2014 Puslaidininki Fizikos Katedra
hab.dr. Vladimiras Gavriušinas
OPTINIAI REIŠKINIAI
PUSLAIDININKIUOSE
V.
V. Gavryushin
Gavryushin "Optical
"Optical Phenomena
Phenomena in
in Semiconductors"
Semiconductors"
Turinys
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1. S VEIKOS POTENCIALAS. B SEN KVANTAVIMAS
Quantum introduction
Particles under interaction – Potential pockets
Continuity conditions for the wavefunction
Sta iakamp kvantin duob
Baigtinio gylio sta iakamp duob
The Delta-Function Potential
Coupled quantum wells
Parabolinis (harmoninis) potencialas. Kvantinis osciliatorius
Kuloninis potencialas. (Eksitonai. Seklios priemaišos)
Apibendrinimas. S veikos potencialas - energijos spektras
sen tankis
sen užpildymas. Kvazidaleli
voka
V. Gavryushin
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2. KVAZIDALEL S KRISTALUOSE
Elektronai ir skyl s
Kristalo elektronin strukt ra
Kronig-Penney model (1D periodic rectangular potential)
Wave functions in periodic lattice
Transliacin simetrija. Kristalo Juostinis spektras
Brillouin zone
Van Hove singularities
Elektronai amorfiniuose kristaluose
Plazmonai
Eksitonai
Fotonai
Fotonai vakuume. Fotonai kristaluose
Fotoniniai kristalai.
Poliaritonai
Lattice Vibrations: Phonons
Dispersijos d snis. Akustiniai ir optiniai fononai
Fononiniai poliaritonai
Fononai supergardel se
Fononai amorfiniuose kristaluose
Elektron-fononin
veika
Poliaronai
Bipolarons. Cooper Pairs. Superconductivity
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
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Turinys
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3, ERDVIŠKAI APRIBOTOS STRUKT ROS. KVANTIN INŽINERIJA
Quantum wells and superlattices
Quantum wires and quantum dots
Excitons and Shallow defects in Quantum Structures
Application of Quantum Structures
4. PERTURBACIJOS TEORIJA. ATRANKOS TAISYKL S
Elektron ir foton
veika
Optini reiškini perturbacijos teorija
Vienkvan iai ir daugiakvan iai reiškiniai
Tverm s d sniai ir atrankos taisykl s
Energin s atrankos taisykl s
Impulso atrankos taisykl s
Grupi teorija ir simetrin s atrankos taisykl s
Simetrini atrankos taisykli nustatymas
Poliarizacini priklausomybi nustatymas
5. ŠVIESOS SUGERTIES
Tarpjuostiniai tiesioginiai šuoliai
Eksitonin sugertis
Defektais surišti eksitonai
Eksiton-fononin
veika
"Urbacho" kraštas
Konfig racin šuoli tikimybi interferencija (Fano efektas)
Tarpjuostiniai netiesioginiai šuoliai
Netiesiogini eksiton sugertis
Priemaišin sugertis
Seklus ir gilus gardel s defektai
Gardelin “fononin ” sugertis
Laisv
kr vinink sugertis
6. ŠVIESOS EMISIJA. LIUMINESCENCIJOS REIŠKINIAI
Tarpjuost laisv kr vinink plazmos rekombinacija
Stimuliuota rekombinacija. Lazeriai
Nespindulin rekombinacija
Spindulin laisv eksiton anihiliacija
Defektais surišt eksiton rekombinacija
Poliariton rekombinacija
Priemaišin s liuminescencijos mechanizmai
Emission under quantum confinement
V. Gavryushin
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7. IŠORINIAI LAUKAI. POVEIKIS SPEKTRAMS
Elektrini lauk poveikis spektrams
Franco-Keldyšo efektas
Štarko efektas
Elektrinio lauko poveikis eksitonams
Elektronai Magnetiniame lauke
Hall effect
Quantum Hall effect
Magnetinio lauko taka. Landau b senos
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8. NETIESIN OPTIKA
Dielektrin skvarba
Savipraskaidr jimas. Kvantiškumo metamorfoz s
Nonlinear absorption
Nonlinear refraction. Selffocusing
Second-Harmonic Generation
Third-Harmonic Generation and Intensity-Dependent Refractive
Index
Induced gratings. Non-degenerate four-wave mixing.
Two-Photon Absorption
Kvantmechaninis netiesin s optikos aprašymas
Difference-Frequency Generation
Optical Parametric Generation,
Fazinis sinchronizmas = Impulso atrankos taisykl s
Optinis detektavimas - Optical rectification in crystals
Dvifoton spektroskopija
Multi-photon imaging. Two-Photon Microscopy
Z-scan technique
Ultraspar ioji spektroskopija - Ultrafast Laser Spectroscopy
The Excite-Probe Technique
Time-Resolved Fluorescence
Techniques for Transillumination Imaging
Optiniai netiesiškumai erdviškai ribotuose aplinkose
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www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
2
Quantum introduction
The wavefunction evolves in time and space according to the
Time-dependent Schrödinger equation.
Hˆ ( r , t ) i
Hˆ (r , t )
t
(r , t )
Energy and Momentum must be
replaced by their equivalent
quantum mechanical operators.
Eˆ (r , t )
Eˆ
i
t
Time-independent Schrödinger equation
Hˆ (r )
E
The solution of proceeds by the method of separation of
variables. Write the wavefunction as a product of a space and
a time components:
(r )
Wave functions of this type are called standing
waves or stationary states, because the
probability density is time-independent:
| ( x, t ) | 2
t
0
(r ) (t ) eik r ei
(r , t )
pˆ
i
t
ei ( k r
t)
r
p
k
A(ri) - elektromagnetinio lauko vektorinis potencialas
A
V. Gavryushin
ec
2
N
exp(i [(
r)
t ])
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3
Quantum introduction
2
T
E
- bangos dažnis laike
= bangos amplitud s
kitimo greitis laike
E
p
2
k
i
t
i
t
i
t
B
2
k
(r , t )
(r ) (t ) eik r e i
t
ei ( k r
t)
B
pˆ
k - banginis skai ius =
= bangos dažnis erdv je
= bangos amplitud s kitimo greitis erdv je
Impulso operatorius:
pˆ
x
k
x
i
grad x ( )
V. Gavryushin
k
i
r
pˆ ( x)
i
i
x
i
r
i
r
r
( x)
The gradient of scalar function
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4
Quantum introduction
- bangos dažnis laike
= bangos amplitud s kitimo greitis laike
The wavefunction evolves in time and space according to the
time-dependent Schrödinger equation.
Hˆ ( r , t ) i
Momentum and kinetic energy
must be replaced by their equivalent
quantum mechanical operators.
(r , t )
t
2
2
U (r , t )
2m
(r , t )
i
t
Eˆ
(r , t )
i
t
Time-independent Schrödinger equation
Hˆ (r )
E
(r )
E
The solution of proceeds by the method of separation of variables.
Write the wavefunction as a product of a space and a time components:
2
2
U (r )
2m
(r )
E
(r )
Wave functions of this type are called standing
waves or stationary states, because the
probability density is time-independent:
| ( x, t ) | 2
t
pˆ
x
k
x
i
grad x ( )
V. Gavryushin
i
0
k - banginis skai ius = k
= bangos dažnis erdv je
= bangos amplitud s
kitimo greitis erdv je: k
Impulso operatorius:
r
pˆ ( x )
i
x
(r ) (t ) eik r ei
(r , t )
pˆ
i
t
ei ( k r
t)
r
2
B
i
r
( x)
The gradient of scalar function
p
k
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5
Heisenberg's position-momentum uncertainty principle
Non-commuting observables cannot share a common basis set of eigenvectors.
[ xˆ , pˆ ]
xˆpˆ
pˆ xˆ
i
Anticommutation of observables leading to the uncertainty relations.
Heisenberg's position-momentum uncertainty principle:
x p
Heisenberg's Energy-time uncertainty principle:
E t
E
px
x
t
Plot of the Heisenberg’s measurement prediction space.
V. Gavryushin
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6
Quantum introduction
Quantization
kn
2
n
n
,
Lx
n 1,2,
Quantization of circular orbits
Waves in a confined geometry have discrete modes
Carbon Nanotubes
!! Born-Carman cyclic periodical
conditions for solids.
V. Gavryushin
k
q
,q
2 q
| Ch |
q 2
,
NC a
0,1, 2, 3,...N 1
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7
Crystal =
Electrons in a
"quantum box"
Quantum introduction
Quantization
L
Waves in a confined
geometry have discrete
modes
2
kn
n
n
,
Lx
dN 2 D
dk
n 1,2,
2 k
a=1A
2
L
2
(E)
1 dN 2 D
L2 dE
m*
3D ( E )
1 dN 3 D
L3 dE
2
2D
3D
dN 3 D
dk
L
2
3
L = 1 cm
2
2 3
m 3 2 Ec ( k ) E g
k- erdv s kvantavimas
k2
2
Ec ( k )
(E)
Eg
k2
2m*
1 dN
V dE
1 dN dk
V dk dE
V. Gavryushin
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8
Quantum introduction
Crystal = Electrons in a "quantum box"
Quantization
Waves in a confined
geometry have discrete
modes
kn
n
,
Lx
2
n
L
n 1,2,
a=1A
L = 1 cm
k- erdv s kvantavimas
3D
3D
V. Gavryushin
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9
S VEIKOS POTENCIALAS. B SEN KVANTAVIMAS.
KVANTAVIMAS. [Quantum introduction]
The gradient of scalar function f is
denoted grad(f), or f , where is the
vector differential operator. The
gradient of f is the vector field whose
components are partial derivatives of f:
Central postulate of wave mechanics is that all of measurable
information about a system is contained in its wavefunction.
The wavefunction evolves in space according to the timeindependent Schrödinger equation
Hˆ (r )
E
(r )
2
2
pˆ
i
k
2
Ekin
2
2m
2
k2
2m
2m
p2
2m
U (r )
(r )
E
(r )
x
mv 2
2
2
p quantum
2
|k |
2
T
E photon
h
1
1
h
T
Wave functions of this type are called stationary
solutions or stationary states, because the
probability density is time-independent:
t
|
( x, t ) |
2
V. Gavryushin
0
xˆ
grad x ( )
2
( x) U ( x) ( x)
2m x 2
Momentum and kinetic energy must be replaced by
their equivalent quantum mechanical operator.
x
E
( x)
Curvature = 2nd derivative
^
H
2
2
2
2m
Kinetic
V (r )
2m
Potential
2
2
2
x2
y2
z2
V (r )
Kinetic energy is associated with the curvature
of wavefunction
Laplace operator:
The Laplacian of a function
is equal to the
divergence of the gradient:
f
2
f
div ( grad ( f ))
And is associated with the curvature of function f.
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10
Examples of Schrödinger equation solution
Parabolinis (harmoninis) potencialas
F ( x)
x
U ( x)
U ( x)
1 2
x
2
F ( x)
x
Huko d snis
Mas s m dalel , veikiama šios j gos,
atlieka harmoninius svyravimus
dažnio: = ( /m)1/2 .
Schrödinger equation
Hˆ ( x)
E
( x)
2
2
2m
2
2
2m x 2
V. Gavryushin
U (r )
(r )
E
(r )
1 2
x
2
( x)
E
( x)
Bandomosios funkcijos metodas
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11
Examples of Schrödinger equation solution
i) Lorentzian type wave function (x)
2
( x)
1
1 x2
Let's consider a quadratic potential
1 2
x
2
U ( x)
2m x
2
2
2m x
( x) U ( x ) ( x )
V. Gavryushin
E ( x)
( x)
H (x) and (x) have different shapes, and hence (x) of is not
an eigenfunction of the Hamiltonian with a quadratic potential.
Really, Lorentzian-type wave function is a solution
for exciton state formed under Coulomb interaction.
2
2
E
It satisfies all the requirements of a good wave function:
it falls off towards infinity, it is normalized,
and both it and its first derivative are continuous for all x
Time-independent Schrödinger equation
2
Hˆ ( x)
2
m
2
( x)
2 3x 2 1
(1 x 2 ) 3
Hˆ ( x)
2
m
2
2
2m x 2
2 3x 2 1
(1 x 2 )3
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U ( x)
( x)
1
x2
2
2 1 x
12
Graphical solution of Schrödinger equation
ii) Gaussian wave function (x)
( x)
Ne
mx2
2
If we had chosen a Gaussian wave function instead of the Lorentzian,
we would have found that H (x) is indeed proportional to (x),
but only if we had picked the right coefficient in the exponent
2
Hˆ ( x)
2
2m x 2
N
2 m
E ( x)
Hˆ ( x)
1 2
x
2
E Ne
E
( x)
e
( x)
mx2
2
mx2
2
Right-hand (red line) and left-hand (blue line) parts
of Schrödinger equation for Gaussian-type wave function (x)
in parabolic potential (black line) for energy state E = 0.5.
So, applying H to (x) on the left-hand side should be the same as a number E
times (x) on the right-hand side? A simple example is provided by
V. Gavryushin
d 2 4x
e
dx 2
16 e 4 x
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13
Sta iakamp kvantin duob
Be galo gili kvantin duob
0, if 0 x Lx ,
, otherwise ( x 0, x
U ( x)
2
E
( x)
d2
U ( x)
2m dx 2
d
2
some function , if 0 x Lx
0, otherwise ( x 0, x Lx )
( x)
Hˆ ( x)
x
Ei
i ( x)
Wavefunctions (x) has a general solution of (+ Euler relations):
( x)
2m
(U
E)
2 m(U
E)
dx 2
2
k
i ( x)
Lx )
Ae
ik x
Be
ik x
A' sin(| k | x) B' cos(| k | x)
x
Oscillatory standing waves
d2 x
dx 2
k2
x
(0) 0
( L) 0
Continuity conditions for wave function
( x0
Euler relations:
exp(ikx) cos(kx) i sin(kx)
exp( ikx)
cos(kx) i sin(kx)
V. Gavryushin
( x)
A sin kx
What values are allowed for A and k?
0)
d ( x0 0)
dx
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( x0
0)
d ( x0 0)
dx
14
2
d2
2m dx 2
Be galo gili kvantin duob
What values are allowed for k and A?
( x)
2
d2
A sin(kx)
2m dx 2
E
( x)
A sin kx
E A sin(kx) What values are allowed for A and k?
2mE
( x) A sin(
x)
E A sin( kx)
2
2m
( x)
Ak 2 sin( kx)
2
k2
2m
E
2mE
k
2
k
1
( x)
B
Energy eigenvalues for the electron in a box
The longest wavelength in box is
n
Or same,
centered to box:
2 Lx
and the higher modes have wavelengths given by
n
2 Lx
n
n ( x)
n
2
x (n 1)
cos
Lx
Lx
2
2
sin n
x
Lx
Lx
L
2
where n = 1, 2, 3, ...
The sine is zero whenever its argument happens to be an
integer multiple of .
kn
2
n
n
,
Lx
n 0, 1, 2
n ( x)
n
x
Lx
An sin
Standing waves are solutions:
kn
2mEn
V. Gavryushin
2
n
En
Lx
2 m Lx
2
n2
e0 n 2
Quantum effects increase with decreasing m* and L2 !
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15
Be galo gili
kvantin duob
Energy levels,
wavefunctions (a)
and probability density
functions (b)
in an infinite quantum well.
Calculated for a 10 nm
wide well containing
an electron with mass m0.
iii) Normalization of the wavefunction for particle in Box
L
1
A
2
Lx
n
x
L
A 2 sin 2
dx
0
L
1
2 n
1 cos
A
x
20
L
2
Properly normalized wave function n(x) of the infinitely
deep square well for a particle in the state with energy En
n ( x)
dx
n
2
sin
x
Lx
Lx
2mEn
2
sin(
x)
Lx
1 L
A2
x
2 0
A2
V. Gavryushin
sin 2 nx L
2 n L
1
sin 2 n
L
2
2 n L
dx
L
0
A2
L
2
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16
OrthoOrtho-normalized wavefunctions
If a set of wavefunctions satisfies Schrödinger equation:
and they are orthogonal:
and they are normalized:
i
j
0
i
i
1
Hˆ ( r )
or
i
( x)
E
(r )
j
( x )dx
0
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
or, coupling the both conditions, - they are ortho-normalized:
i
j
ij
0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0 0
n
2
sin( kn x)
Lx
( x)
n
2
x
sin
Lx
Lx
kn
n
1
1
V. Gavryushin
1
0 0 0 0 0 0 1 0 0
n
,
Lx
2
2
n 0, 1, 2
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1
0
1
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17
Kristalo elektronin strukt ra
n
Electron in
"quantum box"
allowed wavevectors:
n
2
x
sin
Lx
Lx
2
sin(k n x)
Lx
( x)
kn
2
n
n
,
Lx
n 1,2,
aprašantys elektrono jud jim išilgai x ašies; o energij - kaip sum atitinkam
men ,
L
Be galo gili kvantin duob
Dispersion Law of infinite well
2
En
2
n
k
2me
2
2me Lx
2
n
2
e0 n
Wavenumber kmin = 2 /2L= /L, for n =1,
is corresponding to the maximum
possible wave length L in quantum "box".
2
n ( x)
2
sin n
x
Lx
Lx
L
2
Dispersion Law of “1D1D-box”
box”
k- erdv s kvantavimas
V. Gavryushin
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18
Kristalo elektronin strukt ra
n
Electron in
"quantum box"
allowed wavevectors:
2
sin( k n x)
Lx
( x)
2
sin n x
Lx
Lx
kn
2
n
n
,
Lx
n 1,2,
aprašantys elektrono jud jim išilgai x ašies; o energij - kaip sum atitinkam
men ,
L
Dispersion Law of infinite well
2
En
2
n
k
2me
2
2me Lx
2
n2
e0 n 2
Wavenumber kmin = 2 /2L= /L, for n =1,
is corresponding to the maximum
possible wave length L in quantum "box".
3D
k- erdv s kvantavimas
V. Gavryushin
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19
Expectation energy of Hamiltonian
Hˆ
Ek
k
* ˆ
kH
*
k
Ek
k
Hˆ
k dr
Schrödinger equation
k
*
k
k
Ek
*
k
k
dr
Felix Bloch (1905(1905-1983)
Stanford University, Nobel Prize, 1952
Therefore,
Ek
expectation energy of the Hamiltonian:
*
km
Hˆ
*
km
1 0 0 0 0 0 0 0 0
kn
dr
k n dr
1
m Hˆ n
N
Hˆ m ,n
0 1 0 0 0 0 0 0 0
(**)
N
M
i j
L
0 0 1 0 0 0 0 0 0
2
0 0 0 1 0 0 0 0 0
( i x)
( j x) dx
0 0 0 0 1 0 0 0 0
L
0 0 0 0 0 1 0 0 0
2
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
for normalized wave functions.
0 0 0 0 0 0 0 0 1
N
Denominator in (**) have N identical terms, each equal to one, then:
*
km
kn
dr
mn
N
n,m
V. Gavryushin
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20
0.137
Square potential well in MathCAD
0.548
Analytical solution for energy :
Eigenvalues En and eigenfunctions
n(x)
for n = 1,2,3,4 of particle in box:
2
En
2
n
2
k
2
2
h bar ni
E
Lx
2me i 2me
2m
L
gives the same result.
1.234
2
2.193
E
3.427
4.935
6.717
Orthonormalized wave functions:
atomic units
h bar
N
n
particle is an electron
1
9
m
1
i
1 N
Box length
L
j
2
n
cos
x (n 1)
2
Lx
Lx
( x)
6 Bohr
1 N
Hamiltonian Matrix for Particle in an Infinitely Deep Well:
L
n
H
( i x)
i j
h bar
2
k n2
2m
2
n
,
L
n
x
0, 1, 2
2
j
Returns a vector of eigenvalues –
Spur of the square matrix H :
( j x) dx
L
2m
11.103
2
Hˆ
kn
2
8.773
L
2
H
e
0.137
0
0
0
0
0
0
0
0
0
0.548
0
0
0
0
0
0
0
0
0
1.234
0
0
0
0
0
0
0
0
0
2.193
0
0
0
0
0
0
0
0
0
3.427
0
0
0
0
0
0
0
0
0
4.935
0
0
0
0
0
0
0
0
0
6.717
0
0
eigenvals ( H)
0.137
0.548
e
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
1.234
L
0 0 1 0 0 0 0 0 0
2.193
2
0 0 0 1 0 0 0 0 0
3.427
4.935
0
0
0
0
0
0
0
8.773
0
6.717
0
0
0
0
0
0
0
0
11.103
8.773
11.103
V. Gavryushin
Matrix elements
of wavefunctions overlap
M
i j
( i x)
( j x) dx
0 0 0 0 1 0 0 0 0
L
0 0 0 0 0 1 0 0 0
2
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
Orthonormalized wave functions !!
0 0 0 0 0 0 0 0 1
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21
Particle in a 2D Box. Infinite Square Well
Schrödinger equation
Potencialas U(x,y) veikia elektron jud jim tik statmeniems
duobei kryptims; o išilgai duob s, 0z kryptimi, elektron
jud jimas išlieka laisvas. Tod l elektrono bangin funkcij
galime užrašyti kaip sandaug parcialini 1-D funkcij
(r )
(r )
nx
( x)
ny
2
Lx L y Lz
( y)
e
ik z z
1
Lz
e
Hˆ ( x, y )
d2
2m dx 2
ik z z
2
d2
dy 2
E ( x, y )
( x, y )
E U ( x, y )
( x, y )
ny
nx
sin
x sin
y
Lx
Ly
kz2
2 me
2
E
nx
ne
2
nx
V. Gavryushin
n x2
2
ny
n x2
n y2
Lx 2
Ly 2
2me L x
2 2
E
2
2 me
2 2 2
ny
2me L y 2
kz 2
2 me
2
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
22
Particle in a 2D Box.
Infinite Square Well
We can use MathCAD procedures
to calculate wavefunctions of the
particles in 2D-Box.
The un-normalized eigenfunctions would be:
( x y)n m
sin n
x
a
sin m
y
a
To do this you have to choose
to either set up an array of functions
or let m and n be arguments of the function.
The latter is easier if we want just
to plot one state. Define:
( x y n m a)
x
sin n
a
sin m
n := 1, m := 1
n := 2, m := 1
n := 2, m := 2
n := 3, m := 3
y
a
where the parameter a has also been
included as an argument.
Note that we could use a rectangle
instead with two parameters a, b.
a := 1 N := 50 i := 0..N j := 0..N
x
i j
i
a
N
V. Gavryushin
y
i j
j
a
N
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23
Particle in a 3D Box.
Infinite Square Well
More sophisticated view
(r )
nx ( x )
nx ,n y ,n z ( x, y , z )
ny ( y)
23 2
L x L y Lz
sin
2
n x , n y ,n z
2
n
2
x
sin
Lx
Lx
n ( x)
nz ( z )
nx
x
Lx
2
(
nx
sin
2
m xx L x
ny
Ly
ny
2
y
sin
2
m yy L y
nz
z
Lz
nz
2
2
m zz L z
2
)
Electronic structure of a InAs/GaAs
selfself-assembled quantum dot as deduced
from the 3D Schrö
Schrödinger equation written
in a 8 band k·p formalism.
The solution takes in account a realistic
elliptical flat lenslens-shape geometry (see inset).
The representation of the envelope wave
functions considers probability volumes
including 2/3 of the presence probability
of the electron.
WL denotes the wetting layer 2D continuum,
beneath the bulk barrier 3D continuum. [[i]
[[i]]]
[i] pages.ief.u-psud.fr/QDgroup/modeling.html
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
24
Boundary or Continuity conditions for the wave function
Boundary conditions
(i)
(r) has to be continuous for all r:
( x0
0)
( x0
0)
(ii) d /dx has to be continuous for all x
for finite potential U(x):
d ( x0 0)
dx
(iii)
d ( x0 0)
dx
(x) has to be finite outside potential U(x):
lim | ( x ) |
finite
x
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
25
Baigtinio gylio sta iakamp duob
Particle in Finite-Walled Box
2
U ( x)
2
2m x
( x) U ( x) ( x )
2
0
for
x
L 2
U0
for
x
L 2
E ( x)
2
Outside the well :
2
I , III ( x )
x2
2mE
I , III ( x ),
For negative the above differential equation has exponentially
rising and falling solutions.
I , III
( x)
C1e
x
x
C2e
I ( x)
C e
x
III
2
Inside the well :
x2
2
II ( x)
II
( x)
A2 e i
II
( x)
A sin kx
Boundary conditions :
x
x
i x
or
B cos kx
n(
d
dx
V. Gavryushin
D e
2m( E U )
II ( x),
A2' e
( x)
L 2 0)
n ( L 2 0)
n(
L 2 0)
d
dx
n(
L 2 0)
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26
Baigtinio gylio sta iakamp duob
2mE
2m( E U )
Outside the well :
I
( x)
C e
( x)
D e
III
x
x
Confining a particle to a smaller space
requires a larger confinement energy.
Since the wavefunction penetration
effectively "enlarges the box", the
finite well energy levels are lower than
those for the infinite well.
Inside the well :
II ( x )
A2 e i
x
A2' e
i x
Boundary conditions :
n(
d
dx
L 2 0)
n(
L 2 0)
n(
L 2 0)
d
dx
C e
n(
L 2
C e
D e
D e
V. Gavryushin
L 2 0)
A2 e
i L 2
L2
i A2 e
L 2
A2 e i
L 2
i L2
L 2
i A2 e i
A2' e i
i A2' e i
A2' e
L2
L 2
L2
i L 2
i A2' e
i L2
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27
Baigtinio gylio sta iakamp duob
C
C e
L 2
A2 e
L2
C e
i A2 e
L 2
D e
A2 e i
L 2
D e
i L 2
i L2
L 2
i A2 e i
i A2' e i
A2' e
L2
e
L2
i L 2
i A2' e
i )
A2
e
L
2
i
C
2i
A2
e
i )
L
2
i
i L2
A2'
2i
D
A2'
L2
0
0
A2
0
e
0
i e
e
(
i )
e
(
L
2
i
2i
i )
L2
e
e
L2
C
D
i L2
i L2
i L2
e
i ei
L2
i iL
e C
i
L2
ei
i ei
L2
0
i L2
e
i e
i L2
which can only
be satisfied if
and
L
2
i
2i
D
D
i
e
i
There are two possible solutions:
V. Gavryushin
A2’
This yields two equations
for D in terms of C,
Solving for A2 and A2’ in terms of D we obtain
(
L2
e
Solving for A2 and A2’ in terms of C we obtain
(
D
A2' e i L 2
i L
i
i
C
i
i
ei
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
e 2i
L
L
28
Odd states
i
i
ei
L
But we also have
ei
This implies
L
e
2i
,
2
i
i
2
2
2 i
2
L
e
i
e
2i
with
cot
e
,
cot
cot
2
L
2
tan
L
2
However, it had only one unknown , but now contains two unknowns, and .
This means that alone it is not enough; we also have to take into account the interdependence
between and .
We have
tan
k0
2
L
2
2
2mU
2
We can try a
graphical solution.
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
29
Even states
i
i
This implies
We have
ei
e
L
2i
But we also have
ei L ,
cot
L
2
k0
2
2
2
e
i
e
2 i
2
L ,
2
and
2
i
i
2i
e
cot
with
cot
Duob s gylio taka:
cot
L
2
2mU
2
A symmetric square well always has at least one bound state.
An asymmetric square well has no bound state for sufficiently
small values of L.
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
30
Duob s plo io taka:
Duob s gylio taka:
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
31
A triangular potential well in MathCAD
And the same, but with a ramp:
L
2
Hamiltonian Matrix for Particle in an Infinitely Deep Well:
L
2
( i x)
H
i j
( i x)
Hr
i j
h bar
2
2m
2m
2
j
x
L
L
2
( j x) dx
2
( j x) dx
L
2
L
2
j
h bar
L
2
H
0.137
0
0
0
0
0
0
0
0
0
0.548
0
0
0
0
0
0
0
0
0
1.234
0
0
0
0
0
0
0
0
0
2.193
0
0
0
0
0
0
0
0
0
3.427
0
0
0
0
0
0
0
0
0
4.935
0
0
0
0
0
0
0
0
0
6.717
0
0
0
0
0
0
0
0
0
8.773
0
0
0
0
0
0
0
0
0
11.103
Command that returns a vector of eigenvalues for the square matrix H :
3.13708
1.08076
0
0.08646
Hr
0
0.02382
0
0
V. Gavryushin
1.19104
4.2337
1.16722
5.19325
1.19104
0
1.20084
0
0.11028
0
0.12008
0
0.12505
0
0.03362
0.01476
0
1.16722
3.54831
0.0098 1.35308 10
0.08646 1.4988 10
0
1.08076
15
0
0.03859
1.08247 10
15
0.04144
15
1.54043 10
0.11028
1.70697 10
0.02382
15
15
1.16573 10
0.1279
0
15
5.38458 10
1.20581
1.20581
7.9348
1.20866
1.20866
9.71681
1.21045
6.66134 10
1.21045
11.77298
1.21165
1.21165
14.1033
0
1.11022 10
0.1279
2.55351 10
15
0.12969
15
15
1.856
3.245
15
4.385
5.445
0.04144
6.42695
0
eigenvals ( H)
0.01476
2.66454 10
0.03859
0.12505
0
1.20084
1.04083 10
0.0098
3.83027 10
0.03362
0
0.12008
0
15
e
15
e
6.63
8.083
0.12969
15
9.836
11.908
4.21885 10
15
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
14.679
32
A triangular potential well
Normalized electron densities
An |
n
( x l F ) |2
for the first (1) and second (2) subbands
in a triangle potential with the slope F=eEel,
lF
2
2m eEel
13
Mouse click to animated show
2
E n,k
En
2
En
2m
V. Gavryushin
E (k )
1/ 3
3 e
2
En
k2
2m
2/3
Eel2 / 3
1
n
4
2/3
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
33
Parabolinis (harmoninis) potencialas. Kvantinis osciliatorius
F(x) = - x (Huko d snis)
Mas s m dalel , veikiama šios j gos, atlieka harmoninius svyravimus dažnio:
Potencial U(x) galima išreikšti:
F ( x)
x
2
U ( x)
2
2m x
F ( x)
1
m
2
2
2
xˆ 2
( x)
x
U ( x)
E
( x)
1 2
x
2
1
(x) yra:
n ( x)
n
Hn
2 n!
0
x0 )
0
exp
x
mv 2
2
2
x
n
(x
2
2
Sprendimas ieškomas formos (Gaussian function):
= P(x') exp(-x'2/2).
Substituting this function into the Schrödinger equation
and fitting the boundary conditions leads to :
Normuotos tikrines funkcijos
1
m
2
= ( /m)1/2 .
2
P ( x )e
1 2
x
2
2
x P ( x )e
2
(P
2x P
(x
1) P)e
P
2x P
P EP
1 2
x
2
1
x
2
2
E P ( x )e
2
x Pe
1 2
x
2
1
x
2
2
E Pe
1 2
x
2
x0 ) 2
(x
2
2
0
Iš esm s tai yra Gauss'o funkcijos
pramoduliuotos Hermite polinomo
H n ( x)
n
( 1) e
V. Gavryushin
x2
dn
dx n
e
x2
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34
Parabolinis potencialas
Charles Hermite,
1822-1901 France
Ermito polinomai Hn( ) ir harmoninio osciliatoriaus tikrin s funkcijos
En
n
V. Gavryushin
1
2
Hn
( x x0 )
n
( x)
n(
0
( x)
).
2
0
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
35
Kuloninis potencialas.
(Eksitonai. Seklios priemaiš
priemaišos. Vandenilio atomas)
( Hˆ o
U (r ))
U (r )
0
2
2me
E ex
ex
e2
re rh
2
2
e
ex
2mh
2
h
e2
rh
0 re
E ex
ex (re , rh )
Transliacinio ir vidinio jud jim
atskyrimas - Eksitonai
ex (re , rh )
Spherical Polar
Coordinates
Hyperbolic (hydrogen) potential
2
2m
e2
r
x (r
)
Ex
x (r
)
Separating the Hydrogen Equation
V. Gavryushin
x = r sin
cos ,
y
r sin
sin ,
z
r cos .
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36
Kuloninis potencialas.
Hyperbolic (hydrogen) potential
(Eksitonai. Seklios priemaiš
priemaišos. Vandenilio atomas)
2
2m
Sferin s harmonikos. Kampin uždavinio dalis
Laplaso operatorius sferin se koordinat se yra šitoks:
2
2
2
x2
y2
z2
2
2
1
r
r r2
(
1) Yl m ( , )
)
Ex
x (r
)
Separating the Hydrogen Equation
L̂
r
p
r
i
kurio tikrin s funkcijos yra taip vadinamos sferin s harmonikos Ylm( , ),
tenkinan ios judesio kiekio momento operatoriaus kvadrato L2 bei jo projekcijos Lz
tikrini ver i lygtims:
2
x (r
1 1 ˆ2
L
2
2
r
ia atsiranda L- judesio kiekio momento operatorius
Lˆ2 Yl m ( , )
e2
r
Lˆ z Yl m ( , )
Spherical Polar
Coordinates
mYl m ( , )
Sprendiniai šio tipo diferencini lyg i - yra žinomi jungtiniai Ležandr'o polinomai:
m
Pl (u )
2
(1 u )
m
2
dm 1 dl 2
{ l
(u 1) l }
l
m
du 2 l! du
kurie nusako azimutines priklausomybes bangini funkcij kvantuot b sen . Ši polinom
pavidalas nusako sferines harmonikas. Jei m 0, normavimo ir fazinio eim daugikli tikslumu,
sferin s harmonikos Ylm( , ) sutampa su Ležandr'o polinomais:
Ylm ( , )
( 1) m
Jungtiniai Ležandr'o polinomai
Plm(x) pirmoms l kvantinio skai iaus
vert m ir m=0
2l 1 (l - m)! m
Pl (cos ) exp(im )
4 (l + m)!
Sferin s harmonikos yra kompleksin s ir funkcijos, o fizikin prasm - tikimyb
aptikti
kvazidalel
( , )-kryptimi
turi
j
moduli
kvadratai:
2
im
-im
|Ylm( , )| =Ylm( , )Ylm*( , ), kurie (paskutinis daugiklis ia bus e e =1) priklauso
tik nuo -kampo, t.y. turi z-ašin simetrija.
V. Gavryushin
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37
Sferin s harmonikos
2p: |Y10( , )|2
|Y11( , )|2
1s: |Y00( , )|2
|Y20( , )|2
|Y30( , )|2
|Y40( , )|2
V. Gavryushin
|Y21( , )|2
|Y31( , )|2
|Y41( , )|2
|Y22( , )|2
|Y32( , )|2
|Y42( , )|2
Vandenilio Sferin s harmonikos
yra orto-normuotos, -jos nepersikloja erdv je
|Y33( , )|2
|Y43( , )|2
|Y44( , )|2
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38
Radialin s bangin s funkcijos
2
1
Veff
2
nll (
En
)
nll (
)
ia Veff( ) yra efektinis potencialas:
2
Veffl ( )
(
e2
1)
2
2
kurio pavidalas ir apsprendžia bangini funkcij pavidal ir j
priklausomyb nuo orbitinio kvantinio skai iaus l. Išcentrinis
potencialas dar vadinamas išcentriniu barjeru, nes veikia kaip
st mos potencialas, ver iantis dalel jud ti atokiau nuo sferiškai
simetrinio potencialo U( ) centro =0.
Apibendrinti Lagero polinomai Lp(x) keliom p vert m.
Normuota radialin funkcija turi tok pavidal :
nl
( )
2
n 2 aB
3
2
(n l 1)!
[(n l)!]3
2
na B
l
exp(
na B
) L2nl l1 1
2
na B
L(x) - yra apibendrintas Lagero polinomas:
Lmn ( x)
dm
dx
m
Ln ( x)
dm
dx
m
e
x
dn
dx
n
x
n
(e x )
n
n!
x d
e
(e x x n
n
( n m)! dx
m
)
Edmond Laguerre,
1834-1886, France
V. Gavryushin
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39
Lagero polinomai Lp(x)
(x) ir vandenilio tikrin s funkcijos
n(
)
= wave function
2
= probability density
4 r2 2 = radial probability
function
S - Orbitals
Ln( )
V. Gavryushin
n(
)
2
n (
)
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
40
100
200
210
211
300
400
310
410
311
411
320
420
430
421
431
422
432
Vanje-Motto eksiton n=1-4 erdvin
strukt ra: piln
gaubiam
bangini funkcij absoliutini ver
kvadrat |Gnlm(r)|2 atvaizdai
dvima iais paviršiais (rx,ry) 321
erdv je.
Atstumas nuo koordina
pradžios
iki bet kurio paviršiaus taško
atitinka tikimyb s tankiui aptikti tiek
pat r =re-rh nutolusius
bes veikaujan ius elektron ir
322
skyl .
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
41
2
1
2
Radialin s bangin s funkcijos
Veff
nll (
)
En
nll (
)
3D Coulomb potential V(r),
V(r), centrifugal terms
(
ia Veff( ) yra efektinis potencialas:
S – tipo
(l = 0 )
2
Veff ( )
p – tipo
(l = 1 )
2
(
1)
e2
2
2
2 ) (
1) r 2
and effective potentials Veff(r)
(r) for l = 1, 2, 3.
Also is shown corresponding hydrogen bound
states for n = 1, 2, 3, 4.
d – tipo
(l = 2 )
l=2
l=1
Radialin
bangin funkcija
funkcija
Sferin harmonika
harmonika
V. Gavryushin
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42
Apibendrinimas. S veikos potencialas - energijos spektras
V. Gavryushin
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43
Periodic systems
Schrödinger equation
Lattice (translation) vector:
Rn
ni ai
Free electron:
(i = x,y,z)
i
V (r ) V (r R n )
Bloch's theorem
Electron in crystalline materials
Bloch waves:
k
where
u k (r )
Felix Bloch (1905(1905-1983)
(r R ) e
ik R
k
(r )
When location in real space is shifted by R, only
the phase of the wavefunction will be changed.
k
u k (r R )
(r )
k R
e ik R (r R )
Hydrogen atom. 1s-state
2 n
1
1s
a0
exp
r
a0
Given a set of atomic basis functions u(r) for the
unit cell, a set of Bloch basis functions nk(r) is
formed by constructing Bloch sums over the N
cells of the crystal
k (r )
1
N
e ik R n u at (r R n )
n
(Rn = na)
V. Gavryushin
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44
Calculation of energy bands
introduced by Bloch in 1928
TightTight-binding approximation
Hˆ
Ek
k
* ˆ
kH
*
k
*
k
Ek
k
Hˆ
Schrödinger equation
k
k dr
k
*
k
Ek
k
dr
Felix Bloch (1905(1905-1983)
Stanford University, Nobel Prize, 1952
Expectation energy of Hamiltonian
Therefore,
Ek
1 0 0 0 0 0 0 0 0
expectation energy of the Hamiltonian:
*
km
Hˆ
*
km
kn
kn
dr
dr
1
m Hˆ n
N
Hˆ m ,n
N
0 1 0 0 0 0 0 0 0
(**)
M
i j
L
0 0 1 0 0 0 0 0 0
2
0 0 0 1 0 0 0 0 0
( i x)
0 0 0 0 1 0 0 0 0
( j x) dx
L
0 0 0 0 0 1 0 0 0
2
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1
for normalized wave functions.
N
*
km
Denominator in (**) have N identical terms, each equal to one, then:
kn
dr
mn
N
n,m
Hˆ m , n
VUGavryushin
V.
PFK
(r
Rm ) Hˆ
(r
Rn )
k (r )
1
N
eik R n u at (r R n )
n
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45
Tight-binding approximation
Bloch sums over N cells of crystal
1
N
k (r )
Inserting Bloch function we'll find the expectation energy of Hamiltonian:
( Hˆ ) m ,n
i
Rm ) Hˆ
(r
i
(r
eik R n u at (r R n )
n
N
Rn )
k
cnk u n ( r )
(r )
n 1
e ik ( R n
m
Rm )
umi Hˆ u ni
n
u Hˆ uni
u mi
Hˆ
uni
t0
Hamiltonian Matrix for Particle in lattice
if n and m are nearest neighbours. The off
diagonal elements if n m, give us the energy
of interaction between different atomic orbitals
(hopping integral).
Hˆ
Hˆ
t0
0 m ,n
m ,n i
m ,n i
0
Ek
t0
0
0
t0
0
0
...
t0
...
Dispersion law for Particle in lattice :
Ek
0
t0
t0
0
...
0
0
0
t0
i xˆ , yˆ
Such a matrix is called tridiagonal or Toplitz matrix
0
0
...
0
0
t0
t0
0
k
0
t0
0
( Hˆ ) m ,n
k
t0
0
m ,n
0
t0
0
Otherwise, i.e. neglect interactions between
orbitals on non-neighbor atoms.
0
na
Rn
Tight1nn):
):
Tight-binding approximation (1 nearest neighbours
i if n = m they are simply o of an electron in an
uni Hˆ uni
0 atomic orbital.
i
m
1 ik R n
e
N
ckn (r )
t0
0
c1k
c2k
c3k
c1k
c2k
Ek c3k
cNk
cNk
eik Rn
t0
n
Matrix algebra:
Summation over the nearest neighbours
VUGavryushin
PFK
V.
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46
TightTight-binding approximation
1D crystal. One ss-orbital, single atom basis
Ek
0
e ik Rn
t
0
t [ e ik a
e
ik a
]
n
Euler relations:
Summation over the nearest neighbours:
j-1 j j+1
exp(ika)
{Rn} = {(a,0), ( a,0)}
exp( ika)
E (k )
kn
0
2
N 1
0
cos(ka) i sin( ka)
2t cos ka
n
n
En
cos(ka) i sin(ka)
2t cos
,
n
N 1
a ,
n 1,2,..., N
k max
VUGavryushin
V.
PFK
2
2
2a
a
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47
Hydrogen molecule H2
TightTight-binding approximation
Atomic orbital of first atom
Hydrogen atom.
atom. 1s1s-state
1
1
1
1s
exp
a0
r
a0
r r1
exp
Atomic orbital of second atom
1
2
a0
a0
r2
1
2
(r ) c1 1 (r ) c2 2 (r )
2
0
2t cos k n a
kn
En
0
2t cos
n 1, 2; N
E1
E2
n
N 1
2
n
N 1
n
,
1
2
2
1
2
2
1
2
2
1
2
Adjacent atoms:
"antibonding" combination
a ,
2
0
2t cos
0
2t cos
1
2 1
0
t,
0
t,
2
2 1
E
VUGavryushin
V.
PFK
a0
a0
r1
E (k n )
r r2
exp
0
t
All atoms in phase:
"bonding" combination
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48
TightTight-binding approximation
1D crystal. One ss-orbital, single atom basis
j-1 j j+1
Ek
E (k n )
e ik Rn
0
kn
n
0
2t cos k n a
n
2
n
N 1
,
n 1,2,..., N
Hydrogen molecule H2
Adjacent atoms:
"antibonding" combination
E
0
All atoms in phase:
"bonding" combination
VUGavryushin
V.
PFK
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49
Density of states (1D)
E0-2
The density of states is defined as the number of
energy levels (electronic states) per unit interval
of energy:
DOS
dE
dn
1
dn dE
4
N
1
( En )
dn
dEn
4
2 n
sin
N
N
E
2
E0+2
N=4
4
N
N=6
N=20
2 n
N
1 cos 2
N=100
For the ring of N atoms, we have
En
2 cos
0
2
0
N
4
2
(E
0
)2
Energy band E(k)
So,
2 n
N
DOS(E)
E
(E)
dn
dE
1
N
4
2
(E
top
0)
2
1
.
0
The density of states is infinity when
E= o±2 ,
and has its smallest value of
(E)
N
2
when E =
V. Gavryushin
0.
1
bottom
0
1
2
k
3
0
2
4
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6
50
2D square lattice
TightTight-binding approximation
â2
â1
Ek
a1
a2
eik Rn
0
a 0
0 a
Unit
Cell
n
Summation over the nearest neighbours:
Reciprocal lattice
{Rn} = {(a,0), ( a,0), (0,a), (0, a)}
k1
E (k x , k y )
0
(e ik x a
e
ik y a
e
ik x a
e
ik y a
)
k2
2
1a
0
0
1a
Brillouin zone
Dispersion law:
E (k x , k y )
0
Euler relations:
2 [cos( k x a ) cos( k y a )]
exp(ikx) cos(kx) i sin(kx)
exp( ikx)
cos(kx) i sin(kx)
ky
E
ky
kx
VUGavryushin
V.
PFK
kx
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51
2D square lattice
TightTight-binding approximation
â2
â1
Ek
a1
a2
eik Rn
0
a 0
0 a
Unit
Cell
n
Summation over the nearest neighbours:
Reciprocal lattice
{Rn} = {(a,0), ( a,0), (0,a), (0, a)}
k1
E (k x , k y )
0
(e ik x a
e
ik y a
e
ik x a
e
ik y a
)
k2
2
1a
0
0
1a
Dispersion law:
E (k x , k y )
0
Euler relations:
2 [cos( k x a ) cos( k y a )]
exp(ikx) cos(kx) i sin(kx)
exp( ikx)
1st Brillouin zone
cos(kx) i sin(kx)
ky
E
E
ky
kx
VUGavryushin
V.
PFK
kx
kx
ky
Brillouin zone
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52
TightTight-binding approximation
2D square lattice
Dispersion law:
E (k x , k y )
0
2 [cos( k x a x ) cos( k y a y )]
M
X
Brillouin zone
X
V. Gavryushin
M
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53
TightTight-binding approximation
Dispersion law:
E (k x , k y )
2D square lattice
0
2 [cos(k x a ) cos( k y a)]
Dispersion along high symmetry directions.
Point X [( /a,0)] is a saddle point
in reciprocal space.
V. Gavryushin
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54
Tight-binding approximation
2D square lattice
Dispersion law:
E (k x , k y )
0
2 [cos(k x a ) cos(k y a )]
2D: 1 Van Hove singularity in DOS
Dispersion along high symmetry directions.
V. Gavryushin
Density of states
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55
Simple cubic lattice
Tight-binding approximation
1, 6
e ik an
Dispersion law:
E (k )
0
eik x a
e
ik x a
e
ik y a
e
ik y a
eik z a
e
6
k
ik z a
n
2 [cos(ak x ) cos(ak y ) cos(ak z )]
Band structure for 2s(C) orbitals of cubic carbon lattice
6
BZ dispersion scheme for the Sodium
in simple cubic cell.
Contours of constant energy for
spectrum of 2D square lattice
Contours of constant energy for
spectrum of 3D cubic lattice
Contours of constant energy for
spectrum of free electron
VUGavryushin
V.
PFK
At the k-points , X, R and M of
reciprocal space the crystal orbitals are
real, and the corresponding crystal
orbital schemes are shown.
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56
TightTight-binding approximation
Simple cubic lattice
The density of states as a function of
energy showing its functional change
as the energy becomes larger than
E0. The inset illustrates the saddle
point of the energy dispersion.
Contours of
constant
energy for
spectrum of
3D cubic
lattice.
Points X [( /a, /a,0)] and M [ /a,0,0] are the
saddle points in reciprocal space.
Dispersion law:
VUGavryushin
V.
PFK
E (k )
0
2 [cos( ak x ) cos( ak y ) cos( ak z )]
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57
TightTight-binding approximation
Simple cubic lattice
3D
Dispersion law:
2
E kin
2D
k2
2me
E (k )
0
2 [cos(ak x ) cos( ak y ) cos( ak z )]
Contours of constant energy for spectrum of
free electron and cubic lattice.
V. Gavryushin
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58
TightTight-binding approximation
Simple cubic lattice
Contours of constant energy for
spectrum of cubic lattice.
Dispersion law:
E(k )
0
2 [cos(ak x ) cos(ak y ) cos(ak z )]
V. Gavryushin
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59
TightTight-binding approximation
Simple cubic lattice
Contours of constant energy for
spectrum of cubic lattice.
Dispersion law:
E(k )
0
2 [cos(ak x ) cos(ak y ) cos(ak z )]
V. Gavryushin
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60
Fermi-Dirako pasiskirstymas
3D Carrier concentration
Distributions in Metals
Fermi Surfaces
V. Gavryushin
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61
Tight-binding approximation
3D: 2 Van Hove
singularities in
DOS
Tight binding
Density of States
Density of states dependencies (E) for the tighttight-binding dispersion laws ( o=6, =1) for cubic lattices in several
dimensions: 1D (a), 2D (b), 3D (c).
V. Gavryushin
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62
Evoliucija atomini s ir p orbitali
valentin ir laidumo juostas kristale
1D crystal.
Two orbitals (s + p).
Single atom basis
1D examples:
s-orbital band vs.
p-orbital band
2p
1s
Atomic orbitals
Re
s-like states
E.Kaxiras, Atomic and Electronic Structure of Solids, 2003
V. Gavryushin
kl
( x)
p-like states
Bloch’o bangin s funkcijos (sumos)
kl
eikx
( x)
l
x na
n
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63
Brillouin zone
The first Brillouin
zone for facecentered cubic,
diamond and zinc
blende structures.
V. Gavryushin
Leon Brillouin
(1889 – 1969)
"Wave Propagation
in Periodic Structures"
(1946)
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64
Elektronai amorfiniuose kristaluose
Crystalline matter can be characterized by long-range order.
Solid materials that lack long-range order are called amorphous solids or glasses.
The precise definition of an amorphous material is somewhat problematic.
Delokalizuot Bloch'o b sen
vokos arti BZ centro k=0
amorfiniams k nams negalioja ir yra pateiktos išplautomis juostomis.
Linijomis parodytos b senos, kurios atitiktu amorfinio k no kristaliniam analogui.
Juostos prasiskverbia draudžiam energij sritis ir netgi gali persikloti priklausomai nuo Eg dydžio.
Regular (a) and strongly distorted (b) NaCl structure
V. Gavryushin
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65
Fermion b sen tankis
( E)
1 dN
V dE
1 dN dk
V dk dE
Density of electrons in k is constant and equals
the physical length of the sample L divided by 2 for each dimension.
The number of states between k and k + dk in 3, 2 and 1 dimensions then equals:
dN 3 D
dk
(k)
L
3
4 k
dN 2 D
dk
2
L
Bendresn išraiška 3D atvejui, gspin = 2, atrodo taip
3D
2
( Ek )
dS
E
k k
3
S
2
2 k
dN1D
dk
L
d
(k)
1
( k) d
L
d
Surface area of a sphere
Length of a circle
3D
ia integruojama pagal izoenergin pavirši S fermion dispersijoje.
2
Ec ( k )
2m*
k
3D
(E)
2D (E)
1D ( E )
Eg
k2
2mc dEc
Ec ( k ) E g
1 dN 3 D
L3 dE
1 dN 2 D
L2 dE
1 dN1D
L dE
V. Gavryushin
2
dk
2D
k
m
2
2 3
m*3 2 Ec (k ) E g
1D
m
2
m*
2
1
0D
E Emin
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66
Fermion b sen tankis
Density of electrons in k is constant and equals
the physical length of the sample L divided by 2 for each dimension.
The number of states between k and k + dk in 3, 2 and 1 dimensions then equals:
dN 3 D
dk
3D
V. Gavryushin
L
3
4 k
dN 2 D
dk
2
2D
L
2
1D
2 k
dN1D
dk
3D
L
0D
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67
Generalized form for density of states
(d ) (E)
V. Gavryushin
2 Ad
L
d
2m *
2
d
2
( E Eg )
d
1
2
H
E
Eg
1
where the index d = 3, 2, 1
Specifies dimensionality,
3D, 2D, and 1D, of the sample,
Eg is band gap,
m is the effective mass,
Ad = 1, 2 and 4 ,
H(E/Eg - 1) is the Heaviside
step function
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68
2D b sen tankis
2D (E)
m S
H (E
2
En )
n
Energy levels (bottoms of subbands),
density of states, and
energy dispersion
versus k = (ky2 + kz2)1/2
for 2D electron gas
in GaAs quantum well structure
of 10nm width
V. Gavryushin
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69
Influence of the states Degeneracy
Contours of
2D states
Energy
Of states
States for 2D
quantum well
of 10 10 nm width
in GaAs.
E1D(0)
E11 E12, E21
E22
E31,
E13
E32
E ,E
E23 41 14
E33
E42
E24
E43
E34
E44
Degeneracy
1
2
1
2
2
2
1
2
2
1
E1D(0)/
1
2
5
8
10
13
17
18
20
25
32
E1D(0), (eV)
0.112
0.28
0.448
0,56
0,728
0,952
1,008
1,12
1,4
1792
NanoTubes
Van Hove singularities in d.o.s.
Densities of states
for 3D 3D (gray),
2D 2D (blue),
1D 1D (brown)
0D 0D (black)
electron gases in
GaAs conduction
band.
V. Gavryushin
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70
Influence of states
Degeneracy
Degeneracy
(not including spin)
of the lowest 16
energy levels
in quantum well (2D),
quantum wire (1D)
with square crosscross-section
and a quantum cube (0D)
with infinite barriers.
The energy 1 equals the
lowest energy in a quantum
well which has the same
size
2D
State
0D
1
2
3
4
5
1
4
9
16
25
1
1
1
1
1
2
5
8
10
13
Degenerate
States
(1,1)
(2,1),(1,2)
(2,2)
(3,1),(1,3)
(3,2),(2,3)
6
36
1
17
(4,1),(1,4)
2
14
7
8
9
49
64
81
1
1
1
18
20
25
(3,3)
(4,2),(2,4)
(4,3),(3,4)
1
2
2
17
18
19
10
100
1
26
(5,1),(1,5)
2
21
11
12
121
144
1
1
29
32
(5,2),(2,5)
(4,4)
2
1
22
24
13
169
1
34
(5,3),(3,5)
2
27
14
41
(5,4),(4,5)
2
29
15
50
(5,5)
1
30
16
V. Gavryushin
1D
E/
1
n(E) E/
1
n(E) E/
1
2
1
2
2
3
6
9
11
12
33
1
Degenerate States
n(E)
(1,1,1)
(2,1,1),(1,2,1),(1,1,2)
(2,2,1),(1,2,2),(2,1,2)
(3,1,1),(1,3,1),(1,1,3)
(2,2,2)
(3,2,1),(3,1,2),(2,3,1)
(2,1,3),(1,3,2),(1,2,3)
(3,2,2),(2,3,2),(2,2,3)
(4,1,1),(1,4,1),(1,1,4)
(3,3,1),(3,1,3),(1,3,3)
(4,2,1),(4,1,2),(1,4,2)
(2,4,1),(2,1,4),(1,2,4)
(3,3,2),(3,2,3),(2,3,3)
(4,2,2),(2,4,2),(2,2,4)
(3,3,3)
(5,1,1),(1,5,1),(1,1,5)
(4,3,2),(4,2,3),(2,4,3)
(3,4,2),(3,2,4),(2,3,4)
(5,2,1),(5,1,2),(1,5,2)
(2,5,1),(2,1,5),(1,2,5)
(4,4,1),(4,1,4),(1,4,4)
(5,2,2),(2,5,2),(2,2,5)
1
3
3
3
1
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6
3
3
3
6
3
3
1+3
6
6
3+3
71
1D b sen tankis
2
E1D (k )
E n y ,nz
k x2
2me
2 2
2m e
n 2y
L2y
n z2
L2z
2
k x2
2me
1D ( E )
dN
dE
2mc1 2
E
E n y , nz
Density of states
for GaAs 1D
(10 10 nm)
quantum wire
with infinite barriers (black).
3D semiconductor (gray),
2D quantum well (blue).
Eg = 1,5 eV,
me 0.067m
0.067m
V. Gavryushin
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72
0-D b sen tankis
2 2
E n1 ,n2 ,n3
2 me
n x2
L2x
0D (E)
n 2y
L2y
n z2
L2z
1 (n x
2
ny
2
1
L3
(E
E n x ,n y , n z )
n x , n y ,n z
2
nz )
Density of states for
GaAs Quantum Dot
(10 10 10 nm)
with infinite barriers (black).
(black).
Degenerate states are shown
by quantum numbers (nml
(nml))
and satisfy the data of Table.
The height of discrete lines is
proportional for level degeneracy.
The curves for bulk (3D gray),
quantum well (2D blue)
and quantum wire (1D khaki)
are shown also for comparison.
Eg = 1,5 eV, me 0.067m
0.067m
V. Gavryushin
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73
B sen užpildymas
Identical but distinguishable
particles
Examples: Molecular speed
distribution
V. Gavryushin
Identical indistinguishable
particles with integer spin
(bosons).
Examples: Thermal radiation,
Specific heat
Identical indistinguishable
particles with half-integer spin
(fermions).
Examples: Electrons in a metal,
Conduction in semiconductors.
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74
Pasiskirstymo funkcij
fermionams ir bozonams
sutapatinimas su klasikine
Bolcmano-Maksvelo statistika,
elektronin s b sen tankio
funkcijos (E) fone.
V. Gavryushin
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75
Fermi-Dirako pasiskirstymas
Density of states or number of
energy states per unit volume
in the interval dE
N ( E ) dE
( E ) f ( E ) dE
Energy interval
Number of particles
per unit volume with
energy in interval E
and E +dE
V. Gavryushin
The distribution function
or probability that a
particle is in energy state
E
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76
Plazmonai
Plasma oscillations
Plasma waves
Plasmons - are the quanta
of the plasma oscillations
(fluctuations). These
oscillations are due to return
forces of electric fields.
Let us turn to the response of free electrons to an oscillating
electromagnetic wave. The oscillation is described by the equation of
motion ( ma = F ) of the electron
d 2x
m 2
dt
m
dx
eE0e
dt
i t
The first term on the right is a velocity-dependent damping force and the
second term is the driving force with E the applied field.
The steady-state solution of this equation, representing the oscillating
motion of the electron, must be of the form x(t) = x0e-i t.
Substituting this we get for
the amplitude of motion
solution of oscillating form:
V. Gavryushin
x (t )
e
m
1
2
i
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E0 e
i t
77
Plazmonai
Plasma oscillations
Let us consider the oscillating dipole moment
created by each electron p(t) = - e·x(t).
For a sample with N free electrons, this gives a
polarization
Pe (t )
Ne 2
m
exe (t ) N
1
2
E (t )
i
e E (t )
0
e(
) 1
e2 N
1
0m
e
1
2
i
where p is called the
plasma frequency:
2
p
e(
)
e(
) i
e(
) 1
2
p
2
N
0 m
i
If damping is negligible ( << )
imaginary
part of the free-electron contribution
vanishes and the real part becomes:
V. Gavryushin
p
e
e(
) 1
2
p
2
Paul Drude (1863(1863-1906)
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78
Plazmonai
(h
< Eg)
Dielectric function and E(k) behavior for a plasma of free
electrons with zero damping.
The shaded region corresponds to a forbidden band of frequencies.
Electromagnetic waves within this region are strongly attenuated.
e(
) 1
2
p
2
Energy - wavevector relation (left)
and Frequency dependence of the dielectric constant
near a resonance (right).
The shaded region indicates the range of values for
which absorption occurs.
V. Gavryushin
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79
Plazmonai
(h
> Eg)
Burstein–Moss effect
Burstein–
Burstein–Moss effect
in InN crystal layers.
Room-temperature
absorption edge as
function of free
electron concentration.
Elias Burstein
V. Gavryushin
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80
Eksitonai
Nevill Francis Mott
Jacov Frenkel
For Wannier-Mott excitons (large
radius) we use the effective mass
theory for the envelope function.
V. Gavryushin
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81
Eksitonai
The bonding (envelope)
orbital for elec
electron and
hole with different
effective masses:
me < mh.
Eksitono transliacinio ir vidinio jud jim atskyrimas
2
r = re - rh
2m e
2
2
e
2m h
2
2M
R
me re
me
m h rh
mh
2
K2
2M
2
2
R
2
2
2
k2
e2
rh
0 re
2
h
2
r
e2
r
e2
2 |r|
x (R, r )
x ( R, r )
ex (re , rh )
Ex
Ex
Redukuota mas
V. Gavryushin
E ex
ex (re , rh )
x (R, r )
x (R , r )
cv
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1
mv
1
mc
1
82
Hamiltonianas suskyla dviej nepriklausom dali sum ; R ir r yra nepriklausomi
kintamieji, tod l lygties sprendiniu bus dviej parcialini bangini funkcij
sandauga:
viena kuri priklauso tik nuo eksitono svorio
centro koordinat s (transliacin ), o kita - nuo
(
R,
r
)
(
R
)
G
(
r
)
ex
ex
env
elektrono pad ties skyl s atžvilgiu (gaubiamoji
- envelope)
Eksitonai
state ir padaline abi gautos lygties puses iš
2
1
x (R )
2M
2
R
x (R )
1
G (r )
x
, gauname:
2
2
2
r
e2
r
G (r ) = E x
Šios lygties dešin je yra pastovus dydis; kair pus - dviej , skirting kintam
Tokiai lygybei turi galioti s lygos:
2
2M
2
R
2
The 1st term gives Bloch solution
x (R )
= ER
x (R )
(R )
K
2
2
r
, nari suma.
e2
r
u c (re ) u v (rh ) e iKR
where uc,v are central cell functions.
G (r )
E r G (r )
The second leads to the hydrogen atom problem,
Ex = ER + Er
V. Gavryushin
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83
2
Eksitonai
Eex.cont (k )
Eg
k2
2
cv
cv
Redukuota mas
1
mv
1
mc
1
The eigenvalues are
2
E x (K )
En
Er
e4 1
2 2 2 n2
ER
(Eg
K2
2( m e m h )
En )
Hydrogenic series of
exciton levels
EB
n2
Tai apib dina š derin kaip nauj , impulso P = K
kvazidalel , susijusi su mas s centro koordinate R, kuriai
dingas savo dispersijos d snis Ex(K), atitinkantis
kvazidalel s efektin s mas s M = me*+ mh* transliaciniam
jud jimui su kinetine energija ER = 2K2/2M :
Fn 00 (0)
V. Gavryushin
2
V0 1
aB3 n3
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84
2
Redukuota mas
Ec (k ) Ev (k )
Eg
E
E
k2
2mv
2
k2
2mc
2
Eg
2
k2
cv
Redukuota mas
center of mass
cv
C
Ec
kc2/2m c*
Ecv
k2/2
cv
Eg
cv
1
mv
1
1
mc
k
0
V
0
Ev
kv2/2mv*
0
mv *=
Schematic definition of a system with masses m and M in
motion (a), and the corresponding reduced mass model
k system (b), in which a particle with a reduced mass is
moving around the center of mass, indicated by .
b)
V. Gavryushin
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85
Eksitonai
Jonizuot eksiton tolydinio spektro b senos
Sommerfeld’o faktorius
Gk (0)
2
exp( )
sh( )
2
2
1 exp( 2
)
k
kur
EB
x
V. Gavryushin
EB
- Eg
2
EB
2 2
k
1
ka B
Hydrogenic series
of exciton levels
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86
Eksitonai
Uždaros parabolin s
trajektorijos
Hiperbolin s
trajektorijos
aB
V. Gavryushin
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87
Eksitonin šviesos sugertis.
sugertis.
Kontinuumas.
Kontinuumas.
EB
E
2
EB
- Eg
E
cv
B
2 2
k
ex ( )
2
( , eksitonis )
cv ( , vienelektr onis )
ex
1
kaB
Fnlm (0)
2
cv
( )
Fk (reh ) r
eh
2
k
0
exp ( )
sh( )
)
K
F1s (k )
8
VUGavryushin
V.
PFK
2
1 exp ( 2
e
ik r
1s
( r ) dr
aB3 / 2
[1 (ka B ) 2 ]2
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88
Šviesos sugertis
Eksitonin sugertis
En, 0
Fn 00 (0)
Eexb
n2
Eg
2
V0 1
aB3 n3
ex ( )
nlm
(e p ) 0
2
Fnlm ( reh ) r
eh
VUGavryushin
V.
PFK
0
cv
( )
2
k
0
K
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89
Fotonai vakuume
p
k
E
E
Fotonai
2
c
k
c p
Dielektrin skvarba
Fotonai kristaluose
E
c
n
k
c
Re
( )
k
Palyginkite gautus rezultatus foton dispersijai
esant laisv kr vinink plazmai
su dispersijos kreiv mis reliatyvistin s dalel s :
E
m02 c 4
c2 p2
kuris pateiktas skirtingos prigimties dalel ms:
kai rimties mas m 0 (punktyras – "fotonplazmonas")
ir kai m = 0 (ištisin linija - fotonas).
V. Gavryushin
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90
Bozon b sen tankis
(E)
Fotono dispersijos d snis :
k
( n x , n y , nz )
L
L
L
3
v min
E (k )
L
dN 3 D
dk
c~ k
E (k )
c~
1 dN 1 dN dk
V dE V dk dE
2
2
c k
T
/c
phot
dE
dk
c0
4 k2
( L)3
L
3
c~
2
2
~
c3
E
~
c
k
4 k2
( )
4
Surface area of a sphere
dS
4 k
2
3D
(E)
1 dN 3 D
L3 dE
1 dN 3 D dk
L3 dk dE
1 L
L3
3
4 k
2
1
c~
1 L
L3
3
E2 1
4
( c~) 2 c~
4E 2
2 3~ 3
c
Tipin bozon
sen tankio funkcija,
nusakoma j dispersijos d sniu
min
a
kn
n
n
V. Gavryushin
2L
max
2
L
2a
,
n 1,2,
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91
Fotonai
Br
= 2a sin(
Br)
Fotonai
kristaluose kvazidalel s
A regular array of atoms diffracts Xrays when the Bragg condition is
met. For incident X-rays of a given
wavelength different planes reflect
at different Bragg angles.
Photonic
bandgaps
V. Gavryushin
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92
Fotonai
Fotoniniai kristalai
kristalai
Photonic bandgaps
3D representation of the
photonic band structures of (a)
an isotropic homogeneous
nondispersive medium and (b)
a 1D photonic crystal.
Only 2D slices of the wave
vector space are depicted.
V. Gavryushin
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93
Elektroniniai ir
Fotoniniai kristalai
V. Gavryushin
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94
Poliaritonai
EksitonEksiton-fotonin kvazidalel kristaluose
2
Eex (k )
E
Eg
c
n
dE
dk
EB
k2
2(mc mv )
k
c
( )
k
c
( )
(a) Dispersion curve of a ‘bare’ exciton and ‘bare’
photon (straight line with slope of hc/ , i.e. without
considering their mutual interaction.
(b) Schematic of the exciton–polariton dispersion
curves. LT stands for the longitudinal-transverse
exciton splitting.
V. Gavryushin
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95
Foton dispersijos d snis:
Fotonai
( )
Dielektrin skvarba
,0 Hˆ ( sp ) 0, i 0, i Hˆ ( abs ) ,0
2
c
Re
( )
k
2
At the matter:
Ei
i
Re
c
k
n
E
At the vacuum:
( )
,0 Hˆ ( sp ) 0, i 0, i Hˆ ( abs ) ,0
2
i
Ei
dE
dk
c
( )
At the resonance:
A atom undisturbed by external electromagnetic fields has "sharp"
resonance lines, i.e. infinitely long lived quantum states only.
Radiative decay is a consequence of interaction with the
electromagnetic vacuum, here displayed through its density of
states. The natural line width is a consequence of this interaction.
V. Gavryushin
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96
Start with the simplest case of monoatomic linear
chain with only nearest neighbour interaction
Fononai
1-D Monatomic Lattice
Parabolic potential:
F ( x)
Hook’
Hook’s law:
x
2
M
Newton law: F = ma
d un
dt 2
•The force to the right:
(u n
•The force to the left:
(u n u n 1 )
1
Fn
1
2
U ( x)
kx x2
U ( x)
kx x
Longitudinal wave:
a
a
un )
•The total force =
Fn
FnL
FnR
(u n
The force on the nth atom
un
0
(2un
0
un
1
)
(u n
1
un 1 )
0
Un-1
Un
Un+1
un 1 )
If one expands the energy near the equilibrium
point for the nth atom and use elastic
approximation, Newton’s equation becomes
Eqn’s of motion of all atoms are of this
form, only the value of ‘n’ varies
V. Gavryushin
mu n
(un
1
2un un 1 )
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97
1-D Monatomic Lattice
mun
(un
1
Discrete Differentiation
In terms of indices, the first and second
derivatives are written as
2un u n 1 )
• All atoms oscillate with a same amplitude u0 and frequency
• Then we can offer a solution:
Plane wave
dun
dt
.
un
d 2 un
dt 2
..
un
..
un
2
Ae
un
un
i Ae
i
2
2
i kxn0
Ae
i kxn0
t
For the expected harmonic traveling
waves, we can write
t
i kxn0
t
xn0
na
xn na un
V. Gavryushin
.
Undisplaced position
Displaced position
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98
Equation of motion for nth atom
1-D Monatomic Lattice
mun
2
m
Ae
i kxn0
(un
t
i kxn0 1
Ae
kna
m
m
2
2
Ae
Ae
i kna
i kna
t
i kxn0
2 Ae
Ae
Ae
i kna ka
i kna
t
t
i kxn0 1
Ae
e
t
ika
2A e
2A e
i kna
i kna
t
k (n
kna
k(n 1)a
t
t
2un un 1 )
1
t
t
Ae
Ae
1) a
i kna ka
i kna
t
t
e
ika
Cancel Common terms
m 2
V. Gavryushin
eika 2 e
ika
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99
1-D Monatomic Lattice
2
m
m
e
2
ika
e
ika
2
eix e ix
2 cos x
eika e
2 cos ka
2
1 ix
e e
2
cos 2 x
sin 2 x
1
1 cos x
2
ix
2 cos ka 2
2 K (1 cos ka)
m
ika
cos x
sin x
2
4 sin
2
1 cos x
ka
2
Maxim
4
2 ka
sin
m
2
4
ka
sin
m
2
max
V. Gavryushin
4
m
f it is
o
e
u
al
um v
a
a
2
a
1
4c1
M
The result is periodic in
k and the only unique
solutions that are
physically meaningful
correspond to values in
the range:
k
x
2
2 sin 2
k
0
a
a
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2
a
100
Fononai
Fn
Fn
(2un
0
un
0
un 1 )
1
u0e i(
un
Solution: plane wave
Mu' 'n
(2 e
ika
eika )un
2 (1 cos(ka))un
(k)
2 f
2
2
M
sin
vg
Group velocity: v g
V. Gavryushin
Phonons
One-Atomic Linear Chain.
Dispersion relation
Newton law: F=ma
d 2 un
M
dt 2
Lattice Vibrations
Akustiniai fononai
t ki xn )
0
0
a
k
2
vg k
d
dk
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101
Fononai
Linear dispersion
vs k ,
Sound velocity:
vs
Akustiniai
fononai
V. Gavryushin
One-Atomic Linear Chain.
Dispersion relation :
(k )
max sin
C/
max
2
a
k
2
M
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102
Optiniai
Fononai
Fononai
d 2u n
M1 2
dt
1
d 2 vn
M2 2
dt
(u n vn )
1
2
( vn u n )
(u n vn 1)
2
(vn un 1)
Diatomic Chain.
Acoustic and Optical branches.
2
1, 2 ( k )
2
0
2
1
2
1
1
sin 2
2
0
4
(
1
2)
M1 M 2
M 1M 2
2
4
1
2
M 1M 2
sin 2
ka
2
0
4
1
(
1
a
k
2
,
M 1M 2
,
)(
)
M
M
2
1
2
2
M 1M 2
(M1 M 2 )
V. Gavryushin
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103
Optiniai
Fononai
Fononai
Diatomic Chain.
Acoustic and Optical branches.
2
0
2
1, 2 ( k )
1
2
1
2
0
4
1
(
1
1
2
sin 2
a
k
2
,
M 1M 2
,
M2)
2 )( M 1
2
M 1M 2
(M 1 M 2 )
Zone
folding
V. Gavryushin
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104
Fononai
V. Gavryushin
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105
Fononai
The optical modes have frequencies near = 1013 1/s,
which is in the infrared part of spectrum. Thus, when
IR radiation is incident upon a lattice it should be
strongly absorbed in this band of frequencies.
Transmission spectrum for IR radiation incident
upon a very thin NaCl film.
Note the sharp minimum in transmission
(maximum in absorption) at a wavelength of
about 61 10-4 cm. This corresponds to a
frequency = 4.9 1012 1/s.
V. Gavryushin
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106
Fononai
Phonons in a simple 2D square lattice
LA
i
= 2 ki/L
TA
Only First nearest
neighbors
Acoustic Phonon Bands
Phonons in a 2D Mono-atomic Crystal
V. Gavryushin
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107
Fononai
Phonons in a simple 2D square lattice
LA
TA
Saddle point
(hyperbolic dispersion)
Acoustic Phonon Bands in a 2D
MonoMono-atomic square lattice Crystal
Sound
V. Gavryushin
Phonon
Free electron
Lattice electron
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108
Fononai
Acoustic Phonon Bands in a 3D MonoMonoE (k )
atomic simple square lattice crystal
d
m
d
[1 cos(aki )] 2
i 1
m
i
a
sin 2 ( ki )
2
1
No Saddle points in free space !
Saddle points
(with hyperbolic dispersion)
in discrete lattice
Contours of constant energy for
spectrum of 3D cubic lattice.
Sound
V. Gavryushin
Phonon
Free electron
Lattice electron
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109
Acoustic Phonon Bands in a 3D MonoMonoatomic simple cubic lattice crystal
Contours of constant energy for
spectrum of 3D cubic lattice.
Dispersion laws:
Acoustic Phonon Bands:
d
d
E (k )
[1 cos(aki )] 2
m
i 1
m
i
a
sin 2 ( ki )
2
1
Electron Bands in TightTight-binding approximation :
d
E (k )
0
2
cos(aki )
i 1
V. Gavryushin
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110
Acoustic Phonon Bands in a 3D MonoMonoatomic simple cubic lattice crystal
Contours of constant energy for spectrum
Dispersion laws:
Acoustic Phonon Bands:
d
d
E (k )
[1 cos(aki )] 2
m
i 1
m
i
a
sin 2 ( ki )
2
1
Electron Bands in TightTight-binding approximation :
d
E (k )
0
2
cos(aki )
i 1
V. Gavryushin
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111
Elektroniniai ir
Fotoniniai kristalai,
Fononai
Zone folding
method
V. Gavryushin
a’ = 2a
= primitive cell
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112
Fononai
Fononai amorfiniuose kunuose
V. Gavryushin
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113
Fononai supergardel se
A superlattice consisting of
alternating thin layers of two
different materials.
V. Gavryushin
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114
Poliaronai
In this we illustrate how a polaron moves through a 1D
lattice.
A polaron is a quasi-particle formed of a charge plus a
local distortion that follows it.
These atomic displacements are electronically driven they are not thermal phonons.
Polaron. Electron (center
circle) electrostatically
moves the ions of polar
crystal. As a result of these
changes the effective mass
of the electron changes.
V. Gavryushin
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115
Poliaronai
Ek
0 Hˆ e
Ek 0
ph
0 Hˆ e
0
ph
E0
n
Ek 0
{
n n Hˆ e
ph
0
En
k , n Hˆ eabsph k q, n 1 k q, n 1 Hˆ eemph k , n
E0
k ,n
Ek
q
q
k , n Hˆ eemph k q, n 1 k q, n 1 Hˆ eabsph k , n
E0
Ek
Hˆ e
q
q
ph
Mq
m pol
}
C q J k ,k '
~
2 NM
m*
1
nq
q
m* 1
6
nq 1
6
yra bedimensin poliarinio ryšio (Frohlich'o) konstanta:
e2
4
m*
2
LO
1
1
0
Polaron: In polar crystals motion of negatively charged electron
distorts the lattice of positive and negative ions around it.
Electron + Polarization cloud (electron excites longitudinal EM
modes, while pushing the charges out of its way)
= Polaron (has different mass than electron).
V. Gavryushin
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116
Poliaronai
Bipoliaronai
Kuperio poros
Superlaidumas
Quantum model of the bipolaron formation
Coulomb repulsion part
looks like a sombrero
“Mexican hat” potential
Polaron deformation
potentials overlap part
V. Gavryushin
Mechanistic model
of the bipolaron
formation
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117
Poliaronai
Bipoliaronai
Kuperio poros
Superlaidumas
Mechanistic model
Quantum model
of the bipolaron formation
Phonon-Mediated Effective
Attraction between Electrons
Let us first consider the simple
picture were the lattice is
schematically represented as an
elastic layer, while electrons are
“billiard balls” which locally deform
the layer with their “weight”. A
second ball is attracted by the
deformation produced by the first
one, so that putting two balls close
together reduces the elastic
deformation energy. The elastic
layer, thus, mediates an effective
attraction between the balls.
This picture already shows how this
relatively weak “elastic” effect can
overwhelm the repulsive Coulomb
interaction. We want now to
describe the effective attraction
between electrons as an interaction
mediated by the exchange of
“elastic waves” (see Fig). These
waves are the quanta of elastic
lattice deformation and are referred
to as phonons.
E. Arrigoni, Lectures on the Spin Pairing
Mechanism in High-Temperature
Superconductors, Lect. Notes Phys. 712, 47–65
(2007)
V. Gavryushin
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118
Poliaronai
Bipoliaronai
Kuperio poros
Superlaidumas
1947: Bogoliubov (Superfluidity, Bose condensation with shortrange repulsive two-body interaction)
1950: Ginzburg and Landau (Superconductivity, Bose
condensation of charged quasi-particles)
1957: Bardeen, Cooper and Schrieffer identified the Bosons of
Ginzburg-Landau (effective theory)
V. Gavryushin
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119
Optini reiš
reiškini perturbacijos teorija
Elektron ir foton s veika
Pˆi
i
i
e
pˆ A
mc
Hˆ int
i
(1)
(i
f)
ei
pˆ i A(ri )
mi c
e
( A)
i
mc
e
( pA )
mc
Hˆ int
W
pˆ i2
2m i
e 2
m V
2
ei2
i
A(ri) - elektromagnetinio lauko vektorinis potencialas
P - apibendrinti elektron
impulsai
Apibendrinti impulsai Hamiltoniane fig ruoja
elektron kinetin s energijos pavidale:
Pˆi 2
2mi
ei
A(ri )
c
pˆ i
2m i c
2
A
ec
2
N
exp(i [(
r)
t ])
A 2 (ri )
Kvadrupolin s s veikos operatorius
Dipolin s s veikos operatorius:
( pe ) exp(ik r )
f Hˆ int i
2
(E f
Displaced wavefunction of
electrons and holes in a
polarizing electric field E
Ei
)
f
pˆ
i
k
( x)
1,
jei
x
0
0,
jei
x
0
x
V. Gavryushin
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2
Optini reiškini perturbacijos teorija
Elektron ir foton s veika
A
ec
2
N
exp(i [(
r)
t ])
A(r) - elektromagnetinio lauko vektorinis potencialas
Dipolin s s veikos operatorius:
e
( pA )
mc
Hˆ int
W (1) (i
2
f)
e 2
m V
f Hˆ int i
2
( pe ) exp(ik r )
(E f
Ei
)
f
(n)
W FG
2
F Hˆ M M Hˆ G
F Hˆ G
F
EM
M
EG
F Hˆ M n
M1
Mn
1
(EM n 1
1
EG )
( EM 2
E G )( E M1
( x)
V. Gavryushin
2
M 2 Hˆ M 1 M 1 Hˆ G
EG )
(EF
EG )
1,
jei
x
0
0,
jei
x
0
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3
Vienkvan iai ir daugiakvan iai reiškiniai
Vienfotoniai procesai
(1)
WFG
2
Linear Absorption
2
F Hˆ G
( EF
EG )
F
g ,n
f , n -1
n
n
(n -1)
(n + 1 )
f , n+1
g ,n
a)
Dvifotoniai
Dvifotoniai procesai
b)
(2)
WFG
F Hˆ M M Hˆ G
2
F
M
EM
EG
2
( EF
EG )
Two-Photon Absorption
V. Gavryushin
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4
Vienkvan iai ir daugiakvan iai reiškiniai
Trifotoniai procesai
F Hˆ M 2 M 2 Hˆ M 1 M 1 Hˆ G
2
(3)
WFG
F
( EM 2
M1 M 2
EG )( EM 1
2
( EF
EG )
EG )
f , n -3
m 2 , n -2
m1
m 2 , n 1 -2
m2
m 1 , n -1
m 1 , n 1- 1
f , n 1 - 2 ,n 2 + 1
g , n
a)
1+
2+
f
g , n
= E f- E g .
k 1+ k 2+ k 3 = K f- K g .
3
b)
1+
g
2=
3
,
c)
k 1 + k 2= k 3 .
1=
2+
3
,
k 1 = k 2+ k 3 .
Miš
Mišraus kvantiš
kvantiškumo procesai
2
a)
V. Gavryushin
b)
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5
Dielektrin skvarba
Kvantin prigimtis
2
,0 Hˆ ( sp ) 0, i 0, i Hˆ ( abs ) ,0
{
( )
Ei
2
i
,0 Hˆ ( abs ) 2 , i 2 , i Hˆ ( sp ) ,0
Ei
Displaced wavefunctions of electrons
and holes in a polarizing electric field E
}
Foton elektroninio
“klampumo”
klampumo” efektas
Poliaronai
V. Gavryushin
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6
Foton elektroninio
“klampumo”
klampumo” efektas
Dielektrin skvarba
Kvantin prigimtis
2
,0 Hˆ ( sp ) 0, i 0, i Hˆ ( abs ) ,0
{
( )
2
i
R. Goldhahn, ea., Anisotropy of the dielectric
function for wurtzite InN, Superlattices and
Microstructures 36 (2004) 591–597
Ei
,0 Hˆ ( abs ) 2 , i 2 , i Hˆ ( sp ) ,0
Ei
V. Gavryushin
Real part of the dielectric tensor components
for wurtzite InN. The ordinary and
extraordinary components are drawn by the
full and dashed lines, respectively.
}
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7
Foton dispersijos d snis:
Fotonai
( )
Dielektrin skvarba
,0 Hˆ ( sp ) 0, i 0, i Hˆ ( abs ) ,0
2
c
Re
( )
k
2
At the matter:
Ei
i
Re
c
k
n
At the vacuum:
( )
,0 Hˆ ( sp ) 0, i 0, i Hˆ ( abs ) ,0
2
i
Ei
At the resonance:
A atom undisturbed by external electromagnetic fields has "sharp"
resonance lines, i.e. infinitely long lived quantum states only.
Radiative decay is a consequence of interaction with the
electromagnetic vacuum, here displayed through its density of
states. The natural line width is a consequence of this interaction.
V. Gavryushin
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8
Foton dispersijos d snis:
Fotonai
( )
Dielektrin skvarba
,0 Hˆ ( sp ) 0, i 0, i Hˆ ( abs ) ,0
2
( )
2
c
Re
( )
k
Ei
i
Re
c
k
n
,0 Hˆ ( sp ) 0, i 0, i Hˆ ( abs ) ,0
2
i
Ei
At the resonance:
J. S. Blakemore,
Semiconducting and other
major properties of gallium
arsenide, J. Appl. Phys.
53(10). 1982 R123
Symbolic representation of the real part K1 and
imaginary part K2 of the complex dielectric constant for
GaAs, from the "low frequency" regime of K1->Ko well
below the reststrahlen IR region, through to the highly
absorbing visible and ultraviolet regions of interband
transitions.
Artistic license has been used in drawing the dispersion
K, and absorption K2 curves in the reststrahlen range 10
times wider (and with peaks and valleys 10 times
reduced) than reality.
V. Gavryushin
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9
S veikos operatoriai
a A
A
Dipolin s s veikos operatorius:
Hˆ int
e
( pA)
mc
A
e c
Matricini element atskyrimas komponentus
e
mc
Hˆ int
F Hˆ int G
F Hˆ int G
Nk 1
aˆ k
( abs )
F Hˆ int
G
Nk
aˆ k
Nk
Nk
f ( pA ) g a
Pfg n` a n
f ( pA ) g a
n
Pfg n` a n
n` a n
f Hˆ el g
f Hˆ el g
n
1
( emis )
F Hˆ int
G
V. Gavryushin
exp(ik r )
Nk
( abs )
F Hˆ int
G
Nk 1
e
n`
mc
F Hˆ int( emis ) G
2
V
Gimimo a+j ir nykimo aj operatoriai
[a ( pA ) a ( pA )]
f , n` Hˆ el g , n
a A
n` a n
f Hˆ el g
n
1
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f Hˆ el g
10
Lauko teorijos vadas. Antrinis kvantavimas
n( k1 ), n( k 2 ),
j
a+j aj,
j
n1 , n 2 ,
, n( k i ),
,nj,
, n(k N )
n j n1 , n 2 ,
n1 , n 2 ,
,n j,
, nN
,nj,
Gimimo a+j ir nykimo aj operatori
H
E
H
a a
E
n
vokoms
1
2
1
2
n = 0, 1, 2 , ...
V. Gavryushin
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11
Fotoniniai matriciniai elementai
ak
ak
a
N
En+1 = En +
N
a
a a
N
C N2
CN
1
N
En-1 = En -
N 1
N 1
N 1
1
1
N 1
1
N 1
N
Cn
N
N
aa
1
Nk 1
N 1
a a 1 C N* 1C N
N 1 C N2
N
CN
1
N dr
1
N 1
N
1
V. Gavryushin
1
,
a
C N* 1C N
Nk
Nk
Nk 1
N 1
N 1
N 1
N
a a C N* 1C N
N
,
N
Cn
N
a
N 1
Nk
Nk
C N* 1C N
1
N 1
1
1
N 1
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12
Their names follow from the fact that they annihilate, or create
respectively energy quanta of the quantum oscillator
Fotoniniai matriciniai elementai
Annihilation and creation operators
Nk 1
Nk
aˆ k
ˆk
1a
( abs )
F Hˆ int
G
( emis )
F Hˆ int
G
Nk
Nk
Nk
n` a n
n` a n
The action of the bosonic creation operator b+ and adjoint
annihilation operator b in the occupation number space. Note
that b+ can act indefinitely, while b eventually hits |0> and
annihilates it yielding 0.
V. Gavryushin
Nk
e
( pA)
mc
Hˆ int
1
f Hˆ el g
f Hˆ el g
f Hˆ el g
n
n
1
f Hˆ el g
The action of the fermionic creation
operator c+ and the adjoint annihilation
operator c in the occupation number
space. Note that both c+ and c can act at
most twice before annihilating a state
completely.
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13
Tverm s d sniai ir atrankos taisykl s
W
2
(1)
FG
f Hˆ g
2
(E f
Eg )
f
Energin s atrankos taisykl s
2
W (1)
2
c Hˆ el
fot
2
v
(E)
u c (r ) ( pe ) u v (r ) exp[i(
K )r ] d 3 r
[ E c (k )
E v (k )
]
(0.1)
Šuoli tikimyb
C
V
Simetrin atranka
c Hˆ int v
Pcv
c
(r ) Hˆ el
v
Impulso
Energijos
atranka
atranka
(r )d 3r
Bloch’o bangin s funkcijos
c
(r , k c ) uc (r ) exp(ik c r ),
v
(r , k v ) uv (r ) exp(ik v r )
Bloch’o bangin funkcija
Number of allowed quantum states
may be calculated using a summation
over wavevector k-space
N (E)
E E (k )
k
V. Gavryushin
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14
Energijos atranka
impulso atranka
Number of allowed quantum states
may be calculated using a
summation over wavevector k-space
N (E)
E E (k )
In k-space there is only one grid point in every small volume of size: (2 /L)d
k
dk
k
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
L
2
3
d 3k
15
Joint Density of States
Energijos atranka
Density of allowed quantum states
may be calculated using a
summation over wavevector k-space
D( E )
1
V
E E (k )
k
L
2
dk
k
3
d 3k
it is more convenient to transform the integration variable to E.
By expressing d3k in spherical coordinates and manipulating
the dispersion relation one finds:
d 3k
(2 )3
cv
( )
2
V. Gavryushin
2
( m* ) 3 / 2 ( E
2
2
EC )
3
dE
3/ 2
3
Eg
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16
Joint Density of States
Energijos atranka
Density of allowed quantum states
may be calculated using a
summation over wavevector k-space
(E)
1
V
E E (k )
k
2
E (k )
L
2
dk
k
k2
2m
d 3k
3
d 3k
4 k 2 dk
2
1
(E )
dk x dk y dk z
k2
)
2m
2
k dk ( E
2
0
x
,
2
k2
2m
x
3D
2m
k
,
2
x
dk
dx
m
2 2x
,
cv
( )
2
2
(E)
3/ 2
3
Eg
2m
1
2
2
x
0
m 2m
m
2 2x
dx ( E x)
x dx ( E x)
2 3
0
E
V. Gavryushin
2
2
3
( m* ) 3 / 2 E
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17
Tverm s d sniai ir atrankos taisykl s
Hˆ int
Hˆ el
e 2
m V
e
( pA )
mc
fot
( pe ) exp(iq r )
Impulso atrankos taisykl s
2
W (1)
2
c Hˆ el
fot
2
v
(E)
K )r ] d 3 r
u c (r ) ( pe ) u v (r ) exp[i(
[ E c (k )
E v (k )
]
(0.1)
Šuoli tikimyb
C
V
Simetrin atranka
Pcv
pcv
c Hˆ int v
ˆ
c ( r ) H el
uc , k ( r )[ i
c
uv , k (r )]
d 3r
e
i ( kc kv q ) r
V
Pcv
V. Gavryushin
( pe ) cv
Energijos
atranka
atranka
3
r
(
r
)
d
v
v
kc
Impulso
kv
q
0
kv
q
kc
exp[i (
d 3r
V
c
(r , k c ) uc (r ) exp(ikc r ),
v
(r , k v ) uv (r ) exp(ikv r )
Bloch’o bangin s funkcijos
Hˆ int
( pe ) exp(iq r )
k c , kv q
Pcv
3
K )r ] d r
c Hˆ el v
( pe ) cv
uc (r )( pe )uv (r ) exp[ i (
K
( pe ) cv
K )r ] d 3r
0,
jei
K
0
1,
jei
K
0
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18
Šviesos sugertis
Tarpjuos
Tarpjuostin
(1)
sugertis
(e p ) cv
( )
2
cv
( )
cv
c
( )
Impulso atrankos taisykl s
kv
q
kc
kv kc
v
q
2
q 0
k
( )
2
a
kC
q
,(
500 nm, a
0.5 nm)
kv
for Band-to-Band transitions
VUGavryushin
V.
PFK -- V. Gavriuš
Gavriušinas
www.pfk.ff.vu.lt/
www.pfk.ff.vu.lt
/vg ---- vladimiras.gavriusinas@ff.vu.lt
v.gavriusinas@cablenet.lt
www.pfk.ff.vu.lt
19
Tverm s d sniai ir atrankos taisykl s
Pcv
c Hˆ int v
c
(r ) Hˆ el
v
(r )d 3r
Simetrijos atrankos taisykl s
2
W (1)
2
c Hˆ el
fot
v
2
(E)
u c (r ) ( pe ) u v (r ) exp[i(
K )r ] d 3 r
[ E c (k )
E v (k )
]
(0.1)
Šuoli tikimyb
C
V
Simetrin atranka
Bloch’o bangin s funkcijos
c
Impulso
Energijos
atranka
atranka
1s
(r , k c ) uc (r ) exp(ikc r )
s-like states
v
(r , k v ) uv (r ) exp(ikv r )
2p
p-like states
Bloch’o bangin funkcija
Elektron orbitali
periodin funkcija
Gaubiamoji
periodin funkcija
V. Gavryushin
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20
Tverm s d sniai ir atrankos taisykl s
Pcv
c Hˆ int v
c
(r ) Hˆ el
v
(r )d 3r
Simetrijos atrankos taisykl s
2
W (1)
2
c Hˆ el
fot
v
2
(E)
u c (r ) ( pe ) u v (r ) exp[i(
K )r ] d 3 r
[ E c (k )
E v (k )
]
(0.1)
Šuoli tikimyb
C
V
Simetrin atranka
c
(r , k c ) uc (r ) exp(ik c r ),
v
(r , k v ) uv (r ) exp(ik v r )
V. Gavryushin
Energijos
atranka
atranka
c
v
Elektronini juost
(orbitali )
periodin funkcija
Bloch’o bangin s funkcijos
Jei matricin element sudaran
funkcij kompozicija turi savyje
pilnasimetrin (erdviškai homogenin )
komponent (atvaizdavim ), - tai toks
matricinis elementas yra nelygus
nuliui ir juo nusakomi optiniai šuoliai
yra ”leidžiami pagal simetrij ”.
Impulso
pcv
Pastovioji komponent –
pilnasim trin (s-tipo)
Harmonin komponent
– asimetrin s osciliacijos
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21
Simetrini atrankos taisykli nustatymas
Grupi teorija ir simetrin s atrankos taisykl s
Izotropiniai sfalerito tipo kristalai Td
c
ci
v
i
c
i
c
( R)
v
( R)
v
ci i ( R )
i
c
Zincblende Td
(ZnS):
GaAs, GaP,
InAs, InSb,
ZnSe, CdTe …
Dvigubi atv.
Viengubi
atvaizdavimai
Td
E
v
1
4
6S4
6
Bazin s funkcijos
d
8
(4
2
1
1
1
1
1
1
2
1
1
1
-1
-1
3
2
-1
2
0
0
4
3
0
-1
1
-1
5
3
0
-1
-1
1
2
1
0
2
0
1/2
6
2
2
x +y +z (s-tipo)
x,y,z
c
v
6
8
2 0)
1 0 0 0)
(8
8
Valence and conduction bands at zone center in Zincblende
type crystals have symmetries 4 and 1 respectively.
The optical dipole transition operator has symmetry 4 - so by
the matrix elements theorem: dipole transition from valence
band to conduction is allowed.
3C2
(2 1 0
4
6
8C3
6
1 0 0 0)
3
(2
1 2 0 0)
4
(3 0
1 1
5
(3 0
1
1)
1 1)
(p-tipo)
be spin
3
4
5
1/2
7
2
1
0
- 2
0
3/2
4
-1
0
0
0
3/2
8
3/2
1/2
skaitant spin
V. Gavryushin
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22
Simetrijos atrankos taisykl s
Matrix Representations of Symmetry Operations
cos(
1
-
2)
= l21 ir t.t.
Consider how an {x,y,z} vector
is transformed in space
- kreipiamieji kosinusai
Represented in matrix form this gives:
Transformation matrix
V. Gavryushin
If coordinate axes are oriented
as shown in fig.
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23
Simetrijos atrankos taisykl s
Matrix Representations of Symmetry Operations
Consider a counter clockwise
rotation by about the z axis
Transformation matrix:
From trigonometry
we know that
cos(
1
-
2)
= a21 ir t.t.
- kreipiamieji kosinusai
Represented in matrix
form this gives:
The transformation matrix for a clockwise rotation by
is:
2D Pos kio matrica:
The general operator for rotation about the z
axis by is:
3D Pos kio matrica
rotates points in the xy-Cartesian plane
counter-clockwise through an angle about
the origin of the Cartesian coordinate system
V. Gavryushin
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24
Simetrijos atrankos taisykl s
Matrix Representations of Symmetry Operations
Pos kio matrica:
Pos kio matricos
simetrijos operacijoms:
Veidrodinis
atspindis
Identiškumo
operacija
Inversijos
operacija
i
[x, y, z]
[-x, -y, -z]
Mirror Symmetry
V. Gavryushin
y
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25
Simetrijos atrankos taisykl s
Matrix Representations of Symmetry Operations
Pos kio matrica:
Pos kio matricos
simetrijos operacijoms:
Identiškumo
operacija
Inversijos
operacija
Veidrodinis
atspindis
Pos kis kampu 2 /3 aplink aš
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
z
3
26
Notation for Symmetry Operations
Rotation matrix:
If an object is symmetric with
respect to rotation by angle
it is said to have has an
“n-fold rotational axis”
Mirror Symmetry
y
Screw symmetry:
(x’,y’,z’) = Cn (x,y,z) + T ;
Cn is the corresponding point group.
T = translation operator, (0,0,P/n).
V. Gavryushin
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27
Simetrijos atrankos taisykl s
Pos kis kampu 2 /3 aplink aš
z
3
Simetrijos operacijos trikampio grup s C3v
Symmetries of an equilateral triangle
2D Pos kio matrica:
D R
cos
sin
sin
cos
Dvi C3 pos kio
operacijos:
2C3
If an object is symmetric with
respect to rotation by angle
it is said to have has an
“n-fold rotational axis”
V. Gavryushin
C3 pos kis
du kartus: C32
C33 = E
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28
Mirror Symmetry
Simetrijos atrankos taisykl s
Veidrodinis
atspindis i
Simetrijos operacijos trikampio grup s C3v
Symmetries of an equilateral triangle
2D Pos kio matrica:
D R
cos
sin
sin
cos
3
V. Gavryushin
v
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29
Simetrijos atrankos taisykl s
Charakteri lentel s,
Bazin s funkcijos
Character Tables for Point Groups
Irreducible Representation of C3v
C3v
Grup C3v
Character Table for
C3v Point Group
Symmetries of an
equilateral triangle
Instead of using the irreducible representation matrices we
can often just use their characters - i.e. only handle
numbers not matrices.
Transformation matrices for C3v:
A “square” table showing the
characters of all the irreducible
representations of a group is
known as the CHARACTER
TABLE.
We write down a similar table for the
traces (characters) of the representation
matrices, grouped by classes of
symmetry operations
V. Gavryushin
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30
Simetrijos atrankos taisykl s
Charakteri lentel s,
Bazin s funkcijos
The effect of symmetry
elements on mathematical
functions is useful because
orbitals are functions!
Analysis of the symmetry
of a molecule will provide
us with insight into the
orbitals used in bonding.
Character Tables for Point Groups
A “square” table showing the
characters of all the irreducible
representations of a group is known
as the CHARACTER TABLE.
Neredukuotini
atvaizdavimai
Grup C3v
Character Table for
C3v Point Group
Bethe
Basis functions
Mulliken
E
2C3 3
1
A1
1
1
1
z
2
A2
1
1
-1
Rz
3
E
2
-1
0
(x,y); (Rx, Ry)
Symmetries of an
equilateral triangle
2C3
v
3
v
“1” indicates that the operation
leaves the function unchanged:
it is called “symmetric”.
“-1” indicates that the operation
reverses the function: it is called
“anti-symmetric”.
“E” indicates that the representation is doubly-degenerate – this means
that the functions grouped in parentheses must be treated as a pair
and can not be considered individually.
V. Gavryushin
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31
Simetrijos atrankos taisykl s
Character Tables for Point Groups
Character Table for C3v
Point Group
Theorem: A unique decomposition of a reducible representation into irred. reps. can be
obtained from its characters
V. Gavryushin
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32
Irreducible Representations and Basis Functions
•
Symmetry operations in a group can be represented by the way they transform a set of wave
functions into each other.
A given set of functions
a set of transformation matrices.
These transformation matrices contains information on the symmetry of the functions and are known
as representations of the functions.
•
When the transformation matrices for all the symmetry operations for certain functions can be reduced
to smaller square matrices then the representation is said to be reducible.
•
These transformation matrices are not unique since they depend on the functions chosen. However,
the sums of their diagonal elements (trace) are the same for functions of similar symmetry. The trace
is known as the character of the functions corresponding to the various symmetry operations.
•
Elements in the group can be divided into classes.
Elements belonging to the same class have the same character
•
Instead of working with the transformation matrices it is often more intuitive to work with an
appropriate choice of functions (known as basis functions) since the character can be obtained with
any set of basis functions;
e.g. the three functions: {x,y, and z} can be chosen as the basis functions representing all p states
with the angular momentum l=1.
•
Basis functions with distinct symmetry gives rise to different irreducible representations.
•
For a given group the number of irreducible representations is equal to the number of classes
in the group.
V. Gavryushin
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33
Simetrijos atrankos taisykl s
Symmetry of orbitals and functions
Charakteri lentel s.
Bazin s funkcijos
pz orbital has the same symmetry as an arrow pointing along the z-axis.
The effect of symmetry elements on
mathematical functions is useful
because orbitals are functions!
Analysis of the symmetry of a
molecule will provide us with insight
into the orbitals used in bonding.
An H2O molecule has
two mirror planes.
They are both vertical
(i.e. contain the
principal axis), so are
denoted v and v
V. Gavryushin
Neredukuotinis atvaizdavimas
Character Table for
C2v Point Group
Symmetry of Functions
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34
Simetrijos atrankos taisykl s
V. Gavryushin
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35
Simetrijos atrankos taisykl s
Charakteri lentel s.
Bazin s funkcijos
Symmetry of orbitals and functions
pz orbital has the same symmetry as an arrow pointing along the z-axis.
Character
Table for C2v
Point Group
V. Gavryushin
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36
Simetrijos atrankos taisykl s
Charakteri lentel s.
Bazin s funkcijos
Symmetry of orbitals and functions
pz orbital has the same symmetry as an arrow pointing along the z-axis.
Character Table for
C2v Point Group
V. Gavryushin
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37
Simetrijos atrankos taisykl s
Charakteri lentel s.
Bazin s funkcijos
Symmetry of orbitals and functions
Rotation about the n axis, Rn, can be treated in a similar way.
The z axis is pointing
out of the screen!
If the rotation is still in the same
direction (e.g. counter clock-wise),
then the result is considered
symmetric.
If the rotation is in the opposite
direction (i.e. clock-wise), then the
result is considered
anti-symmetric.
Character Table for
C2v Point Group
V. Gavryushin
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38
bcc lattice as a
superposition of two
simple cubic lattices
Simetrijos atrankos taisykl s
Charakteri lentel s.
Bazin s funkcijos
Viengubi
atvaizdavimai
Dvigubi atv.
Neredukuotinis atvaizdavimas
E
GaAs, CdS, ZnO, …
Sfalerito tipo (Td) kubini kristal charakteri
3C2
6S4
Bazin s funkcijos
6 d
8C3
1
1
1
1
1
1
2
1
1
1
-1
-1
3
2
-1
2
0
0
4
3
0
-1
1
-1
5
3
0
-1
6
2
1
7
2
8
4
(R) lentel
x2+y2+z2 (s-tipo)
x,y,z
(p-tipo)
be spin-orbitin s s veikos
-1
1
0
2
0
1/2
1
0
- 2
0
3/2
-1
0
0
0
3/2
1/2
3/2
1/2
skaitant spin-orbitin s veik
Viengubos ir dvigubos grup s.
Spin-orbitin s s veikos taka atrankos taisykl ms
Jeigu optiniai šuoliai yra leidžiami pagal simetrij
spinoriniams atvaizdavimams, bet yra draustini nagrin jant
juos viengubuose atvaizdavimuose, tai galima teigti, kad
toki proces tikimyb bus tarpinio didumo tarp simetrijos
leistin ir draustin šuoliu ("silpnai leistini").
V. Gavryushin
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39
Simetrijos atrankos taisykl s
Charakteri lentel s.
Bazin s funkcijos
bcc lattice as a
superposition of two
simple cubic lattices
Character Table for Td group
GaAs, CdS, ZnO, …
Ge, Si
(R) lentel
Neredukuotinis atvaizdavimas
Oh kubini kristal charakteri
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
40
Simetrijos atrankos taisykl s
Pfg
Neredukuotini atvaizdavimai
R
a11
a12
a1n
a 21
a 22
a 2n
a m1
a m2
a mn
Redukuotinas
atvaizdavimas
a11
a 21
a12
a 22
0
0
b13
b23
b31
b33
0
0
( r ) H el
g
( r )d 3 r
0
0
c nn
1
0
0
0
2
0
0
0
0
n
- Neredukuotini
atvaizdavimai
n
The Irreducible Representation is the combination of
symmetry representations in the point group that sum to
give the Reducible Representation.
Elektron-fotonin s s veikos operatorius
paprastai yra dipolinio tipo (pe)
p (r )dr
V. Gavryushin
f
0
b11 b12
b21 b22
b32
f H el g
0
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41
Direct Product and Direct Sum
Direct Sum Representations
block-diagonalized form
Kronecker sum has the important properties:
detC = detA·detB
and trC = trA + trB (det = determinant and tr = trace).
Any element in the direct sum of two vector spaces of matrices
can be represented as a direct sum of two matrices.
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
42
Direct Product
Product Representations
Block diagonal matrix multiplication
The choice of irreducible
representations to form the
direct product should be unique.
The Kronecker product has important property:
If F is the direct product of a number of matrices
A, B,C, ..., then
trF = (trA)·(trB)·.....
e
f
i
e ( R)
f
( R)
ci
i
V. Gavryushin
i ( R)
Each block is multiplied independently, i.e.
ci
i
Symbolically the decomposition of the product can
be written as: a
..
b =
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43
Direct Product and Direct Sum
•
Suppose the functions f1..fn belongs to a n-dimensional irreducible representation
a while the functions g1,..gm belongs to a m-dimensional irreducible
representation b.
The products figj then form a n x m dimensional representation. This new
representation is known as the direct product of a and b .
The symbol for this direct product is: a b .
In case this direct product forms a reducible representation then we can
decompose it into the sum of irreducible representations :
etc.
This sum of irreducible representations is know as the direct sum of
..
The symbol of direct sum is: .
•
Thus symbolically the decomposition of the product of fi and gj can be written as:
..
ci i
a
b =
e
f
•
Notice that the dimension of the direct sum has to be equal to the dimension of
the direct product i.e. n+m.
i
e ( R)
f
( R)
ci
i
V. Gavryushin
i ( R)
The choice of irreducible representations to
form the direct product should be unique.
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44
Wif
(1)
2
f Hˆ int i
2
(E f
)
Ei
e
( pA )
mc
Hˆ int
Applications of Character Tables and Group Theory
pˆ
i
k
f
•
Matrix Element Theorem
– the matrix element M =
|P|
of the interaction operator P
is zero unless the direct product of the irreducible representations
of all three contains the identity representation
c
v
Allowed
transitions
1
”Tiesiogine sandauga”
sandauga” (Kronekerio)
Kronekerio)
Pfg
f H el g
f ( r ) H el
3
(
)
r
r
d
g
f
g
a11
a21
j
a12
a22
a1n
a2 n
1
0
0
2
am1 am 2
amn
0
0
Redukuotinas
atvaizdavimas
V. Gavryushin
i
1
0
0
0
n
Neredukuotinas
atvaizdavimas
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45
Applications of Character Tables and Group Theory
Wif
2
(1)
2
f Hˆ int i
(E f
)
Ei
f
•
Matrix Element Theorem
c
v
1
– Alternate statement:
the matrix element is zero unless the direct product of the irreducible
representations of
and
contains the irreducible representation of
interaction operator H .
c
Allowed
transitions
v
nes
c
v
1
1
1
kuri seka iš bangini funkcij normavimo
ir to, kad 1 yra pilnasimetrin , t.y. Kronekerio
sandauga iš jos nieko nekei ia
V. Gavryushin
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46
Grupi teorija ir simetrin s atrankos taisykl s
rodymas naudingos taisykl s:
1
Normavimo proced ra
p – tipo,
tipo,
kuri seka iš bangini funkcij normavimo
ir to, kad 1 yra pilnasimetrin , t.y. Kronekerio
sandauga iš jos nieko nekei ia
(x)
Ortonormavimo s lyga
f g
p – tipo,
tipo,
f
3
(r )
g
g
g
f
f
*(x)
p (r )
- irreducible representation of
dipole interaction operator P
3
(
)
r
d
r 1
p
||
=1
V. Gavryushin
(r )d r
1, if
0, if
1
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47
Grupi teorija ir simetrin s atrankos taisykl s
s
Hint
Pfg
f H el g
s
f ( r ) H el
3
r
d
(
)
r
g
Hint
d
p
c
v
1
j
1
Forbidden
transitions
Allowed
transitions
V. Gavryushin
f
g
i
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48
Simetrini atrankos taisykli nustatymas
Grupi teorija ir simetrin s atrankos taisykl s
Izotropiniai sfalerito tipo kristalai Td
c
ci
v
i
c
i
c
( R)
v
( R)
v
ci i ( R )
i
c
Zincblende Td
(ZnS):
GaAs, GaP,
InAs, InSb,
ZnSe, CdTe …
Dvigubi atv.
Viengubi
atvaizdavimai
Td
E
v
1
4
6S4
6
Bazin s funkcijos
d
8
(4
2
1
1
1
1
1
1
2
1
1
1
-1
-1
3
2
-1
2
0
0
4
3
0
-1
1
-1
5
3
0
-1
-1
1
2
1
0
2
0
1/2
6
2
2
x +y +z (s-tipo)
x,y,z
c
v
6
8
1/2
7
2
1
0
- 2
0
4
-1
0
0
0
3/2
8
1 0 0 0)
3
(2
1 2 0 0)
4
(3 0
1 1
5
(3 0
1
1)
1 1)
(p-tipo)
be spin
3/2
2 0)
1 0 0 0)
(8
8
Valence and conduction bands at zone center in Zincblende
type crystals have symmetries 4 and 1 respectively.
The optical dipole transition operator has symmetry 4 - so by
the matrix elements theorem: dipole transition from valence
band to conduction is allowed.
3C2
(2 1 0
4
6
8C3
6
3
4
5
Allowed
v
c
transitions
3/2
1/2
skaitant spin
V. Gavryushin
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49
Simetrini atrankos taisykli nustatymas
Grupi teorija ir simetrin s atrankos taisykl s
Izotropiniai sfalerito tipo kristalai Td
7
6
2
5
4
c
Dvigubi atv.
Viengubi
atvaizdavimai
Zincblende Td
(ZnS):
GaAs, GaP,
InAs, InSb,
ZnSe, CdTe …
(2 1 0
6
v
7
1
4
4
6
7
Valence and conduction bands at zone center in Zincblende
type crystals have symmetries 4 and 1 respectively.
The optical dipole transition operator has symmetry 4 - so by
the matrix elements theorem: dipole transition from valence
band to conduction is allowed.
Td
E
1
1
1
1
1
1
2
1
1
1
-1
-1
3
2
-1
2
0
0
4
3
0
-1
1
-1
5
3
0
-1
-1
1
2
1
0
2
0
1/2
6
8C3
3C2
6S4
6
2
5
2 0)
(2 1 0
2
0)
(4 1 0
2 0)
(1 1
1
1
1)
(3 0
1
1
1)
Bazin s funkcijos
d
x2+y2+z2 (s-tipo)
c
x,y,z
v
(p-tipo)
be spin
7
6
2
5
1/2
7
2
1
0
- 2
0
3/2
4
-1
0
0
0
3/2
8
3/2
Forbidden
v
c
transitions
1/2
skaitant spin
V. Gavryushin
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50
Simetrini atrankos taisykli nustatymas
Grupi teorija ir simetrin s atrankos taisykl s
Anizotropiniai heksagoniai kristalai C6v
Dvigubi atv.
Viengubi nered. Atvaiz.
7
c
0
1
3
(2
0
1
9
(2
0
2
0
0 0)
1
(1
1
1
1
(4
0
2
0
0 0)
2
(1
1
1
1
5
(2
9
3 0 0)
(4
7
7
5
(2
2
1
1
0 0)
6
(2
2
1
1
0 0)
C6v
E
C2
2C3
2C6
1
1
1
1
1
1
1
2
1
1
1
1
-1
-1
3
1
-1
1
-1
1
-1
4
1
-1
1
-1
-1
1
5
2
-2
-1
1
0
0
6
2
2
-1
-1
0
0
7
2
0
1
3
0
0
8
2
0
1
- 3
0
0
9
2
0
-2
0
0
0
V. Gavryushin
3
d
7
2
1
1
v
Bazin s funkcijos
0)
1 )
1
1
By inspection of the character table it is
not too difficult to see that the only
possible combination is:
3
0
v
1)
0 0)
7
9
7
7
5
1
6
2
5
z; x2+y2+z2
x,y
1/2
3/2
1/2
3/2
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51
Eksiton b sen simetrija
ex
env
c
v
p-like
s-like
aB – Boro radiusas
Schematic illustration of exciton states with p-like symmetry. The relative size of a localized orbital indicates roughly the relative
probability of the electron being found there, if the hole is taken to be at the origin. "+" and "-"show relative phases. States in
(a) and (b) are s-like in the hydrogenic model, but acquire over-all P character because of the hole in (a) or the electron in (b).
The state shown in (c) is p-like in the hydrogenic model to begin with.
2p
1s
FurjeFurje-atvaizdas
ex
env
f
c
g
env
env
ex
c
c
v
v
1
1
v
1
arba
ex
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
52
Grupi teorija ir simetrin s atrankos taisykl s
Wif
(1)
2
v Hˆ int c
2
( Ec
”Tiesiogine sandauga”
sandauga” (Kronekerio)
Kronekerio)
)
Ev
e
f
c Hˆ el v
c
( r ) Hˆ int
c
v
v
i
(r )d 3r
j
1
Tarpjuostini šuoli atranka:
c
v
arba
1
a11
a12
f
a1n
a21
a22
a2 n
0
2
am1 am 2
amn
0
0
Redukuotinas
atvaizdavimas
i
j
1
0
0
0
0
n
Neredukuotinas
atvaizdavimas
Eksiton b sen simetrija:
c
v
ex
env
f
c
g
env
env
ex
c
c
v
v
v
1
1
1
arba
ex
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
53
Grupi teorija ir simetrin s atrankos taisykl s
W fi( 2 )
f Hˆ m m Hˆ i
2
f
m
Ef
Dvifotoniai šuoliai
2
(E f
Em
c
c Hˆ int v
Ei )
m
m
v
c
c
v
i
c
v
( r ) d 3r
1
j
Tarpjuostini šuoli atranka:
v
v
1
c
( r ) Hˆ int
1
Dvifotoni tarpjuostini šuoli atranka:
Alternatyvios atrankos taisykl s
c
v
1
Dvifotoni eksitoni šuoli atranka:
ex
env
c
f
i
env
env
V. Gavryushin
v
c
c
v
v
ex
1
1
1
arba
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
ex
1
54
Poliarizacini priklausomybi nustatymas
Pavyzdžiui, eksitoniai dvifotoniai suger iai turime atrankos taisykles:
ex
1
ci
2
tod l ir poliarizacin kampin priklausomyb (e1,e2) tur s min
kristalo elementari kampini funkcij tiesin s kombinacijos pavidal :
i
i
(eˆ1 , eˆ2 ) | M Gfg (eˆ1 , eˆ2 ) | 2 | M Gfg (
| M iG G (eˆ1 , eˆ2 ) | 2
2
)
|
2
1
i
Dabar galime pateikti pavyzdžius.
Vienfotoniai suger iai eksitonines
priklausomyb s bus toks:
1
x2
G1(1) (eˆ)
y2
l2
| M (fg1) (eˆ) | 2
exc
z2
m2
M 12
+
1
n2
senas Oh grup s kristaluose atitikimas bazin s funkcijos ir kampin s
env
(cos 2
c
cos 2
v
cos 2 ) 1
const
Dvifotoniai suger iai atitinkamai gautume:
exc
G1( 2) (eˆ1 , eˆ2 )
env
l1l 2
| M (fg1) (eˆ1 , eˆ2 ) | 2
V. Gavryushin
m1m2
c
n1n2
M 12 (eˆ1 , eˆ2 ) 2
v
1
(eˆ1 eˆ2 )
2
cos(eˆ1 , eˆ2 )
M 12 cos 2 (eˆ1 , eˆ 2 )
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
55
Šviesos
sugertis
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
56
Šviesos sugertis
Tarpjuos
Tarpjuostin
Pcv
sugertis
( pe ) cv
( pe ) cv
c
( )
K )r
d 3r
E
Impulso atrankos taisykl s
kv
(1)
K
E
ei(
( )
(e p ) cv
2
cv
q
kc
( )
cv
kv kc
v
q
2
k
q 0
( )
2
a
kC
q
,(
500 nm, a
0.5 nm)
kv
for Band-to-Band transitions
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
57
Šviesos sugertis
Tarpjuos
Tarpjuostin
sugertis
Kittel C. Introduction to Solid
State Physics, 8Ed, 2005
Impulso atrankos taisykl s
for Band-to-Band transitions
Photon dispersion curve:
E
kv
kc
c
q
q
Absorption of a photon of energy and negligible wavevector
takes an electron from E in the filled valence band to Q in the
conduction band. If ke was the wavevector of the electron at
E, it becomes the wavevector of the electron at Q. The total
wavevector of the valence band after the absorption is ke, and
this is the wavevector we must ascribe to the hole if we
describe the valence band as occupied by one hole. Thus kh =
- ke; the wavevector of the hole is the same as the wavevector
of the electron which remains at G. For the entire system the
total wavevector after the absorption of the photon is ke + kh =
0, so that the total wavevector is unchanged by the absorption
of the photon and the creation of a free electron and free hole.
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
58
Šviesos sugertis
Vienelektronis juost modelis.
Parabolinis artutinumas
W (i
Tarpjuostiniai
tiesioginiai
šuoliai
f)
2
f Hˆ int i
2
(E f
Ei
)
f
cv (k )
d 3k
3
ZB (2 )
( E c (k c ) E v ( k v )
)
(1)
( )
(e p) cv
2
cv (
)
cv
E (k )
E (k
k0 )
2
E (k 0 )
V. Gavryushin
2m
(k x2 k y2 k z2 )
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
59
Šviesos sugertis
(1)
cv
W
2
c
[ Ec
W
W
N
v,N
F| =
c,N
1 Hˆ
( N 1)
2
F Hˆ intabs G
|G = |
( kc ), N
( abs )
F Hˆ int
G
(1)
cv
2
(1)
FG
(EF
EG )
EG = Ev(kv) + N
EF = Ec(kc) + (N -1)
-1|
abs
int
( Ev
2
v
(kv ), N
N
)]
N Pˆ fg( el )
c
2
Hˆ el
v
( Ec ( k c ) Ev ( k v )
)
2
Wcv(1)
uc (r )(e p)uv (r )e i ( kv
I
kc q ) r
dr
( Ec ( k c ) Ev ( k v )
)
V
Wk(1)
I (e p ) cv
2
( Ec ( k c ) E v ( k v )
kc
)
| q(
| K ZB |
V. Gavryushin
1 m) |
a
kv
q;
|q | 0
n c
2 n
3A
10 8 cm
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
10 4 cm
1
1
60
Šviesos sugertis
Bugerio d snis:
I (d )
d
I (0)e
ZB ( 2
(1)
2
( )
( )
I
W(
( )
)
W
W(
I
)
2
2 3
(
cv )
32
(
cv
Eg
)
d 3k
Wk(
3
BZ ( 2 )
( )
v
)
E g )1 2
2
Eg
k2
2mv
2
k2
2mc
2
Eg
2
k2
cv
Ec (k ) Ev (k )
Joint Density of States
1
mc
(e p) cv
( )
dI
dz
dI
dz
Ec (k ) Ev (k )
Redukuota mas
cv
d
( E c ( k c ) E v (k v )
)3
k
1
mv
dz
c
d 3k
cv ( k )
dI
I
ln I ( z ) |0d
1
2
cv
( )
cv
32
cv
2
(e p) cv (
E g )1 2
cv
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
61
Joint Density of States
Energijos atranka
Density of allowed quantum states
may be calculated using a
summation over wavevector k-space
(E)
1
V
E E (k )
k
2
E (k )
L
2
dk
k
k2
2m
d 3k
3
d 3k
4 k 2 dk
2
1
(E )
dk x dk y dk z
k2
)
2m
2
k dk ( E
2
0
x
,
2
k2
2m
x
3D
2m
k
,
2
x
dk
dx
m
2 2x
,
cv
( )
2
2
(E)
3/ 2
3
Eg
2m
1
2
2
x
0
m 2m
m
2 2x
dx ( E x)
x dx ( E x)
2 3
0
E
V. Gavryushin
2
2
3
( m* ) 3 / 2 E
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
62
Redukuota mas
Šviesos sugertis
cv
( )
d 3k
( Ec ( kc ) Ev (kv )
3
(
2
)
ZB
)
cv
( )
2
1
2
2
32
cv
2
Eg
2 2
Ec ( k ) E v ( k )
k
Eg
2
cv
Ec (k ) Ev (k )
Redukuota mas :
cv
1
mv
1
mc
1
E
E
cv
C
Ec
kc2/2mc*
Ecv
k2/2
cv
Eg
k
k
0
0
V
a)
V. Gavryushin
Ev
kv2/2mv*
0
mv*=
b)
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
63
Šviesos sugertis
Band tails
The probability for the conduction band edge
energy to occur at an energy EC,
Gaussian :
E
1
2
p ( Ec )
- root-mean-square fluctuation of band edges
Unperturbed
density of states in
conduction band
c( )
Kane
c
Kane
c
(E)
2m
1
2
2
2
3/ 2
1
2
*
c
1
2m
2
2
Convolution
method :
*
c
e
2
1 Ec
2
32
Ec
2
E
( E)
c
( E ) p( Ec )dEc
E
E Ec e
1 Ec
2
2
dEc
Persiklojimo integralas
“Kane”
Kane” band edge
(Gaussian broadening)
VUGavryushin
V.
PFK -- V. Gavriuš
Gavriušinas
www.pfk.ff.vu.lt/
www.pfk.ff.vu.lt
/vg ---- vladimiras.gavriusinas@ff.vu.lt
v.gavriusinas@cablenet.lt
www.pfk.ff.vu.lt
64
Šviesos sugertis
Band tails
Convolution of the unperturbed
density of states and Gaussian
broadening function:
Kane
c
(E)
1
mc*
5/ 2
2
E
(T )
3/ 2 E
E Ec e
mc a B
mv E cA
1 Ec
2
2
dEc
2
coth
2 kT
Gaussian
broadening
function :
1
2
p ( Ec )
Unperturbed
density of states in
conduction band
c
( )
2mc*
1
2
2
e
1 Ec
2
2
32
2
Ec
Kane band edge
(Gaussian broadening)
VUGavryushin
V.
PFK -- V. Gavriuš
Gavriušinas
www.pfk.ff.vu.lt/
www.pfk.ff.vu.lt
/vg ---- vladimiras.gavriusinas@ff.vu.lt
v.gavriusinas@cablenet.lt
www.pfk.ff.vu.lt
65
Šviesos sugertis
Kane
c
(E)
1
5/ 2
"Urbacho"
Urbacho" kraš
kraštas
*
c
2
m
3/ 2 E
E Ec e
E
VUGavryushin
V.
PFK -- V. Gavriuš
Gavriušinas
1 Ec
2
(T )
mc a B
mv E cA
2
dEc
2
coth
2 kT
www.pfk.ff.vu.lt/
www.pfk.ff.vu.lt
/vg ---- vladimiras.gavriusinas@ff.vu.lt
v.gavriusinas@cablenet.lt
www.pfk.ff.vu.lt
66
Grupi teorija ir simetrin s atrankos taisykl s
s
Hint
Pfg
f H el g
s
3
r
d
(
)
r
g
f ( r ) H el
Hint
d
p
c
Allowed
transitions
v
Forbidden
transitions
c
VUGavryushin
V.
PFK -- V. Gavriuš
Gavriušinas
v
i
www.pfk.ff.vu.lt/
www.pfk.ff.vu.lt
/vg ---- vladimiras.gavriusinas@ff.vu.lt
v.gavriusinas@cablenet.lt
www.pfk.ff.vu.lt
j
1
67
Šviesos sugertis
Simetrijos draudž
draudžiami šuoliai
Teilero eilut
p (k )
p (k m (0))
p cv (k )
pcv (0)
m0
p (k m ( K ))[k k m (0)]
k
cv
2
(1)
draust
( )
cv
k
(e p) cv
5
cv ( )
cv (
2
Eg )
( )
(e p) cv
cv
2
32
cv
cv ( )
(e p) cv
2
Cu2O
Eg
cv
Allowed
v
c
transitions
VUGavryushin
V.
PFK -- V. Gavriuš
Gavriušinas
2
cv
k 0
Forbidden
v
c
transitions
(1)
leist
3
GaAs
www.pfk.ff.vu.lt/
www.pfk.ff.vu.lt
/vg ---- vladimiras.gavriusinas@ff.vu.lt
v.gavriusinas@cablenet.lt
www.pfk.ff.vu.lt
68
Evoliucija atomini s orbitali
laidumo juost kristale
p cv (k )
p cv (0)
m0
k
cv
pˆ
i
r
1s
Bloch’o bangin s funkcijos (sumos)
Atomic orbitals
kl
eikx
( x)
l
x na
n
2p
Re
s-like states
kl
( x)
p-like states
E.Kaxiras, Atomic and Electronic Structure of Solids, 2003
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
69
Šviesos sugertis
Simetrijos draudž
draudžiami šuoliai
2
(1)
draust
( )
cv
k
(e p) cv
5
cv ( )
cv (
2
Eg )
( )
(e p) cv
cv
2
2
cv
k 0
Forbidden
v
c
transitions
(1)
leist
3
32
cv
cv ( )
(e p) cv
2
Cu2O
Eg
cv
Allowed
v
c
transitions
GaAs
Cu2O
Absorption spectra of Cu2O
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
70
bcc lattice as a
superposition of two
simple cubic lattices
Simetrijos atrankos taisykl s
Cu2O has
inversion symmetry
atvaizdavimai
Character Table for Td group GaAs, CdS, ZnO, …
Td
E
1
1
1
1
1
1
2
1
1
1
-1
-1
3
2
-1
2
0
0
4
3
0
-1
1
-1
5
3
0
-1
-1
1
2
1
0
8C3
3C2
6S4
6
Bazin s funkcijos
d
0
x2+y2+z2 (s-tipo)
x,y,z
(p-tipo)
be spin
1/2
Oh kubini kristal charakteri
(R) lentel
Neredukuotinis atvaizdavimas
Ge, Si, Cu2O
V. Gavryushin
4
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
71
Šviesos sugertis
Td
Izotropiniai sfalerito tipo kristalai Td
Simetrini atrankos taisykli nustatymas:
nustatymas:
atvaizdavimai
Zincblende Td
(ZnS):
GaAs, GaP,
InAs, InSb,
ZnSe, CdTe …
c
v
1
4
Allowed
v
c
transitions
4
Valence and conduction bands at zone center in Zincblende
type crystals have symmetries 4 and 1 respectively.
The optical dipole transition operator has symmetry 4 - so
by the matrix elements theorem: dipole transition from
valence band to conduction is allowed.
Td
E
1
1
1
1
1
1
2
1
1
1
-1
-1
3
2
-1
2
0
0
4
3
0
-1
1
-1
5
3
0
-1
-1
1
2
1
0
8C3
3C2
6S4
6
Bazin s funkcijos
d
0
x2+y2+z2 (s-tipo)
(1)
leist
( )
32
cv
(e p) cv
2
Eg
GaAs
x,y,z
(p-tipo)
be spin
1/2
4
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
72
Šviesos sugertis
GaAs
Simetrijos draudž
draudžiami šuoliai
GaAs
Cu2O
1
(1)
leist
( )
32
cv
(e p) cv
2
Eg
5
Simetrini atrankos taisykli nustatymas:
nustatymas:
c
5
1
5
v
Forbidden
v
c
transitions
Cu2O
Matrix elements theorem:
4
c
(1)
draust
V. Gavryushin
( )
5
2
cv
(
Eg )
3
v
1
2
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
73
Šviesos sugertis
Cu2O
Simetrijos draudž
draudžiami šuoliai
Spinpin-orbit coupling
Simetrini atrankos taisykli nustatymas:
nustatymas:
c
v
4
7
8
6
6
2
5
3
4
(1)
draust
Forbidden
v
c
both SO
transitions
5
5
( )
cv (
2
Eg )
3
2
cv
Cu2O has inversion symmetry.
Conduction and valence bands have same parity,
so dipole transitions to ss-states are forbidden.
This series is due to transitions to np-states and
as a result there is no 1s
1s line.
line.
V. Gavryushin
Cu2O
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
74
Šviesos sugertis
Oh
7
6
Simetrijos draudž
draudžiami šuoliai
Oh
2
5
Cu2O
g
Oh
Jg
+
{Td} {Td}
-
{Td} -{Td}
Cu2O has inversion symmetry.
Conduction and valence bands
have same parity, so dipole
transitions are forbidden.
Simetrini atrankos taisykli nustatymas:
nustatymas:
6
c
v
7
Td
4
7
6
2
6
7
5
2
Dvigubi atv.
Viengubi
atvaizdavimai
5
Td
E
1
1
1
1
1
1
2
1
1
1
-1
-1
3
2
-1
2
0
0
4
3
0
-1
1
-1
5
3
0
-1
-1
1
2
1
0
2
0
1/2
6
2
1
0
- 2
0
3/2
7
4
-1
0
0
0
3/2
8
8C3
3C2
6S4
6
(2 1 0
2 0)
(2 1 0
2
0)
(4 1 0
2 0)
(1 1
1
1
1)
(3 0
1
1
1)
Bazin s funkcijos
d
x2+y2+z2 (s-tipo)
x,y,z
Forbidden
v
c
transitions
(p-tipo)
be spin
Cu2O
1/2
3/2
1/2
skaitant spin
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
75
Šviesos sugertis
Oh
8
6
Simetrijos draudž
draudžiami šuoliai
3
4
Cu2O has inversion symmetry.
Conduction and valence bands
have same parity, so dipole
transitions are forbidden.
Cu2O
5
Simetrini atrankos taisykli nustatymas:
nustatymas:
c
4
Td
Viengubi
atvaizdavimai
Dvigubi atv.
6
(2 1 0
8
(4
v
6
8
6
3
4
5
Td
E
1
1
1
1
1
1
2
1
1
1
-1
-1
3
2
-1
2
0
0
4
3
0
-1
1
-1
5
3
0
-1
-1
1
2
1
0
2
0
1/2
6
2
1
0
- 2
0
3/2
7
4
-1
0
0
0
3/2
8
8C3
3C2
6S4
6
(8
8
2 0)
1 0 0 0)
1 0 0 0)
3
(2
1 2 0 0)
4
(3 0
1 1
5
(3 0
1
1)
1 1)
Bazin s funkcijos
d
x2+y2+z2 (s-tipo)
x,y,z
Forbidden
v
c
transitions
(p-tipo)
be spin
Cu2O
1/2
3/2
1/2
skaitant spin
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
76
Šviesos sugertis
Eksitonin sugertis
ex ( )
nlm
(e p ) 0
2
k
0
2
Fnlm (reh ) r
eh
0
cv
( )
K
Interacting particles
(excitonic effects)
VUGavryushin
V.
PFK -- V. Gavriuš
Gavriušinas
www.pfk.ff.vu.lt/
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v.gavriusinas@cablenet.lt
www.pfk.ff.vu.lt
77
Šviesos sugertis
Eksitonin sugertis
2
aB
2
EB
2
cv
0
e2
cv
1
aB2
Bloch wavefunction
for free electrons:
(r )
c ,k
1
u c , k ( r ) e ik r
V
Exciton
wavefunction:
nlm
ex , K
(r )
1
Fnlm (reh )eiK r
V
Envelop (gaubiamoji)
function:
VUGavryushin
V.
PFK -- V. Gavriuš
Gavriušinas
www.pfk.ff.vu.lt/
www.pfk.ff.vu.lt
/vg ---- vladimiras.gavriusinas@ff.vu.lt
v.gavriusinas@cablenet.lt
www.pfk.ff.vu.lt
78
Šviesos sugertis
ex
Eksitonin sugertis
| ep | 0
ex
(r ) pe e
ik r
Fnlm (k ) pˆ cv
dr
Fnlm (k )
Furje atvaizdas:
atvaizdas:
vandenilinio tipo 1s-b sena:
3
F1s (r )
F1s ( k )
F1s (r )e
1
e
aB3
d 3r
e
r / aB
e
Fnlm (r )dr
r / aB
Envelop (gaubiamoji)
function:
r 2 sin d d dr
ex
3
B 0
r / aB
e
2
r dr e
ik r cos
sin d
0
2
(
[e
1
ik ) r
aB
ik
aB3
8
aB3 / 2
[1 (kaB ) 2 ]2
(r ) u k ( r ) Fenv (r )
d
0
(
e
1
ik ) r
aB
2
1
ika B3 / 2 (aB1 ik ) 2
]rdr
0
F1s(k)
(k)
(e p ) 0
ex ( )
I
k
0
VUGavryushin
V.
PFK -- V. Gavriuš
Gavriušinas
nlm
(a B1 ik ) 2
(e p ) 0
e
ik r
Furje atvaizdas
F1s (r ) dr
8
k
Fnlm (0) uck (r ) (e p ) uvk (r )
Wex ( )
1
F1s (k )
0
nlm
1/ 2
ik r
2
1
a
ik r cos
ik r
( aB )
e
2
k
0
Spherical Polar
Coordinates
Fnlm (0) pˆ cv (0)
2
Fnlm ( reh ) r
eh
0
cv
( )
aB3 / 2
[1 (kaB ) 2 ]2
K
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79
Eksiton b sen simetrija
ex
env
c
v
p-like
s-like
aB – Boro radiusas
Schematic illustration of exciton states with p-like symmetry. The relative size of a localized orbital indicates roughly the relative
probability of the electron being found there, if the hole is taken to be at the origin. "+" and "-"show relative phases. States in
(a) and (b) are s-like in the hydrogenic model, but acquire over-all P character because of the hole in (a) or the electron in (b).
The state shown in (c) is p-like in the hydrogenic model to begin with.
2p
1s
FurjeFurje-atvaizdas
ex
env
f
c
g
env
env
ex
c
c
v
v
1
1
v
1
arba
ex
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
80
Šviesos sugertis
Eksitonin sugertis
ex
( )
En, 0
Fnlm (0)
Eg
Fn 00 (0)
2
cv
( )
K
Eexb
n2
V0 1
aB3 n3
2
GaAs
Cu2O
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
81
Šviesos sugertis
Eksitonin sugertis
ex
(
( ) C
ex (
g
lim
E
0
n 1
0)
ex (
g
0)
V0
3 3 ( E
aBn
EB n2 )
En, 0
1
2 a B3 E B [1 exp( 2
V0
2 a B3 E B
Eg
EB n 2 )
a B3 n 3 E B
n 1
Allowed
transitions
Eg
EB
E g )]
Sommerfeld’
Sommerfeld’o faktorius
Eexb
n2
Fn 00 (0)
2
V0 1
aB3 n3
GaAs
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
82
Šviesos sugertis
En, 0
Eksitonin sugertis
Fn 00 (0)
V0 1
aB3 n3
2
2
r
V0 1
aB5 n3
Fnp x (0)
ex ( )
C
[
5
aB EB n
Eg
Eexb
n2
Cu2O has inversion symmetry. Conduction
Conduction and valence
bands have same parity, so dipole transitions to ss-states
are forbidden. This series is due to transitions to npstates and as a result there is no 1s
1s line.
line.
1
n5
(
1
GaAs sugerties spektras (T=1,2K
).
(T=1,2K).
Dipoliai leistini pagal simetrij šuoliai.
donorais suriš
surišt eksiton linija
V0 n 2 1
aB5
n5
Eg
Forbidden
transitions
n2 1
)
2
n
n5
EB
cv (
)(1
2[1 exp( 2
2
)
]
)]
Cu2O kristalo sugerties spektras. Dipoliai draustini šuoliai
uoliai. Spektras
prasideda nuo 2p
2p linijos ir formuojamas pp-tipo b sen . Spektras
persikloja su netiesiogini šuoli kontinuumu, (punktyras
punktyras)) – Fano
efektas – linij asimetrija.
asimetrija.
Cu2O
GaAs
beeksiton sugertis
V. Gavryushin
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83
Šviesos sugertis
Eksitonin sugertis
Cu2O has inversion symmetry.
Conduction and valence bands have same parity,
parity, so
dipole transitions to ss-states are forbidden.
This series is due to transitions to np-states and as a
result there is no 1s
1s line.
line.
2
r
V0 1
aB5 n3
Fnp x (0)
ex ( )
C
a B5 E B
[
V0 n 2 1
aB5
n5
1
n5
(
Eg
n 1
EB
n2
)
n2 1
n5
cv (
)(1
2[1 exp( 2
2
)
]
)]
Cu2O kristalo sugerties spektras. Dipoliai draustini šuoliai
uoliai.
Spektras prasideda nuo 2p
2p linijos ir formuojamas pp-tipo b sen .
Spektras persikloja su netiesiogini šuoli kontinuumu (punktyras
punktyras)).
Cu2O
Cu2O
Fano efektas –
linij asimetrija
ex
1s:
2p:
Forbidden
v
c
transitions
V. Gavryushin
Forbidden 1s
exciton line
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84
Šviesos sugertis
Eksitonin sugertis.
sugertis.
Kontinuumas.
Kontinuumas.
Sommerfeld’
Sommerfeld’o faktorius
ex ( )
Fnlm (0)
2
cv
( )
K
2
( , eksitonis )
cv ( , vienelektr onis )
ex
Fn 00 (0)
2
2
( E g (0)
k
EB
E
2
cv
2
2
k2
cv
)
Fk (reh ) r
V0
1
2 aB3 EB 1 exp( 2
eh
2
k
0
exp ( )
sh( )
2
1 exp ( 2
)
)
EB
- Eg
EB
k2
V. Gavryushin
1
kaB
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85
Šviesos sugertis
Eksitonin sugertis
Sud tingos energin s strukt ros
taka spektrams
Fano efektas – linij asimetrija.
asimetrija.
Fano Interference in
Low-Dimensional
Semiconductors
Konfig racin šuoli tikimybi interferencija
(Fano efektas)
W (1 2)
C | M1
M 2 | 2 C | M 12
W (1 2) W (1) W (2)
V. Gavryushin
| M 1 |2
M 22
2M 1 M 2 |
| M 2 |2
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86
Šviesos sugertis
Konfig racin šuoli
tikimybi interferencija
(Fano efektas)
W
2
F
V. Gavryushin
Fr Fc Fc H 0
Fc Fr Fr H 0
EFc
EFr
E0 i
c
E0 i
2
( EF
E0 )
r
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87
Šviesos sugertis
Konfig racin šuoli tikimybi
interferencija (Fano efektas)
2
W
F
( EF
W
Fr Fc Fc H 0
Fc Fr Fr H 0
EFc
EFr
E0 i
c
E0 i
Fr Fc Fc H 0
2
2
2
1
2
c
Fr H 0
q
Wcont 1
E Fr
i
Wcont
q 2Wcont
( EFr
)2
V. Gavryushin
q2
2q
)2
( E Fr
Lorenco f
2
r
)2
r
2
r
E Fr
i
r
continuumas
2qWcont ( E Fr
( E Fr
i
Fano parametras
Wcont 1
r
c
Fc H 0 EFr
2
Re W
r
E0 )
F
W
2
)
2
r
ja
antisimetrinis
Lorencas
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
88
Šviesos sugertis
Konfig racin šuoli tikimybi
interferencija (Fano efektas)
Fano efektas – linij asimetrija.
asimetrija.
Illustration of the Fano formula as a superposition of the
Lorentzian lineshape of the discrete level with a flat
continuous background.
V. Gavryushin
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89
Šviesos sugertis
Fano efektas
Eksitonin sugertis
ex ( )
Fnlm (0)
2
cv
( )
K
Interband absorption of the first
hole/electron miniband for
GaAs/AlGaAs superlattice
Spektras persikloja su netiesiogini
šuoli kontinuumu, (punktyras
punktyras)) –
Fano efektas – linij asimetrija.
asimetrija.
Cu2O
Cu2O kristalo sugerties spektras.
Dipoliai draustini šuoliai
uoliai. Spektras prasideda
nuo 2p
2p linijos ir formuojamas pp-tipo b sen .
Cu2O has inversion symmetry.
symmetry.
Conduction and valence bands have same parity, so
dipole transitions to ss-states are forbidden. This series is
due to transitions to np-states and as a result there is no
1s line.
line.
V. Gavryushin
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90
Šviesos sugertis
Oh, Td
Body Centered Cubic (bcc)
crystal lattice as a
superposition of two simple
Brave cubic lattices
Td
Oh
Oh
V. Gavryushin
g
Jg
+
{Td} {Td}
-
{Td} -{Td}
+ Inversijos centras
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91
Šviesos sugertis
phonon
Tarpjuostiniai netiesioginiai šuoliai
ph
indirect
( Mˆ ph ) ck2 ,i Mˆ i ,vk1
( )
c ,v q ,
(2)
FG
W
F Hˆ i i Hˆ G
2
F
i
Ei
EG
Ecv (k1 )
i
Mˆ i ,vk
optical 2
e pcv (0)
( Ec (k 2 ) Ev (k1 )
ph
)
2
( EF
EG )
(q ) n
( Mˆ ph ) q
D
q
( )
( q ) nq 1 ( )
Phonon
Absorption
Emission
2 V
Phonons
are bosons.
bosons.
BoseBoseEinš
Einšteino
statistika:
statistika:
1
nq
exp
V. Gavryushin
k BT
1
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92
Tarpjuostiniai netiesioginiai šuoliai
ph
indirect
Mˆ
( Mˆ ph ) ck2 ,i Mˆ i ,vk1
( )
c ,v q ,
opt
i , vk1
optical 2
phonon
i
( Ec (k 2 ) Ev (k1 )
Ecv (k1 )
e pcv (0)
( Mˆ ph ) q
2
2
q
nq
( )
nq 1 ( )
abs.
2
em.
(2 ) 3
v
2
3
d k
(2 )3
ZB
2
( 2mv ) 3 / 2
exp
k BT
BoseBose-Einš
Einšteino
statistika
d
v
ZB
Paraboliniam art jimui
1
ph
ind
e pcv
( )
(E
2
2
q
dir
g
)
(nq
2
E gind
q,
ind
( )
( )
1
)
2 ZB
v
c
(
c
v
) d
c
Egind
E gind
(
q
c
8
0
ZB
ph
ind
1
2
e pcv (0)
(E
(
dir
g
q
)d c d
v
q
c
V. Gavryushin
v
3
1
nq
)
ph
2
2
q
)
E gind
1
2
(nq
2
q,
q
1
)(
2
Egind
q
(
E gind
q
)2
)2
)2
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93
Šviesos sugertis
BoseBose-Einš
Einšteino
statistika:
statistika:
1
nq
exp
ph
ind
( )
ind
ph
ind
k BT
( )
V. Gavryushin
( E gdir
(
e pcv (0)
( ,T )
2
2
q
)2
E gind
( nq
q,
2
)
q
1
2
2
1
)(
2
2
q
)2
( E gdir
1
e pcv (0)
Phonon
Absorption
Tarpjuostiniai netiesioginiai šuoliai
E gind
E gind
(
q
q
exp
q
q
k BT
1
)2
Emission
(
E gind
1 exp
q
)2
q
k BT
)2
Low-level absorption spectrum
of high purity Si and Ge at
various temperatures
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94
Šviesos sugertis
Netiesiogini eksiton sugertis
2
ex ph
ind
( )
(E
1
nq
exp
k BT
2
Gex (0) e pcv (0)
dir
g
)
2
q
(nq
2
q,
1
2
Mˆ iexc
, vk1
1
2
dkex ( E ( kex )
)
3
2 ZB ( 2 )
2
1
2
ex ph
ind
( ,T )
( E gdir
2
2
q
3/ 2
cv
E Bdir
)2
E gind
q
exp
3
(2 )3
v
Gex0 e pcv
Gex (0)e pcv (0)
EB
q
k BT
d k
ZB
q
1
2
(2 ) 3
)
q
2
(2
E gind
1 exp
cv
3
)3 / 2
cv
EB
q
q
k BT
Leidž
Leidžiami eksitoneksiton-fononiniai šuoliai
exc.
ind
( )
E gind
(
12
)
q
EB
E Bindir
Draudž
Draudžiami eksitoneksiton-fononiniai šuoliai
ex . forb.
ind
( )
(
E gind
q
)3 2
Sugerties juostos !
N ra rezonansini eksitonini linij .
V. Gavryushin
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95
Šviesos sugertis
Netiesiogini eksiton sugertis
N ra rezonansini eksitonini linij .
Sugerties juostos !
2
ex ph
ind
Gex0 e pcv
( ,T )
( E gdir
2
2
q
3/ 2
cv
E Bdir
)2
E gind
q
exp
EB
q
k BT
E gind
q
1
1 exp
EB
q
q
k BT
Leidž
Leidžiami eksitoneksiton-fononiniai šuoliai
exc.
ind
( )
(
E
ind
g
12
EB
q
)
LO LO
Draudž
Draudžiami eksitoneksiton-fononiniai šuoliai
ex. forb.
ind
( )
(
E gind
32
)
q
E Bindir
Tarpjuostiniai elektronelektron-fononiniai šuoliai
ind
( )
V. Gavryushin
(
E gind
q
)2
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96
Šviesos sugertis
Tarpjuostiniai netiesioginiai šuoliai
ind
( )
E gind
(
q
)2
Netiesiogini eksiton sugertis
exc phon
ind
( )
(
E gind
EB
q
)1 2
N ra rezonansini eksitonini linij .
Sugerties juostos !
E Bindir
LO LO
E Bindir
V. Gavryushin
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97
ex ph
ind
( ,T )
( E gdir
1
EBdir
E gind
)2
q
exp
EB
q
k BT
E gind
q
1
1 exp
EB
q
q
k BT
-LO
Netiesiogini
eksiton
sugertis
O
+L
E Bdir
LO LO
E Bindir
V. Gavryushin
The exciton states do not
appear as peaks as in direct
gap materials but instead as
onset energies, since the
participation of phonons
allows one to reach the
whole density of states.
N ra rezonansini
eksitonini linij .
Sugerties juostos !
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98
Šviesos sugertis
Priemaiš
Priemaišin sugertis
a) Tarpmazgin priemaiš
priemaiša
b) Kraš
Krašto dislokacija
c) Nuosavas tarpmazginis defektas
d) Vakansija
V. Gavryushin
e) Priemaiš
Priemaiš precipitatas
f) Dislokacin kilpa vakansinio tipo
g) Dislokacin kilpa tarpmazginio tipo
h) Pakaitos priemaiš
priemaiša
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99
Šviesos sugertis
Priemaiš
Priemaišin sugertis
Seklus
centrai
V. Gavryushin
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100
Priemaiš
Priemaišin sugertis
Seklus
centrai
Bohr radius:
radius:
kA
1
aA
2
EA
k A2
2mv
2
2mv
2
1
aA
Free (delocalised)
delocalised) carrier :
Bound (localised)
localised) carrier :
u k ( r ) FD ( r )
D (r )
F1s (r )
Envelop (gaubiamoji)
function:
function:
3
( aB )
1/ 2
e
r / aB
1s bound state = Gaussian function
V. Gavryushin
Bloch function
c
(r , k c ) uc (r ) exp(ikc r ),
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101
Seklios
(Kulonin s)
priemaiš
priemaišos
Priemaiš
Priemaišin sugertis
F1s (k )
e
ik r
F1s ( r ) dr
a B3 / 2
[1 ( kaB ) 2 ]2
8
Bound state =
Gaussian function
Furje atvaizdas
F1s ( r )
2
EA
k A2
2 mv
2
2 mv
1
aA
2
3
( aB )
1/ 2
e
r / aB
Envelop
(gaubiamoji)
function:
function:
Gilus
centrai
V. Gavryushin
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102
Šviesos sugertis
Priemaiš
Priemaišin sugertis
A
( r ) uv , 0 (r ) FA ( r )
Envelop
(gaubiamoji)
function:
function:
F1s ( r )
3
( aB )
1/ 2
r / aB
e
2
Bohr radius:
radius:
aB
0
*
2
e m
2
1
2mv* aB2
EA
pˆ cA
FA e p c, k
A
( r ) pe uck ( r ) e
Furje atvaizdas:
atvaizdas:
abs
A
( )
( )
V. Gavryushin
A
( N A n A ) | pˆ cv (0) |2 |
NA
mc a 5A
2
[( / a A ) 2
(k )
A
ik r
EA
2mc (
A
ik r
e
( k ) |2
dr
A
c
( k ) pˆ cv
(r ) dr
(E)
F1s (k )
e
ik r
F1s (r ) dr
8
a B3 / 2
[1 (kaB ) 2 ]2
1s Furje atvaizdas
E A )]4
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103
Šviesos sugertis
A
( )
Priemaiš
Priemaišin sugertis
NA
mc a 5A
2
[( / a A ) 2
EA
2mc (
E A )]4
Seklios
priemaiš
priemaišos
F1s (k ) 8
aB3 / 2
[1 (kaB ) 2 ]2
2
1
2mc* aB2
ED
2
aB
2
0
*
c
e m
Bohr radius:
radius:
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
104
Šviesos sugertis
Band tails
V. Gavryushin
Seklus
centrai
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105
Šviesos sugertis
The probability for the conduction band edge
energy to occur at an energy EC,
Band tails
Gaussian :
E
1
2
p ( Ec )
c( )
Kane
c
(E)
2m
1
2
2
2
3/ 2
1
2
*
c
1
2m
2
2
Convolution
method :
Kane
c
e
- root-mean-square fluctuation of band edges
Unperturbed
density of states in
conduction band
*
c
2
1 Ec
2
32
Ec
2
E
( E)
c
( E ) p( Ec )dEc
E
E Ec e
1 Ec
2
2
dEc
“Kane”
Kane” band edge
(Gaussian broadening)
V. Gavryushin
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106
Šviesos
sugertis
Priemaiš
Priemaišin sugertis
Konfiguracini Koordina i modelis.
S veika su fononais.
“FrankFrank-Kondon'o”
Kondon'o”
nuostoliai
E FC
E n ,i
Ei (Q) M
2
ph
q ,i ( n
S
ph
1 2)
(Q Qg ) 2 2
Configuration diagram for localised
electronic states. Vibrational-electronic
transitions are indicated by the vertical
arrows, together with an absorption and
emission spectrum.
“FrankFrank-Kondon'o”
Kondon'o” nuostoliai
E FC
S
V. Gavryushin
ph
M
2
ph
(Qe Qg ) 2 2
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107
Priemaiš
Priemaišin sugertis ir emisija.
Konfiguracini Koordina i modelis.
S veika su fononais.
2
L( )
n
q
n
ri
n
(
( Ee , n
E g , m ))
Born-Oppenheimer
approximation
n
2
Lmn ( )
n
m
m
n
1
( x)
2 n n!
0
Hn
2
mn
( x x0 )
(
( Ee , m
E g ,n ))
( x x0 ) 2
2 02
exp
0
Lmn
The intensity of phonon assisted transitions is
described by a Poisson distribution
Wo
m(
n
ph )
Sm
Lo m ( )
e
m!
L0
m
S
( )
m
n
S is the Huang-Rhys
coupling parameter
1
2
exp
lim W0 m
S
V. Gavryushin
2
(
E0 m
2
1
2 S
ph
)2
2
exp
(m S ) 2
2S 2
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108
Comparence
Comparence of
of the
the quantum
quantum states
states of
of Shallow
Shallow and
and Deep
Deep Levels
Levels
It is useful to compare wave
functions of the shallow
hydrogenlike defects H and of
the deep defects with -shaped
potential of Lucowsky .
Coulomb wave
2
H
(r ) dr
1 C
2
r
2 r
e
dr
r
R0
3
2
2 r
C2 e
r 2 dr sin d
0
C2 ( e
2 r
2
EI
2
2m
2m E I
, so,
2
It is possible to be convinced by direct substitution, that wave
function is self function of Shrodinger equation with potential of
Coulomb type
and
- type
e2
r
VH (r )
d
0
The specific size -1 (the radius of localization) of the defect can
be entered from reasons of a principle of uncertainty, using the
value of defect binding energy
0
2
m
V (r )
2
1s-state of hydrogenlike defect and Deep defect with -potential
r 2 dr ) [ cos ] 0 2
3
0
2
4 C2 e
2 r
(
r
2
2r
4 2
2
)
8 3
- type wavefunction
2
1 C2
(r ) dr
e
2 r
r2
e 2r 2
C
r dr sin d
2
0 r
0
dr 3
2
2
3
0
H
C2
(r )
e
r
r
e
(r )
2
r
Factors of wave functions are defined by their normalisation.
Matrix elements between of the Blox delocalized and localized
defect states :
M cDH
d
r (r )
(r )
k
H
(r )dr 3
M cD
k
(r )
(r )dr 3
0
2
4 C2 e
2 r
4 C
(e
2
dr
0
4 C2
(0 1)
2
V. Gavryushin
2
C2
2 r
8
)0
(
2
4
2 2
k )
2
k2
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109
Comparence
Comparence of
of the
the quantum
quantum states
states of
of Shallow
Shallow and
and Deep
Deep Levels
Levels
It is useful to
compare wave
functions of the
shallow
hydrogenlike defects
H and of the deep
defects with shaped potential of
Lucowsky .
Boro diametras
Bangin s funkcijos modulio kvadratas: |
|2
Radialin s bang.f-s modulio kvadratas: rk2|
|2
Continuum line - Coulomb wavefunction
Dotted line - -type wavefunction
Matricinio elemento A c modulio
kvadratas: |M c |2
V. Gavryushin
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110
Comparence
Comparence of
of the
the quantum
quantum states
states of
of -core
core Deep
Deep Levels
Levels with
with different
different bound
bound
energy
energy
- type wavefunction
Conduction band
Giliasniam centrui atitinka labiau
išplitusi k-erdv je b sena.
Valence band
V. Gavryushin
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111
Lucowsky spectrum
for Deep Levels
The momentum matrix element for the optical transitions from
the localized state
to the delocalized Bloch type state
k =uk(r) exp(ikr) is easily obtained:
k
Eeff
( )
8
pˆ
2
Eo
1/ 2
4 2e2
3nc mo2
2
1/ 3
I
E
2m
k
pˆ
mo
k
m
2 2
k
EI
2m
3/ 4
2
k2
2m
2
k
EI
Deep level absorption spectrum in the simple -potential
approximation by Lucowsky for deep centers in a “rigid” lattice
Luc
( )
Eeff
2
E0
16 e2 E1/I 2 (
3n m*c
(
Allowed transitions
V. Gavryushin
)3
G. Lucowsky, Solid State Commun.
3, 299 (1965).
Luc
Forbidden transitions
EI )3/ 2
( )
EI ) N / 2
EI1/ 2 (
(
)3
where N=1 for allowed transitions
and N=3 for forbidden transitions
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112
Deep Levels absorption spectra
in Lucowsky model of -potential
Allowed transitions
The family of the symmetry allowed (b)
and forbidden (c) absorption spectra for
deep levels with different binding energy:
1- 0.5eV, 2- 1eV ir 3- 2eV.
Absorption spectra by deep levels for the
Lucowsky transitions, allowed (1) and
forbidden (2) by symmetry.
Allowed transitions
Forbidden transitions
Forbidden transitions
V. Gavryushin
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113
Theoretical spectra of absorption cross-section
Deep level absorption spectrum in the simple -potential approximation by
Lucowsky for deep centers in a “rigid” lattice
•
Lucowsky
model
Convolution of
unperturbed spectra
and Gaussian
broadening function:
Luc
( )
Kopylov Pikhtin
model
G. Lucowsky, Solid State Commun. 3, 299 (1965).
)3
(
where N=1 for allowed transitions
and N=3 for forbidden transitions
Deep level absorption spectra can be described also in the framework of -potential
model with regard to electron–phonon interaction. The spectrum of the
photoneutralization cross-section and its temperature variations can be obtained :
cA
•
EI ) N / 2
EI1/ 2 (
0
(T , )
A 1
cA (T , )
0
A 1
x 1
exp
x2
x 1
exp
x2
(x
)2
(x
E vA
dx
A
mc 1
(
mv EcA
mv
E cA
mc
EvA ) 1)2
dx
A
A. Kopylov and A. Pikhtin,
Sov. Phys. Semicond. 10, 7 (1976)
A
- spectrum smoothing parameter due to electron–phonon interaction
A
(T )
m
aA c
mv EcA
2
coth
2kT
1
aA - dimensionless electron–phonon coupling constant,
- energy of local lattice oscillations,
- photon energy
V. Gavryushin
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114
Deep
Deep Levels
Levels absorption
absorption spectra
spectra influenced
influenced
by
by electron-phonon
electron-phonon coupling
coupling
Absorption spectra changes on the binding energy of
deep levels EvA for the Kopylov- Pikhtin model at
T=250K.
Temperature dependence of impurity absorption
spectra determined by influence of electron-phonon
coupling.
Smoothing parameter
A
temperature dependence
•
V. Gavryushin
Curves:
1- 100K
2- 300K
3- 500K
4- 1000K
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115
Gardelin “fononin ” sugertis
Crystal of Td symmetry
with 2 sublattices:
TO phonon resonance
This generates optical phonons
or transverse electric field waves
with dipole p.
q
2
B
B
2
q
• TO phonon modes of ionic crystals: positive
and negative ions move in opposite transverse
directions.
• This generates transverse electric field waves
V. Gavryushin
• Light resonates with these modes when
the wave vectors and frequencies match:
q
k
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116
Gardel s “fononin ” sugertis
PhononPhonon-polariton
r(
)
(
2
TO
)
0
2
TO
2
• Polariton = coupled TO phonon polarization wave and photon wave
• anticrossing of photon and TO phonon modes when ~ TO
• modifies the dispersion of both photons and TO phonons
E
• Lyddane-Sachs-Teller relationship:
LO
TO
V. Gavryushin
k
k
• Polariton dispersion relation:
2
2
0
c k
c0
2
0
c2
1
2
2
TO
c2
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2
LO
2
TO
2
2
117
Gardel s “fononinis” atspindis
r(
)
(
0
)
Infrared reflectivity
2
TO
2
TO
2
100% reflectivity between
TO
and
LO
• reflectivity band observed between TO and LO
• reflectivity less than 100% due to damping
• damping caused by anharmonic phonon decay.
Typical lifetime ~ 10 ps
V. Gavryushin
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118
Gardel s “fononin ” sugertis
V. Gavryushin
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119
Laisv j kr vinink sugertis
Tarpjuost (selektyvin ) laisvakr
clas (
V. Gavryushin
)
n
sugertis
2
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120
Tarpjuost (selektyvin ) laisvakr v sugertis
ij ( )
ij
n
( )
fi ( , T )
leid
12
n
ij ( )
2
2
3/2
ij
3
( k BT )
( , T ) Cl
(
Mˆ ij ( )
ij
3/ 2
( ) fi ( ,T )
Ei (k )
)
k BT
2
( k BT )
3/ 2
(
m *1
1;
m *2
V. Gavryushin
ij
)1 / 2
exp(
n Mˆ ij ( )
2
12
)1/ 2 exp(
1
12
k BT
N
exp
)
k T
.
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121
Tarpjuost
(selektyvin )
laisvakr v sugertis
leid
12
m *1
1;
m *2
V. Gavryushin
( , T ) Cl
1
n Mˆ ij ( )
2
( k BT )
N
exp
(
3/ 2
k T
12
)1/ 2 exp(
12
k BT
)
.
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122
Tarpjuost
(selektyvin )
laisvakr v sugertis
V. Gavryushin
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123
Šviesos emisija.
Liuminescencija puslaidininkiuose
Tarpjuost laisv j kr vinink
plazmos rekombinacija
Šviesos atspindys
laisv j kr vinink
plazmos
Intraband relaxation times:
0.3 ps (300 fs)
a) Exciton formation illustrated in terms of the
“hot exciton cascade”: after excitation, hot exciton
cooling occurs by optical large curved arrows
and, as the exciton falls to the bottom of the well,
acoustic small curved arrows phonon emission.
b) THz measurements suggest that the motion of
electrons and holes remains uncorrelated for
much of the cooling process. In this case, a more
realistic description involves cooling through the
electron-hole continuum, where the uncorrelated
electron and holes have, on average, no net
center of mass momentum K.
The observation that this process occurs on relatively long time scales in bulk ZnO is remarkable
given the hypothesis that the emissive exciton is formed through a “hot exciton cascade” [1]. In this
picture, photon absorption followed by rapid emission of optical phonons by photocarriers leads to the
formation of hot excitons, which subsequently cool to the emissive K=0 state by slow acoustic phonon
emission Fig.a. Since emission of optical phonons by photocarriers in semiconductors typically occurs
on subpicosecond time scales [2], the slow disappearance of free charges observed here suggests
that the rate determining step for exciton formation is acoustic phonon emission.
[1] D. Kovalev, B. Averboukh, D. Volm, B. K. Meyer, H. Amano, and I. Akasaki, Phys. Rev. B 54, 2518 1996.
[2] J. R. Goldman and J. A. Prybyla, Phys. Rev. Lett. 72, 1364 1994
V. Gavryushin
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124
Šviesos emisija.
Liuminescencija puslaidininkiuose
Tarpjuost laisv j kr vinink
plazmos rekombinacija
direct
V. Gavryushin
indirect
semiconductors
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125
Šviesos emisija.
Liuminescencija puslaidininkiuose
Summary of a photoluminescence experiment showing a
basic experimental setup and schematic of a lowlow-power
luminescence spectrum in the classicalclassical-statistics regime.
Tarpjuost laisv j kr vinink
plazmos rekombinacija
V. Gavryushin
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126
Šviesos emisija.
Liuminescencija puslaidininkiuose
(a) Radiative recombination of an
electron-hole pair by the emission of
photon with energy
~ Eg.
(b) Non-radiative recombination.
Energy released during the
electron-hole recombination is
converted to phonons.
Electrons and holes combine radiatively (with a photon)
or nonradiative (with a phonons)
Adapted from lecture summary #20_1 from Dr. Mitin’s EE240 Lecture
Tarpjuost laisv j kr vinink
plazmos rekombinacija
(1)
FG
W
2
F Hˆ
| G = | c,N ;
F|=
V. Gavryushin
v,N
+1| ;
2
emis
int
G
( EF
EG )
EG = Ec(kc) + N
EF = Ev(kv) + (N +1)
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127
Šviesos emisija.
Liuminescencija puslaidininkiuose
W
(1)
FG
2
emis
F Hˆ int
G
| G = | c,N
F|=
v,N
+1|
2
( EF
(1)
vc
W
EG )
Tarpjuost laisv j kr vinink
plazmos rekombinacija
EG = Ec(kc) + N
EF = Ev(kv) + (N +1)
2
v
emis
1 Hˆ int
(k v ), N
N
Wcvemis
V. Gavryushin
2
c
emis
1 Hˆ int
N
(N
(kc ), N
[ Ev ( N 1)
N
1) (e p ) cv
1
2
fot
v
Hˆ el
( Ec
N
)]
c
( ) ( Ec ( k c ) E v ( k v )
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)
128
Šviesos emisija.
Liuminescencija puslaidininkiuose
Wcvemis
(N
1) (e p) cv
2
fot
Tarpjuost laisv j kr vinink
plazmos rekombinacija
( ) ( Ec ( k c ) Ev ( kv )
)
=1
2
Wcvemis
(I
1) u v (r )(e p )uc ( r ) e i (
kv k c q ) r
dr
fot
( ) ( Ec ( k c ) Ev ( k v )
)
V
Wkemis
(I
1) (e p ) cv
2
fot
( ) ( Ec ( k c ) Ev ( k v )
cv ( )
V. Gavryushin
)
Lspont ( )
kc
n
R( )
kv
q fot ;
n2
2
Rcorr pˆ cv2
| q fot | 0
fot
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cv
( )
129
Šviesos emisija.
Liuminescencija puslaidininkiuose
Wvc (k )
(N
1)
ˆ
v ( k v ) H el
Tarpjuost laisv j kr vinink
plazmos rekombinacija
2
c
(kc )
( Ev ( k v ) Ec ( k c )
2
Lspont
pcv f C ( E ) fV ( E )
V h ,l , s E k
cv
( Ek )
fot
( Ek
)
Spontanin
pontanin emisija
h ) dEk
free-space photon mode density
2
fot
( )
n2
3
2 h c 2 vg
FermiFermi-Dirako
statistika:
statistika:
fC
1
E EF
exp C
k BT
V. Gavryushin
e
EC E F
k BT
1
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130
Šviesos emisija.
Liuminescencija puslaidininkiuose
Wvc
(N
I
z
1)
v
Stimuliuota emisija
Tarpjuost laisv j kr vinink
plazmos rekombinacija
2
Hˆ el
c
( Ev ( k v ) Ec ( k c )
)
Wcv
L
I
dz
z
I ( L)
0
I
z
L
0
I
I
N pcv
I pcv
pcv
2
E
L
2
z
E
0
L
I 0 exp( L )
Bugerio d snis,
neigiami nuostoliai
V. Gavryushin
I (L )
2
ln I ( L) ln( I 0 )
I ( L)
(E)
stiprinimas
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131
Šviesos emisija.
Liuminescencija puslaidininkiuose
Stimuliuota rekombinacija.
Lazeriai
N2
N1
V. Gavryushin
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132
Šviesos emisija.
Liuminescencija puslaidininkiuose
Stimuliuota rekombinacija.
Lazeriai
V. Gavryushin
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133
E
Šviesos emisija.
Liuminescencija puslaidininkiuose
phonons
Exsiton
continuum
Spindulin laisv
eksiton anihiliacija
Binding
energy
hvlaser
hvluminescence
k
V. Gavryushin
www.pfk.ff.vu.lt -- v.gavriusinas@cablenet.lt
134
E
Šviesos emisija.
Liuminescencija puslaidininkiuose
phonons
Exsiton
continuum
Spindulin laisv
eksiton anihiliacija
Binding
energy
hvlaser
hvluminescence
k
In fact the emission spectrum can be calculated from the absorption
spectrum using the so-called van Roosbroeck-Shockley relation [*],
which reads
where (hw) is the absorption coefficient at the energy hw, n is the real
refraction index, and EF= EeF-EhF is the energy difference between the
electron and hole quasi-Fermi levels. The van Roosbroeck-Shockley
relation has been derived using detailed balance
arguments assuming thermal equilibrium, but it can be
shown to have a more general validity.
Empirically, the PL spectrum can be deduced from the absorption coefficient making use of
the van Roosbroeck and Shockley relation (*). Describing the exciton absorption through a
suitable function S(hw), which gives the shape of the absorption peak:
[*] H. B. Bebb and E. W. Williams,
Semiconductors and Semimetals, ed
by R. K. Willardson and A. C. Beer
(Academic, New York, 1972), V. 8, p.
181.
For direct exciton absorption and recombination, neglecting polariton effects, the shape function has been given in two
limiting cases: for weak phonon coupling a Lorentzian function is obtained. For strong exciton phonon coupling, a
Gaussian shape function is obtained. Additional extrinsic broadening of the exciton resonance is caused by crystal
imperfections, strains, and impurities, which result in a Gaussian shape of the exciton line.
V. Gavryushin
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135
Šviesos emisija.
Liuminescencija puslaidininkiuose
EksitonEksiton-fononin rekombinacija
Recombination processes of Wannier excitons in a direct band-gap
semiconductor. K is the wave vector associated with the center-ofmass momentum of the exciton; Egap is the semiconductor band gap
energy, and Eex is the exciton binding energy.
phon is the energy of
an optical phonon, which is nearly independent of momentum.
LA
2 LO
(
1/ 2
kin
exp(
kin
)
nx (
kin
k BT
kin
) W(
kin
)
)
Fig shows the phonon-assisted recombination process of excitons in a
direct-gap semiconductor. Only excitons with low momentum can
recombine via the direct recombination process, while an exciton at any
momentum can recombine via the phonon-assisted process, with the
phonon taking up any excess momentum.
The energy of the emitted photon is equal to the total energy of the exciton minus the energy of the optical phonon, which is nearly constant. The energy
spectrum of the phonon-assisted luminescence therefore gives the kinetic energy distribution of the excitons directly. If we take the matrix element as nearly
independent of the exciton momentum, then the intensity of the light emitted at a given energy is directly proportional to the number of excitons with the
corresponding kinetic energy h2K2/2m.
V. Gavryushin
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136
EksitonEksiton-fononin
rekombinacija
V. Gavryushin
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137
EksitonEksiton-fononin rekombinacija
In Fig. we show the appearance of the LO-phonon satellites
schematically. If we neglect the bottleneck region and
homogenous broadening for the moment, we can deduce
with the Boltzmann occupation probability the distribution of
the excitons as a function of their kinetic energy Ekin
The lineshape of the luminescence of the m-th LOphonon replica is then given by [*]:
Schematic drawing of the decay mechanisms of the
exciton-mLO phonon emission processes
where E0 is the energy of the dipole allowed, transverse
exciton at k = 0.
S. Permogorov, in Excitons, ed. by E.I. Rasha, M.D. Sturge. Modern Problems in
Condensed Matter Sciences, vol. 2 (North Holland, Amsterdam 1982), p. 177
R. Hauschild et. al., Phys. Status Solidi C 3, 976 (2006)
C. Klingshirn, Phys. Status Solidi B 244, 3027 (2007)
V. Gavryushin
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138
Eksiton Ože
rekombinacija - "tarp"tarpeksitonin " sklaida
The P bands in the luminescence
spectra of ZnO at 10K
Polariton dispersion showing the P-band inelastic collision process.
The P2 and P collisions are shown, as well as a schematic of the
luminescence from the P-band.
The peak position of the P-band [i] is shifted from the A-exciton
(EA(T)) by the binding energy of the exciton (Eb) as follows:
The 3/2kBT term is a measure of how many excitons are
available to contribute to the scattering.
[i] Klingshirn CF, Semiconductor Optics, (Springer, New York, 1997).
V. Gavryushin
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139
Defektais suriš
surišt eksiton
rekombinacija
V. Gavryushin
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140
Defektais suriš
surišt eksiton rekombinacija
V. Gavryushin
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141
Poliariton
erdvin
dispersija
Poliariton rekombinacija
E
E
V. Gavryushin
c0
k
Re
( )
c k
c0
k
n( )
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142
Poliariton rekombinacija
E
c k
V. Gavryushin
Poliariton
erdvin
dispersija
c0
k
n( )
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143
Eksiton - fononin
EDK rekombinacija
V. Gavryushin
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144
Eksiton - fononin
EDK rekombinacija
V. Gavryushin
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145
Eksiton - fononin rekombinacija
Fig schematically shows the relationship between absorption and emission
for excitons. In b the dispersion of the excitons is shown, with the A-exciton
at the bottom of the gap. As predicted by Bebb, the peak position of the
phonon-assisted luminescence is shifted from its
LO position in the gap.
Absorption and emission (c,d) in ZnO is shown in black, where strong
absorption of the A-exciton leads to weak emission.
V. Gavryushin
A schematic drawing of the dispersion (a) and the spectra
of the absorption (b), reflection (c), and luminescence (d) of
a high quality, direct-gap semiconductor at low
temperatures in the region of the exciton resonances
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146
2
Priemaiš
Priemaišin s
liuminescencijos
mechanizmai
EA
L( )
Rekombinacija "juosta"juosta-centras"
( N A n A )nC |
nC
|
V. Gavryushin
A
( r ) |2
Furje atvaizdas
A
k A2
2mv
2
1
aA
2mv
2
( k ) |2
c
( E ) fT ( E )
f T ( EC )
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e
EC E F
k BT
147
DonorDonor-akceptorin liuminescencija
DA
( R)
Eg
ED
EA
e2
rDA
DA
( )
e2
rDA
ZnSe:N
WDA (rDA ) Wr
V. Gavryushin
0 exp
2rDA
aD
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148
Dielektrin skvarba
Kvantin prigimtis
2
,0 Hˆ ( sp ) 0, i 0, i Hˆ ( abs ) ,0
{
( )
Ei
2
i
,0 Hˆ ( abs ) 2 , i 2 , i Hˆ ( sp ) ,0
Ei
Displaced wavefunctions of electrons
and holes in a polarizing electric field E
}
Foton elektroninio
“klampumo”
klampumo” efektas
Poliaronai
V. Gavryushin
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149
Foton elektroninio
“klampumo”
klampumo” efektas
Dielektrin skvarba
Kvantin prigimtis
2
,0 Hˆ ( sp ) 0, i 0, i Hˆ ( abs ) ,0
{
( )
2
i
R. Goldhahn, ea., Anisotropy of the dielectric
function for wurtzite InN, Superlattices and
Microstructures 36 (2004) 591–597
Ei
,0 Hˆ ( abs ) 2 , i 2 , i Hˆ ( sp ) ,0
Ei
V. Gavryushin
Real part of the dielectric tensor components
for wurtzite InN. The ordinary and
extraordinary components are drawn by the
full and dashed lines, respectively.
}
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150
Foton dispersijos d snis:
Fotonai
( )
Dielektrin skvarba
,0 Hˆ ( sp ) 0, i 0, i Hˆ ( abs ) ,0
2
c
Re
( )
k
2
At the matter:
Ei
i
Re
c
k
n
E
At the vacuum:
( )
,0 Hˆ ( sp ) 0, i 0, i Hˆ ( abs ) ,0
2
i
Ei
dE
dk
c
( )
At the resonance:
A atom undisturbed by external electromagnetic fields has "sharp"
resonance lines, i.e. infinitely long lived quantum states only.
Radiative decay is a consequence of interaction with the
electromagnetic vacuum, here displayed through its density of
states. The natural line width is a consequence of this interaction.
V. Gavryushin
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151
Foton dispersijos d snis:
Fotonai
( )
Dielektrin skvarba
nQM (
( )
Ei
i
2
1
1
nQ( )
Re
Eg
T)
k
,0 Hˆ ( sp ) 0, i 0, i Hˆ ( abs ) ,0
2
( )
c
Ei
i
Re
2
,0 Hˆ ( sp ) 0, i 0, i Hˆ ( abs ) ,0
2
c
k
n
25 (
At the resonance:
Eg)
1
Eg
2
1
( T)
2
( Eg
)
2
2
( T)
2
( Eg
)
( T)
2
SiC refraction index: Lorentz broadening
20
Eg
Eg
SiC refraction index: Lorentz broadening
2
20
Eg
15
Refraction index, a.u.
Refraction index, a.u.
15
10
5
0
5
0
1
2
Photon energy, eV
V. Gavryushin
10
3
4
0
2.5
3
3.5
Photon energy, eV
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152
Foton dispersijos d snis:
Fotonai
( )
Dielektrin skvarba
,0 Hˆ ( sp ) 0, i 0, i Hˆ ( abs ) ,0
2
( )
2
c
Re
( )
k
Ei
i
Re
c
k
n
,0 Hˆ ( sp ) 0, i 0, i Hˆ ( abs ) ,0
2
i
Ei
At the resonance:
J. S. Blakemore,
Semiconducting and other
major properties of gallium
arsenide, J. Appl. Phys.
53(10). 1982 R123
Symbolic representation of the real part K1 and
imaginary part K2 of the complex dielectric constant for
GaAs, from the "low frequency" regime of K1->Ko well
below the reststrahlen IR region, through to the highly
absorbing visible and ultraviolet regions of interband
transitions.
Artistic license has been used in drawing the dispersion
K, and absorption K2 curves in the reststrahlen range 10
times wider (and with peaks and valleys 10 times
reduced) than reality.
V. Gavryushin
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153
Netiesin optika
Savipraskaidr jimas.
jimas.
Kvantiš
Kvantiškumo metamorfoz s
The momentum conserved photon absorption process.
Instantaneous photon absorption generate nonequilibrium distributions
of electrons (blue) and holes (red), which subsequently thermalize
through ultrafast (
0.3 ps) intraband carrier-carrier and carrier-optical
phonon scattering to form Fermi-Dirac (Boltzmann) distributions.
Slow speed
of light !!
Drastically
growth of n !!
V. Gavryushin
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154
Netiesin optika
Noncollinear SHG phasephase-matching
The dispersion relation of the lowest free exciton
resonance in CuCl (a) and the group velocity determined
from the time-of-flight of picosecond laser pulses and from
the dispersion relation (b) [*]
Y. Masumoto et al., J. Phys.
Soc. Jpn. 47, 1844 (1979)
Poliariton grupinio grei io dispersijos spektroskopija
Poliariton grupinio grei io dispersijos CuCl kristaluose
ultraultra-spar ioji spektroskopija.
a) Poliariton apatin s (LBP) ir virš
viršutin s (UBP) šak
dispersijos kreiv s;
b) Poliariton grupinio grei io vg/c
vg/c priklausomyb nuo
enerener-gijos: taš
taškai eksperimentas, tolydin kreive paskai iuota LBPLBP-šakos iš
išvestines kreiv , punktyras UBPUBP-šakos.
V. Gavryushin
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155
Netiesin optika
Polariton group velocity
R.G. Ulbrich and G.W. Fehrenbach, Phys. Rev. Lett. 43, 963 (1979)
Polariton group velocity
Time-resolved transmission through a thin
(3.7 m) GaAs layer
group velocity
Possible evidence for breakdown of
spatial coherence close to transverse
frequency.
Poliariton grupinio grei io dispersijos
spektroskopija
V. Gavryushin
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156
Netiesin optika
Pusiauklasikinis
fenomenologinis aprašymas
Elektrin medžiagos indukcija, kuria operuoja Maksvelo lygtyse: D = E = E+4 P = (1+4
P (t )
1E
2E
2
3E
3
a)
V. Gavryushin
)E
P= 1E
b) P= 1E + 2E2
( 2>0)
c) P= 1E + 3E3
( 3<0)
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157
Netiesin optika
I)
Pusiauklasikinis
fenomenologinis apraš
aprašymas
P (t )
1E
2E
2
Pirma patyrin kime kvadratiškai netiesin aplink P= 1E+ 2E2 šviesos lauke:
E
Pasinaudojus trigonometrija (cos2x=½(1+cos2x)), indukuotai poliarizacijai gausime:
P(t )
1E
1E
P0
cos t
cos t
P
2E
2
2
2
3E
3
E 0 cos t
cos 2 t
E2
2
2
E 2 cos 2 t
P2 .
II) Kubin je netiesin je aplinkoje
cos3x = 1/3(3cosx+cos3x;
cosx cosy = ½(cos(x+y)+cos(x-y))
tur sime:
P (t )
1 E cos t
1
2
2
E 2 ( 1E
1
2
2
E 2 cos 2 t
P0
V. Gavryushin
2
2
cos
E
t
2
3
4
1
4
3
3
3
3
cos
E
t
3
E 3 ) cos t
E 3 cos 3 t
P cos t P2 cos 2 t P3 cos 3 t
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158
Netiesin optika
E
Šviesos savifokusavimas
Augant šviesos intensyvumui l žio rodiklis gali
prad ti kisti. D l to dar kart panagrin kime
aplink su kubiniu netiesiškumu:
P (t )
E
3E
1
P(t )
1E
( 1E
3
4
cos t
3E
3
3E
3
E cos t
3
cos 3 t
1
4
) cos t
3E
3
cos 3 t
P cos t P3 cos 3 t.
Mus domina pirmas narys, kuris formuoja elektrin s indukcijos D medžiagin lygt :
D
E
(1 4
Reiškia:
E
1)
4 P
3
3E
(1 4
2
0
3
1
3E
3
3E
2
)E
2
L žio rodikliui tod l gausime išraišk :
n
0
1
3
3
0
n2
r
E2
0 (1
3
2
3
E2 )
n0
n2 E 2
0
0
n2 0
I
r
I
V. Gavryushin
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159
Netiesin optika
Kvantmechaninis netiesin s optikos aprašymas
Trifotoniai reiškiniai
W
( 3)
AGG
G H
2
c , v M1 M 2
V. Gavryushin
( em )
2
M2
( E M2
M2 H
( abs)
1
M1 M1 H
E G )( E M1
EG )
( abs )
1
2
G
(E F
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EG )
160
DifferenceDifference-Frequency Generation:
Optical Parametric Generation, Amplification, Oscillation
1
1
3
2
Optical Parametric Amplification (OPA)
1
3
2
mirror
mirror
Optical Parametric Oscillation (OPO)
1
2=
3
1
1
"signal"
2
"idler"
3
3
Optical Parametric Generation
Parametric Down-Conversion
(Difference-frequency generation)
V. Gavryushin
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161
Netiesin optika
Antros optin s harmonikos generavimas
G H 2( em) M 2
2
( 3)
WAGG
( E M2
c , v M1 M 2
2
( 3)
W FG
( Ev
( N1
2)
1
2
M 2 H1( abs) M 1 M 1 H1( abs) G
E G )( E M1
( N 2 1)
2
(EF
EG )
) ( Ev
N1
N2
1
2
)
v ,c
v, N 1
2, N 2
[ El
( abs )
F Hˆ int
G
n
m ,l
1 Hˆ 2em l , N 1
( N1
Pˆ fg( el )
2)
1
2, N 2 l , N1
) (Em
( emis )
F Hˆ int
G
( 3)
WFG
n
( 3)
WFG
2
N1 ( N 2 1)
m,l
Stimulated
V. Gavryushin
3
L
] [( E m
( N1 1)
1
) ( Ev
N1
1
(2
1
2
)]
1 Pˆ fg(el )
( El
Em
u n (r ) e ikr
2
1 )( E m
Ev
1)
2
)
uc (r )(e p)uv ( r ) d 3r
(e p) cv
V
( p e2 ) vl ( p e1 ) lm ( p e1 ) mv
( El E v 2 1 )( E m E v
Spontaneous
1
v Pˆ2 l l Pˆ1 m m Pˆ1 v
N 1 N 1 ( N 2 1)
1
2, N 2 Hˆ 1ab m, N1 1, N 2 m, N1 1, N 2 Hˆ 1ab v, N1 , N 2
( N 1 1)
m ,l
(r )
n,k
EG )
2
2
1)
exp i (2k1
k 2 )r d 3 r
(2
Fazinio sinchronizmo daugyklis
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1
k
2
2 k1
)
k2
162
Antros optin s harmonikos generavimas
( 3)
WFG
2
N1 ( N 2 1)
m ,l
2HG fazinio "nesinchronizmo" s lygomis
1
exp i k r d 3 r
)
L
0
i kzz 2
0
dz
I2
dz
z
I
Mˆ
2
1
( 3)
AHG
1 exp(i k z z )
4
i kz
I 2 ( L)
V. Gavryushin
2 k1
)
k2
L
2
(2
n( ) n( 2 )
2
I M
2
)
e
1 exp(i k z L) 1
4
i kz
0
( 3)
AHG
1
i kr
2
dz
0
L 2
2
sin 2 ( 21 k z L)
( kz )2
1 1 cos( k z L) 1
2
( kz )2
k
k
2
( 3)
W AHG
L
c
2
1
2
I 2 ( L)
Lk
(2
Fazinis sinchronizmas = Impulso atrankos taisykl s
I2
z
e
2
2
( p e2 ) vl ( p e1 ) lm ( p e1 ) mv
( E l Ev 2 1 ) ( E m E v
sin 2 ( 21 kL)
L
( 21 kL) 2
2
2
2
1
a) Antros harmonikos spontanin s generacijos
b) intensyvin priklausomyb I2 (L)=f(I (0)).
b) Antros harmonikos spontanin s generacijos
priklausomyb nuo sklidimo kelio I2 (z)
netiesin je aplinkoje (arba nuo kristalo storio).
storio).
Paž
Pažym tas proceso koherentiš
koherentiškumo ilgis Lk.
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163
Netiesin optika
Antros optin s harmonikos generavimas
2lcoh
lcoh
2lcoh
lc
V. Gavryushin
k
4n
n2
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164
Netiesin optika
Antros optin s harmonikos generavimas
Efficiency of the frequency doubling as a function of phase matching
V. Gavryushin
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165
Netiesin optika
Antros optin s harmonikos generavimas
2HG fazinio sinchronizmo s lygomis
2
2
( 3)
WFG
2
N1 ( N 2 1)
m,l
( p e2 ) vl ( p e1 ) lm ( p e1 ) mv
( El Ev 2 1 ) ( Em Ev
1)
exp i ( 2k1 k 2 )r d 3r
(2
1
2)
Fazinis sinchronizmas
I2
z
I2
z
L
0
I2
I2
I 2 ( L)
( 3)
W AHG
2
I2 I
2
1
M
2
( 3)
AHG
2
I
2
1
M
( 3)
2
e
i kzr
k
1
L
2
2
2
e
1
i kzr
2
I 2 (0) exp CI L M
0
2
dz
k
0
2
1
2
( 3)
AHG
0
2
2
Bugerio d snis,
neigiami nuostoliai
V. Gavryushin
2
1
stiprinimas
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166
Netiesin optika
sinchroninis generavimas
nesinchroninis
generavimas
Antros harmonikos nesinchroninis
generavimas
optinis detektavimas
Optinis detektavimas Optical rectification in crystals
Optical rectification
V. Gavryushin
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167