Modelowanie Nanostruktur

Modelowanie Nanostruktur

Semester Zimowy 2011/2012 Wykład

Jacek A. Majewski

Chair of Condensed Matter Physics Institute of Theoretical Physics Faculty of Physics, University of Warsaw

E-mail: Jacek.Majewski@fuw.edu.pl

Email: jacek.majewski@fuw.edu.pl

Optical Properties of Semiconductor Quantum Structures

Electrodynamics of heterostructures Light propagation perpendicular to the layers Propagation of light along the layers Light absorption in quantum wells Interband transitions

Intraband (intersubband) transitions

Modelowanie Nanostruktur, 2011/2012

Jacek A. Majewski

Wykład 10 – 13 XII 2011

OPTICAL PHENOMENA – BULK SYSTEMS

Optical phenomena

These processes can be described as

 the action of a high-frequency macroscopic field on a solid

 the interaction between the elementary excitation of the solid and the quanta of the electromagnetic field – the photons

The macroscopic interaction of electromagnetic fields with matter is already contained in the Maxwell equations

Transverse, electromagnetic waves

LIGHT

DE

BH

 1 Material equations For nonmagnetic solids

The material constant then contains all information about the interaction of the fields with the matter



Absorption Reflection Dispersion

DE Is this equation valid for high-frequency fields?

Optical phenomena

These processes can be described as

 the action of a high-frequency macroscopic field on a solid

 the interaction between the elementary excitation of the solid and the quanta of the electromagnetic field – the photons

The macroscopic interaction of electromagnetic fields with matter is already contained in the Maxwell equations

Transverse, electromagnetic waves

LICHT

DE

BH

 1 Material equations For nonmagnetic solids

The material constant then contains all information about the interaction of the fields with the matter



Absorption Reflection Dispersion

DE Is this equation valid for high-frequency fields?

Absorption Coefficient & Measurement of Optical Constants

| |2

I EB E

( ) 0 ^z

I z I e^ 2 4

c

 

  

dI I

dz  The intensity of light is proportional to

with Absorption

coefficient

One can deduce that

I

satisfies the equation

Physical implification: Absorption coefficient is the amount of radiation energy absorbed by a unit volume of solid under unit radiation intensity (photon flux energy) in unit time

Empirical formula for semiconductors: n E⁴ _Gap77 n[2.3 4.6] Energy gap in eV

for T. S. Moss, “Optical Properties of Semiconductors”,

Butterworths Scientific (1959)

The optical constants are often determined by measuring i) Transmition

ii) Reflection

Semiconductor

I0

Ir ^t

' I (1 ) 0

It R I

0

IrRI

n ( )    ( )

Electron-Photon Interaction – Theory of Direct Transitions

The simplest interaction process – the absorption of a photon by an electron The electron changes its energy and momentum by the energy and momentum of the photon absorbed

Our aim is to calculate the absorption coefficient (or ) due to such elementary process



2

Absorption coefficient = the energy absorbed (per unit volume & time) the incident energy (per unit volume & time) ( )

W -the number of photons absorbed per unit volume per unit time -the energy absorbed

( ) W 

( , ) 0 exp[ ( )] . .

A r t A e i q r t c c

We represent the incident light by its vector potential

The incident energy flux = uc

uv n 1 ^{2 2}

( )

u 8 E H

  

( ) W ucn

 



v c

 n

Absorption Coefficient and Number of Absorbed Photons

( , ) ( , ) 1 A( , )

E r t r t r t

c t

 

  

 B r t( , )  A r t( , )curlA r t( , ) Coulomb gauge:



( , )r t 0 divA r t( , )0

0 exp[ ( )] . .

E i A e i q r t c c c

 

   

* 2 2

2 0

4 4

u E E A

c

  

 

   1 ² ₀^{2 2} 1 ₀²

4 4

uc A n A

n n c c

  

 

 

n2

n n n

  

2 0

( ) 4 c ( )

W n A

   

  ₂ ²

2 0

( ) cn ( ) 4 c ( )

W A

     

 

 

Interaction of light with solids - QM description

To determine we start from the Schrödinger equation W( )

1 (ˆ )2 ( ) ( , ) ( , )

2

e r t

p A V r r t i

m c t

 

    

  

  pˆ  i

ˆ ˆ0 ˆ ' HH H

2

0 1 ˆ

ˆ ( )

H 2 p V r

 m  ˆ ' e ˆ

H A p

mc 

ˆ ˆ

p A   A p idivA

Since divA r t( , )0 (chosen gauge)

ˆ ˆ

p A  A p

A2- small in comparison to A pˆ

0 0 0

Hˆ i t

  

 Hˆ⁰_n( , )k r E k_n( )_n( , )k r nk _n( , )k r

( ) _n( , ) exp[ _n( ) ]

nk

t a k t i E k t nk

 





We now expand in terms of the solutions of the unperturbed problem 

Bloch states

  || exp[i _m( ) ] '

E k t mk Schr. Eq. 

Quantum mechanical calculation of absorption coefficient

1 ˆ

( ', ) ( , ) exp [ ( ') ( )] ' '

m n m n

nk

a k t a k t i E k E k t mk H nk

i

 





  

( , 0) 1

a kj  a k_n( , 0)0

(0) jk

 

( , , ', '; , )

W j k j k  t

^{j k' '}

2

|a_j'( ', ) |k t

(1)

' '

1 ˆ

( ', ) exp [ _j( ') _j( )] ' ' '

j

a k t i E k E k t j k H jk

i

 

   

2

2 ' 0

1 ˆ

( , , ', '; , ) exp [ ( ') ( )] ' ' ' ' '

t

j j

W j k j k t  



i E k E k t  j k H jk dt Let us assume: and for all other

It means, at time t = 0 the electron is described by (is in state ) jk

The probability that at time t it will be in state is then equal to

In the first approximation

Next we examine the matrix element j k H' ' ˆ' jk

Electron-Photon Interaction – Theory of Direct Transitions

The simplest interaction process – the absorption of a photon by an electron

Absorption coefficient = the energy absorbed (per unit volume & time) the incident energy (per unit volume & time) ( )

W -the number of photons absorbed per unit volume per unit time -the energy absorbed

( ) W 

The total number of transitions per unit volume and time.

3 3 '

1 2

( ) ( , ', ; , )

jj (2 ) BZ

W d k W j j k t

 t 









2 0

( ) 4 c ( )

W n A

   

  ₂ ²

2 0

( ) cn ( ) 4 c ( )

W A

     

 

 

Direct Transitions in Semiconductors

are particularly important if the valence band maximum and conduction band minimum lie at the same

k

.

cb vb

Such direct transitions are the lowest transition of the absorption spectrum and determine the shape of the absorption edge.

We assume that we can describe the extrema by an effective mass

2 2 2 2 2 2

' * *

'

( ) ( )

2 2 2

j j Gap Gap

j j comb

k k k

E k E k E E

m m

      

The reduced effective mass of electron and hole

* * * *

'

1 1 1 1 1

comb mj mj mc mv

    

3 / 2

1/ 2

' 2 2

2

( ) 1 ( )

2

comb

jj Gap

g    E



 

     ( ) ( E_Gap)^{1/ 2} The combined density of states

Direct Optical Transitions: Absorption & Emission

Probability of photon absorption

Probability of photon emission

( , , ; , ) | _if( ) |2 ( _f( ) _i( ) ) W i f k_ t t M k  E k E k  

( , , ; , ) | _if( ) |2 ( _f( ) _i( ) ) W i f k_ t t M k  E k E k  

( ) 21 [ _i(1 _f) _i(1 _f)]

kfi

W W f f W f f

  t



_   _  Number of absorbed photons

2( ) W( )

  

( ) 2 | _vc( ) |2 ( _c( ) _v( ) )[ ( _v( )) ( _c( ))]

kcv

W  



M k  E k E k   f E k f E k ( ) 2 | _vc( ) |2 ( _c( ) _v( ) )[ _v(1 _c) _c(1 _v)]

kcv

W  



M k  E k E k   f f f f

Direct Optical Transitions: Absorption & Emission

1 2

| _vc( ) | ( _c( ) _v( ) )

W M k E k E k

t _    

2

2

2

1 | ( ) | ( ( ) ( ) )

| ( ) | ( [ ( ) ( ) ])

| ( ) | ( ( ) ( ) ) 1

cv v c

vc c v

vc c v

W M k E k E k

t

M k E k E k

M k E k E k W

t

 

 

 





   

    

   

Absorption

(Stimulated) Emission

cb vb

cb vb

f  c i  v

i  c f  v





Excitons – General Theory

cb vb 

e

h

We consider an insulating crystal at

T = 0 K

Ground State:

Valence band – fully occupied

Conduction band – completely empty Excited State:

Valence band – electron

vk

anihilated (hole created) Conduction band – electron

ck

created

Coulomb interaction between electron & hole

When this interaction is strong enough, electron & hole bind together

Electron + hole pair = EXCITON

Optical spectra of a semiconductor near the fundamental edge

A b s o rp ti o n c o e ffi c ie n t

Photon Energy

Coulomb enhancement

1s

2s

Energy Gap



Red line – with electron-hole interaction (excitonic effects) Black line – without electron-hole interaction

(  E_Gap)

NANOSTRUCTURES – QUANTUM WELLS

Electrodynamics in Homogeneous Systems

( )

0 0

( , ) cos( ) ^{i q r t} . .

E r t eE q r t eE e^{  } c c

2 2

0

1 1

4 ^r ( ) 8 ^r

w  E t  E

 

 

2 0

1

8 ^r

I  E

 

In the simplest homogeneous case

The energy of the wave can be characterized by the density of the electromagnetic energy

The intensity of the wave = the energy flux through the unit area perpendicular to the wave vector

q

Optical properties of Quantum Structures

Specific features of optical processes originate from two basic physical peculiarities

Spatial nonuniformity causes specific characteristics

of the interaction of light with matter, including light propagation, absorption, etc..

Electrons in quantum structures have energy spectra different from these of electrons in bulk materials.

Electrodynamics of Heterostructures

E

Gap

A B A

In heterostructures, both the refractive index

and the band gap, i.e., the fundamental edge of absorption,

vary spacially.

This changes the light propagation and character of the interaction of light with matter.

For example, a layer with a larger refractive index causes Partial reflection of electromagnetic waves propagating through the layer,

localization of electromagnetic modes propagating along the layer.

The electromagnetic fields (modes) in quantum structures are substantially different from plane waves.

Spatial modulation of the bandgap leads to nonuniform absorption and emission of light.

Electrodynamics of Heterostructures

Sizes of quantum structures are always much less than the wavelength of the light in the spectral region of interest



100 200

L   A   1000 A

TWO CASES of light interaction with a quantum-well layer

Light propagation

perpendicularly to the layers

Propagation along the layers

( ) E z

z

( ) E z

z

Electrodynamics of Heterostructures Light propagation perpendicularly to the layers

Electromagnetic field depends on the

z

coordinate only

1 *

( , ) [ ( ) ( ) ]

2

i t i t

E z t  F z e^ F z e^ ( , ) 1[ ( ) . .]

2

A z t  A z ei t^ c c

( ) ( )

F z A z c



in out out

I I

 I ( )

E z

z

Heterostructure embedded in the optical resonator

Lz

Introduction of a local absorption coefficient is meaningless in this case.

Loss or gain of the light energy can be characterized as the change of the light intensity after the light passes through the layer

to the initial intensity

Electrodynamics of Heterostructures Light propagation perpendicularly to the layers

Another useful characteristic of the interaction of light with matter is the decay (gain) of the energy of standing waves.

The standing waves are formed by the optical resonator in which a heterostructure is embedded.

The decrement (increment) of the mode is

1 dN R N dt N

  

N – the number of photons of the mode under considerations R – total rate of photon absorption (emission)

2 ,

| | ˆ' | | ( _{f i} )[ ( _i) ( _f)]

i f

R



i H f   E E   f E f E ˆ ' e ˆ

H A p

mc 

Electrodynamics of Heterostructures Light propagation perpendicularly to the layers

2 2

,

ˆ

| ( ) | | | _z| | ( _{f i} )[ ( _i) ( _f)]

i f

R A z_



i p f  E E   f E f E

z

_

 L

Since , we can rewrite

R

in the form

- the position of the center of the quantum-well layer The number of photons N of the fixed mode can be calculated as the total energy of this mode divided by



2 3 2

1 ( ) | ( ) | ( ) | ( ) |

8 ^r 8 ^r

N  z F z d r S  z F z dz

   









2 2

2 ,

| ( ) | 1 | |ˆ | | ( )[ ( ) ( )]

( ) | ( ) | _{i f} ^{z f i i f}

r

F z i p f E E f E f E

z F z dzS

  





    



R

 N

z r

L

 

 c



Electrodynamics of Heterostructures Light propagation perpendicularly to the layers

The simplest case of a narrow well, the changes in the refractive index can be neglected and the standing wave is

( ) 0cos cos

E z eF qz



t

F z ( )  eF

₀

cos qz e

- light polarization

q

– a discrete number

z

q n

L



2 2

,

1 ˆ

cos _w | | _z| | ( _{f i} )[ ( _i) ( _f)]

z i f

qz i p f E E f E f E



SL



 



 







depends on the position of the quantum-well layer!

Electrodynamics of Heterostructures Propagation of light along the layers

( ) E z

z

L

z

( ) cos( )

E  eF z q r    t

q

lies in the

x,y

plane

A narrow-bandgap layer localizes the electromagnetic modes.

The amplitudes of these modes depend on z and decay far away from the layer.

These modes are called the waveguide modes.

Form factor of the mode

For the waveguide mode, one can define the electromagnetic energy per unit area of the waveguide plane as

2 2

1 1

( ) ( ) ( ) ( )

4 ^r 8 ^r

w  z E z dz  z F z dz

 









The number of photons of this mode is N_wwS/( )

Electrodynamics of Heterostructures Propagation of light along the layers

Waveguides modes contd.

One can introduce the total intensity of the waveguide mode:

I  wc

eff

eff / r

c c  - is the group velocity of the mode under consideration Now one can define the gain (absorption) coefficient

for the waveguide mode

1 dI

 I dx

2 2

2 2

,

4 | ( ) | 1 | | ˆ| | ( )[ ( ) ( )]

( ) | ( ) | r

f i i f

i f r

F z L

i e p f E E f E f E

c z F z dzSL

  

  

 



     



2 2

| ( ) | ( ) | ( ) |

r

F z L z F z dz

  



Optical confinement factor

characterizes a portion of the light energy accumulated within the active layer where the photo-transitions take place

 1 is always smaller than 1,

The better the optical confinement, the larger the light absorption or gain.

Light Absorption by Confined Electrons

EGap

W B

Ec 1

Ec 2

Ec

Ev 1

Ev 2

Ev

k//



B

A three-layered semiconductor structure (QW) of type I

Band diagram

Two subbands are shown for both bands.

Parabolic dispersion relations are assumed for both the conduction and the simple valence bands.

The first obvious conclusion can be drawn, i.e., that there must be a shift of the interband spectra toward a short-wavelength region

(1) 1

1 1

0

QW Bulk

c v g Gap

E E E E

      

Light Absorption in Quantum Wells

2 ,

| | ˆ | | (

_{f i}

)[ (

_i

) (

_f

)]

i f

R    i e p f    E  E   f E  f E

Bulk: i  vk f  ck

// //

// 0

( ) 1 ( ) ^v ( )

v

ik r

v vn

vk n

i r u r e z

S



^



 

// //

// 0

( ) 1 ( ) ^c ( )

c

ik r

c cn

ck n

f r u r e z

S



^



 

, , //_v

i  v n k f  c n k, , _//c QWs:

- periodic parts of Bloch functions for - two-dimensional (in-plane) wave vectors - the envelope wave functions of the quantized transverse motion

&

0( ) 0( )

v c

u r u r

&

//v //c

k k

0 0

k 

&

( ) ( )

vn z cn z

 

S

- the area of the quantum well layer

Light Absorption by Confined Electrons

0

( ) ( )

i i

i  u r F r f  u

_f0

( ) r F

_f

( ) r

Periodic, strongly oscillating functions

Momentum Matrix Element

Smooth functions

0 0 0 0 0

ˆ _{i i} ˆ _{f f i i f}( ˆ _f ) _f ( ˆ _f)

i e p f  u F e p u F  u F F e pu u e pF

3 * *

0 0

3 * *

0 0

ˆ [ ( ) ˆ ( )][ ( ) ( )]

[ ( ) ( )][ ( ) ˆ ( )]

i f i f

i f i f

i e p f d r u r e pu r F r F r d r u r u r F r e pF r

   

 





3 * 3 *

0 0 0 0

ˆ _i ˆ _{f i}( ) _f( ) _{i f i}( ) ˆ _f( )

i e p f  u e p u



d rF r F r  u u



d rF r e pF r bulk

pif



if

3 * 3 *

ˆ

^bulk_{if i}

( )

_f

( )

_{if i}

( ) ˆ

_f

( ) i e p f   p  d rF r F r    d rF r e pF  r

Light Absorption in Quantum Wells Interband Optical Transitions

3 *

ˆ _vc^bulk _i ( ) _f( ) i e p f p



d rF r F r For interband transitions:

  ^{ }

// // // //

// // //

3 * 3 *

( )

2 *

//

( ) ( ) ( ) ( )

( ) ( )

v c

c v

ik r ik r

i f vn cm

i k k r

vn cm

d rF r F r d re z e z

d r e dz z z

 

 

  

 

 



 

 

//_v, //_c

k k



vn cm

 

// , //

ˆ

v c

bulk

vc k k vn cm

i e p f   p   

//v //c

k  k

The selection rule for two-dimensional wave vectors

This selection rule differs from the selection rule for bulk crystals as a result of the lack of translational symmetry in the z-direction.

Compared with the bulk case, a new factor appears in the matrix element, namely, the overlap integral of the envelope functions from different bands.

Light Absorption in Quantum Wells

Using the matrix elements , one can calculate all characteristics, for geometries presented previously.

i e p fˆ , , ,

  

// //

// //

2 2

,

, ,

// //

| | | |

( ( ) ( ) )[ ( ) ( )]

v c

v c

bulk

vc vn cm k k

n m k k

cm c vn v vn cm

p

E k E k f E f E

    

 



   

 

2 2 2

// // //

,

| _vc^bulk| | _{vn cm} | ( _cm( ) _vn( ) )[ ( _vn) ( _cm)]

n m

p d k E k E k f E f E

 



 



    

The energy-conservation law following from the function 

// //

( ) ( )

cm vn

E k  E k  

shows that photo-transitions can involve different subbands from both the valence and the conduction bands.

Light Absorption in Quantum Wells Case of parabolic subbands

2 2

0 //

// *

( )

cm cm 2

e

E k E k m

  _// ^{0 2 //2}

( ) *

vn vn 2

h

E k E k m

 

2 2

0 0 //

// // * *

2 2

0 0 //

1 1

( ) ( )

2

2

cm vn cm vn

e h

cm vn

E k E k E E k

m m

E E k



 

      

   ^{* *}

1 1 1

e h

m m

 

0 0

// 2

2 ( _{cm vn})

k   E E

0 0

cm vn

E E

  

// //

( ) ( )

cm vn

E k E k  

The magnitude of the two-dimensional wave vector corresponding to vertical transitions between

(vn)

and

(cm)

subbands Transitions are possible only when

Light Absorption in Quantum Wells Case of parabolic subbands

 

2

2 2 2 2

// // // // 2 // //

( ( ) ( ) ) 2 ( )

cm vn 2

E k E k k k  k k

   



 

      

2 2 // // // //

// //

//

( ) ( )

( )

2

k k k k

k k

k

 

     

2 2 0 0

2 ,

// //

| | | | ( )

[ ( ( )) ( ( ))]

bulk

vc vn cm cm vn

n m

vn cm

p E E

L

f E k f E k

      

 ^{  }

 



0 0

( ) 2 ( )

opt

nm  Ecm Evn

   

  

The optical density of states

are the step-like functions, which are consistent with the results on the density of states of two-dimensional electrons (holes).

Integration



d k² _//

opt( )

nm 

Light Absorption in Quantum Wells Case of parabolic subbands

The function also has a steplike shape

  ( )

Bulk

300 A QW 100 A QW

0 100 200

Absorption coefficient

Photon energy above bandgap [meV]

Light Absorption in Quantum Wells Case of parabolic subbands

  ( )

can be represented as sum over all pairs of the subbands involved in the photo-transitions

2 2

// //

( , , ) | | | | ( )

[ ( ( )) ( ( ))]

bulk opt

sub vc vn cm nm

vn cm

n m const p

f E k f E k

 

 

    



 

,

( ) _sub( , , )

n m

  



 n m

Each contribution is proportional to the overlap integral of the envelope functions

( , , )

sub n m

 

&

( ) ( )

vn z cm z

 

These overlap integrals result in new selection rules

Light Absorption in Quantum Wells Selection Rules for Interband Transitions

VB 1 2 3

1 2 3 CB

1-1 3-3 2-1 2-3

1

v 2

v 3

v 1

c 2

c 3

c

Envelope functions

|  

_{vn cn}

|

2

 1

“allowed” transitions for

|  _{vn cm} |2 1 nm “forbidden” transitions

The quantization of the electron energy leads to major changes in the selection rules and in the intensity of the band-to-band transitions

Intraband Transitions in Quantum Structures

We start by recalling that in an ideal bulk crystal, intraband photo-transitions are impossible because of the energy- and the momentum-conservation laws

&

( ) ( ') ' 0

E k  E k   k  k   q

cannot be satisfied simultaneously.

Intraband photo-transitions in a bulk can be induced only by phonons, impurities, and other crystal imperfections.

In contrast to bulk materials, intraband photo-transitions occur in semiconductor heterostructures.

Intraband Transitions in Quantum Structures

c1 c2 c3 CB

Quantum Well

3 * 3 *

0 0 0 0

ˆ _i ˆ _{f i}( ) _f( ) _{i f i}( ) ˆ _f( )

i e p f  u e p u



d rF r F r  u u



d rF r e pF r , , //

i  c n k f  c n k, ', //^'

// //

// 0

( ) 1 _c ( ) ^{ik r} _cn( )

i ck n r u r e z

S

 ^ 

 

0 0 1

c c

u u 

0 ˆ 0 0

c c

u e p u 

'

// // // //

3 *

ˆ _cn( ) ^{ik r} ˆ ^{ik r} _cn'( ) i e p f 



d r z e^{ } e pe ^  z

Intraband Transitions in Quantum Wells

 

 

'

// // //

'

// // //

( )

' ' 2 *

// '

( )

2 *

// '

ˆ ( ) ( ) ( )

( )ˆ ( )

i k k r

x x y y cn cn

i k k r

z cn z cn

i e p f e k e k d r e dz z z

e d r e dz z p z

 

 

 

 

 

     

 

 

  

 

 

for

*

( ) '( ) 0 '

cn cn

dz z  z  nn

'



// // //

' // //

( )

2

// ,

i k k r

d r e ^{ } k k



' // //

' *

// // , '

ˆ ' _{z cn}( )ˆ_{z cn}( )

cnk e p cn k 



k ke



dz



z p



z

'

// //

k k

If the vector of the polarization of light lies in plane of the quantum well layer (

x,y

– plane), the matrix element is zero and photo-transitions are impossible.

e

If the vectror has a

z

-component ( ), intersubband processes take place.

e  0

' ' '

// //

' '

// // // //

' '

' '

' '

ˆ ( ) ( ˆ ˆ ˆ ) ( )

( ) ( ) ˆ ( )

x ik yy

ik r ik x

cn x x y y z z cn

ik r ik r

x x x x cn z z cn

e pe z e p e p e p e e z

e k e k e z e e p z

 

 



 

    

  