinetic energy: m
kE
2
22
Electrons in solid
*2
)( 20
2
m
kkE
Dispersion near band
extremum
(isotropic and parabolic):
Velocity or group velocity:
)()( ruer k
ikr
k Wave function: Wave function: ikr
k eV
r1
)(
*
)( 0
m
kkv
Group velocity: )(1
kEv k
Velocity at band extremum:
Dynamics (F – force): Dynamics in a band:
Fmdt
dv 1
2
2
2
2
1
*
1;
*
1
111
k
E
mF
mdt
dv
FEFvdt
dE
dt
dvkkkk
(if m* isotropic and parabolic) Force equation:
dt
dk
dt
dpF
Force equation:
dt
dkF Fv
dt
dkE
dt
kdEk
)(
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NNSE 508 EM Lecture #9
17 Holes
• It is convenient to treat top of the
uppermost valence band as hole states
• Wavevector of a hole = total wavevector of
the valence band (=zero) minus
wavevector of removed electron:
• Energy of a hole. Energy of the system
increases as missing electron wavevector
increases:
• Mass of a hole. Positive! (Electron
effective mass is negative!)
• Group velocity of a hole is the same as of
the missing electron
• Charge of a hole. Positive!
eh kk 0
Hole energy:
Missing electron energy: )()( eehh kEkE
*
22
2)(
e
evee
m
kEkE
*
22
2)(
h
hvhh
m
kEkE
**eh mm
eee eh
eeekhhkh vkEkEv )(1
)(1
hh
e
edt
dk
edt
dk
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NNSE 508 EM Lecture #9
18
Example: electron-hole pairs in semiconductors
Electron (e-)
Si atom
Hole (h+)
Eg
E
c
Ev
EHP generation : Minimum energy required to break
covalent bonding is Eg.
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NNSE 508 EM Lecture #9
19
Charge carriers in a crystal
V
E Si atom
qEmaF
hole
qEmaF
electron
Charge carriers in a crystal
are not completely free.
Need to use effective mass
NOT REST MASS !!!
+ -