Photoluminescence of GaAs/AlGaAs quantum wells

LAPPEENRANTA UNIVERSITY OF TECHNOLOGY FACULTY OF TECHNOLOGY

Master’s Degree Programme in Technomathematics and Technical Physics

PHOTOLUMINESCENCE OF GaAs/AlGaAs QUANTUM WELLS

Examiners: Professor, Erkki Lähderanta Alexander Lashkul, Ph. D.

Supervisors: Professor, Erkki Lähderanta Alexander Lashkul, Ph. D.

Lappeenranta 22.05.2008 Andrey Krasivichev

Ruskonlahdenkatu 13-15 C 4 53850 Lappeenranta

Phone: +358449468944

ABSTRACT

LAPPEENRANTA UNIVERSITY OF TECHNOLOGY FACULTY OF TECHNOLOGY

Master’s Degree Programme in Technomathematics and Technical Physics Andrey Krasivichev

Photoluminescence of GaAs/AlGaAs quantum wells

Master’s thesis 2008

53 pages, 17 figures, 1 appendix

Examiners: Professor Erkki Lähderanta Alexander Lashkul, Ph. D.

Keywords: Quantum well, photoluminescence, polarization, uniaxial deformation, energy band splitting.

In this thesis is studied the influence of uniaxial deformation of GaAs/AlGaAs quantum well structures to photoluminescence. Uniaxial deformation was applied along [110] and polarization ratio of photoluminescence at T = 77 K and 300 K was measured. Also the physical origin of photoluminescence lines in spectrum was determined and the energy band splitting value between states of heavy and light holes was estimated.

It was found that the dependencies of polarization ratio on uniaxial deformation for bulk GaAs and GaAs/AlGaAs are different. Two observed lines in photoluminescence spectrum are induced by free electron recombination to energy sublevels of valence band corresponding to heavy and light holes. Those sublevels are splited due to the combination of size quantization and external pressure. The quantum splitting energy value was estimated. Also was shown a method, which allows to determine the energy splitting value of sublevels at room temperature and at comparatively low uniaxial deformation, when the other method for determining of the splitting becomes impossible.

TABLE OF CONTENTS

1 INTRODUCTION……….5

1.1 Main principles of the theory of elasticity………...…………5

1.2 The Hooke's law and coefficients of elasticity.………...6

2 GENERAL OPTICAL PROPERTIES OF SEMICONDUCTORS….……..8

2.1 Absorption of light by semiconductors………8

2.2 Interaction between electromagnetic wave and electron……….10

2.3 Connection between the amplitude of vector-potential of light wave and photon density……….13

2.4 Complex valence band structure and Luttinger Hamiltonian….……….14

3 INFLUENCE OF DEFORMATION TO OPTICAL PROPERTIES OF SEMICONDUCTORS……….……….19

3.1 Modification of dielectric permeability tensor induced by deformation……….………19

3.2 Evolution of nondegenerate conduction band under uniaxial deformation……….……20

3.3 Shifting and splitting of subbands of valence energy band in the diamond-like semiconductors……….……22

3.4 Polarized photoluminescence in strained diamond-like semiconductors……….……..23

4 LOW DIMENSION QUANTIZATION OF ELECTRON STATES IN QUANTUM WELLS………...24

5 POLARIZATION RELATION OF PHOTOLUMINESCENCE IN QUANTUM WELLS AS FUNCTION OF STRAIN..………27

6 THERMOMETRY AT LOW TEMPERATURES………..30

6.1 Contact phenomena……….……….30

6.2 Type T thermocouples…...……….………..34

7 SAMPLE PREPARATION AND MEASURING METHODS……….…...37

7.1 Characterization of investigated samples……….…...37

7.2 Sample preparation for measurements……….…...37

7.3 Adjustment of photoelectronic multiplier with the help of laser……....38

7.4 Principles of the deformation device……….….38

7.5 Calculations of the polarization caused by the used optical system…...40

7.6 The measuring setup and experimental procedure……….…….…41

8 RESULTS AND DISCUSSION……….………..43

9 CONCLUSIONS……….………..51

REFERENCES……….52

APPENDIX 1………53

1 INTRODUCTION

Investigations of the photoluminescence (PL) under uniaxial deformation (UD) are effective methods to study different semiconductor parameters. UD induces the energy band splitting in semiconductors [1] and thus the linear polarized PL appears [1].

Energy band of heavy and light holes split due to two-dimensional (2D) quantization in quantum wells (QW) [1]. This splitting explains polarized emission of light, which propagates perpendicularly to the QW axis of growth, i.e. in the QW plane. Usually the QW layer is situated near the surface of the structure and is very thin. That is why the investigation of PL in QW plane is technically complicated.

The PL light, which propagates along the QW axis of growth, is not polarized in term of UD. The PL becomes polarized when UD is applied in heterostructure plane [1]. Linear polarization index is defined by the energy band splitting value induced by external UD and initial splitting of states of heavy and light holes. The situation is completely different from bulk crystals, where polarization value of PL depends on the energy band splitting caused by deformation and temperature dependent energy band population.

The aims of this work are

• Determination of physical nature of PL lines by experimental investigation of PL polarization under UD.

• Estimation of the 2D energy band splitting value between states of heavy and light holes.

1.1 Main principles of the theory of elasticity

Solid becomes deformed when the external strain is applied. Without external forces the segment of solid is characterized by radius-vector r. When external forces are applied, segment moves to position r’(r). Obviously changing of all coordinate values by the same constant ^u

( )

^{r =r'(r)−r=const} means just movement of the solid. But if different solid parts gain different shift it can be considered as strain or as rotation. Only strain will be taken into account below.

The deformation of solid is characterized by the second order deformation tensor:













∂ +∂

∂

≈ ∂













∂

∂

∂ +∂

∂ +∂

∂

= ∂

α β β α β

α α

β β αβ α

ε r

u r u r

u r u r u r

u _{l l}

2 1 2

1 (1.1)

where α, β, l are indexes of elements of tensor. Quadratic components can be neglected due to the small values of deformation.

When the solid sample is not deformed, all its parts are in mechanical equilibrium. This means that any volume inside the sample has resultant force from other volumes equal to zero. When a strain is applied to solid sample the returning forces appear inside the sample volume to return itself into the initial state. Those forces are called as internal stresses. Obviously, those forces are equal to zero in strain-free sample. The nature of the internal stresses is atomic or molecular, thus the area of internal stress forces is smaller than strain length u(r). The density of forces is described by second order tensor. This tensor σˆ is called as strain tensor:

σˆ div

F_i = or ⁼

∑

^∂_∂

k k

ik

i x

F σ

(1.2)

where _ij

j

i x

F σ

∂

= ∂ is the force, which is applied to the fragment of solid body from

all other solid body’s volume.

1.2 The Hooke's law and coefficients of elasticity

Shape of solid body changes in term of a strain. There is an elastic limit that shows the maximal value of reversible strain. Below this maximal value the deformation is reversible and the shape can be restored.

According to the Hooke’s law when the strain is small the deformation is proportional to the applied stress. The deformation and stress have tensor dependence, which is:

εij = sijkl σkl , (1.3) σij =cijkl εkl , (1.4) where sijkl, cijkl are components of the tensor of rank 4 of elastic flexibility and elastic inflexibility, correspondingly.

Equations (1.3) and (1.4) correspond to nine equations with nine components on the right side. Thus there are 81 components totally, but the quantity of independent components is smaller. This is because of simplification s_ijkl = s_jikl, s_ijkl = s_ijlk and c_ijkl=c_jikl, c_ijkl=c_ijlk due to symmetry of the deformation tensor and strain tensor. Finally one gets 36 independent components instead of 81.

2 GENERAL OPTICAL PROPERTIES OF SEMICONDUCTORS

2.1 Absorption of light by semiconductors

Absorption of light by semiconductors can be due to different phenomena.

Absorption can take place in the wide range of frequencies. At low-frequency light is absorbed by free charge carriers and by oscillations of the lattice. For infrared, visible and ultraviolet frequency range the absorption is connected to transitions of electrons between different electron states in energy bands (see Fig. 2.1). The X-ray absorption can be connected to exciting of electrons from deep localized atomic shells.

Figure 2.1. Energy band structure of semiconductors.

In this work will be taken into consideration only absorption connected with transition of electrons between valence and conduction bands. The photon energy can not be less than the energy gap Eg for band-band electron transitions in case when the valence band is completely occupied and conduction band is empty.

Hamiltonian for the electron motion in periodical crystal potential and external electromagnetic field is:

( )

^r

m V c A p e

H r

)r r

)  +



 

 +

= 2

2

, (2.1)

where p)r

, e and m are momentum operator, electron charge and electron mass, correspondingly, Ar

is the vector-potential of an electromagnetic wave





∂

− ∂

= t

A E c

r 1 r

, c is light velocity and ^V

( )

^rr is periodic potential of crystal lattice. If the light intensity is not high, the vector-potential Ar

can be considered as small. So:

{ }















 + +

 =



 

 + ^{2 2} 1 2 ₂ ²

cp A e p

A p c p e

cA p e

r rr

)r r

,

2 /

8 1



 



= 

=

= c

S p e p c

eE cp

Z eA π

ω

ω , (2.2)

where ω is frequency of electromagnetic wave, S = E² c/(8π) is the density of light flux and Z is a denoted component.

For estimation of Z value the intensity equal to 1 W/cm² can be taken [2]. It is approximately ten times more than flux from the Sun on the extreme border of atmosphere and can be easily obtained in laboratory conditions. The typical momentum value for light with frequency of 10^-15 s^-1 (or energy of photon 0.6 eV) will be

s m kg c

S

p_l e  ≈ ⋅ ⋅



 



≈  ⁻³¹

2 / 1

10 8π 4

ω . At the same time typical momentum for

moving of electron in atomic potential (or electron with energy of ~1 eV) is approximately million times more than p_l. Even in the case of low energies at liquid helium temperature, the typical momentum of electron will be thousand times more than p_l.

Hence the analysis of interaction between electron and wave of light in terms of perturbation theory is applicable. But nowadays exist powerful radiation sources under which illumination the perturbation theory is non-applicable. So, the interaction between electron and wave of light can be described by perturbation operator

( )

⁼⁻

( )

^∇

= A

mc p ie

mc A

H) e r)r h r

' . (2.3)

2.2 Interaction between electromagnetic wave and electron

According to the “golden rule” of quantum mechanics the probability W of electron transition between some state in valence band and some state of continuous spectrum in conduction band is described by:

(

^ω

)

^ν

π δ

ν

ν E E d

M

W 2 ² _{C, V,i}

+

−

= h

h , (2.4)

where Mv is matrix element of transition, EC,v and EV,i are energy values of conduction band in final state (index v) and conduction band in initial state (index i) and δ is delta-function.

General formula for matrix element of perturbation is

( ) ( ) {

^{− −}

}( )

^∇

−

= A i t r e

mc e

H) ih rr r

κ ω exp

' ₀ . (2.5)

where κr and ω refer to electromagnetic wave.

Exact presentation of wave function of electron for states in valence and conduction band in general is unknown. But the answer can be found even in this case. According to Bloch theorem the electron wave function in periodical potential of crystal lattice can be written as:

( )

¹ , ^{( )exp}

{ ( ) }

^,

, u r ik r

r N _{Ck C}

k

C _C

r r r

r _r

r =

ψ (2.6a)

( )

¹ , ^{( )exp}

{ ( ) }

^,

, u r i k r

N

r _{Vk V}

k

V _V

r r r

r _r

r =

ψ (2.6b)

where ^unk_n

( )

^r

r r

, is periodic function, which has period equal to period of crystal lattice; c and v shows the type of the band (conduction or valence band); kr

is electron wave vector, which corresponds to its quasi-momentum p kr

r h

= and N is

amount of elementary cells in crystal. Each Brillouin zone has its own dispersion law ^εn

( )

^{kr .}

Figure 2.2. GaAs as an example of energy bands.

The matrix element of optical transition between valence band state and conduction band state is

∫

Ω

= CkC V kV

V

C A d ru ep u

mc k e

V H k

C r ) r ^r r)r ^r

,

* ,

3 ( )

, '

, . (2.7)

Finally, matrix element of transition and matrix element of momentum operator, which can be estimated with the help of Bloch amplitude of valence band state and conduction band state, are directly proportional

∫

Ω

∇

−

=

V

C Vk

k

CV i d ruC u

pr h ^{r r}

,

* ,

3 . (2.8)

When the matrix element of transition is known it is possible to find absorption coefficient (or probability of optical transition). It is needed to replace summarizing of quantum states with different nondimensional indices by integration of final (or initial) states with wave vector

( )

^π_π

( ( ) )

^δ

(

^ε

( ) ( )

^{ε ω}

)

µµ rrµµ r h

h  − −



 



= ^V _mc^eA

∑∫

^{d k ep k k k}

W_{CV C V}

2 '

' 3

2

2 3

2 . (2.9)

The quantity of these states is directly proportional to the crystal volume.

There is a connection between probability of optical transition and absorption coefficient. W is equal to amount of photons, which are absorbed inside the crystal volume per unit of time. The flux density of photons in plane wave of light is equal N_phv=N_phc n, where c is light velocity and n is environmental refractive index. Thus the absorption coefficient is

( )

( )

^δ

(

^ε

( ) ( )

^{ε ω}

)

ω α π

µ

µ r)rµµ r r r h

−

−

=

=_VN^{W n}_{c m}^e_cn

∑∫

^{d k ep k C k V k}

ph

cV 2

' ,

' , 3 2

1 2

. (2.10)

The situation differs for different types of semiconductors. For gallium arsenide (see Fig. 2.2) bottom of conduction band and maxima of the valence band are situated in the center of Brillouin zone. The valence band consists of two subbands, which are two times spin degenerated. Finally, there is four times spin degeneracy in the center of Brillouin zone. For some simplification it is useful to consider that conduction band bottom and valence band maxima are not degenerated. So, absorption coefficient is equal to zero if energy of the absorbed photon is less than energy gap E_g and the absorption coefficient is in proportion to

Eg

− ω

h if the photon energy higher thanE_g. The tensor ⁼

∑

,'0

,

) (

,' ) (

' , k

p m

p

C r

µ µ

µβ αµ

αβ µ is unidentified. It is possible to present it with the help of matrix elements of momentum operator. It is needed to summarize over all valleys, which are characterizing the fundamental absorption edge line. But there is another way to estimate it with the help of crystal symmetry. In crystals with cubic symmetry the second order tensor will degenerate into scalar

β αβ Cδα,

C = .

2.3 Connection between the amplitude of vector-potential of light wave and photon density

The electromagnetic field can be described [2] by ideal monochromatic plane wave with wave vector κr

( )

^{r t Ae}

{

ⁱ

[ ( )

^{r t}

] }

Ar r = r κrr −ω exp

, ₀ . (2.11)

Electric field of the light wave is transversal, i.e. r κr

⊥

E and is equal

( )

^A

c i t A t c

r

E r r

r r = ω

∂

− ∂

= 1

, . (2.12)

Real vector-potential is the sum

( )

^{r t A}

{

ⁱ

[ ( )

^{r t}

] }

^A

{

ⁱ

[ ( )

^{r t}

] }

^A

{ [ ( )

^{r t}

] }

A r = κrr −ω + − κrr −ω = κrr −ω cos

2 exp

exp

, _{0 0}^* ₀ (2.13)

Finally, the electric field intensity is

[ ( )

^{r t}

]

c A

E = ω κrr −ω sin

2 ₀ , (2.14)

and for magnetic field [2]

( ) ^{[ ( ) ]}

^A

^{[ ( )}

^{r t}

^]

c t n r A

A rot

H = r = κ κrr −ω = ω κrr −ω sin

2 sin

2 _{0 0} . (2.15)

At the same time energy density of such wave is described by

[

^{2 2 2}

]

^{2 0 2 2}

^{[ ( ) ]}

^{2 0 2}

2 sin 1

4 4 1 8

1 A r t A

c H n

E

n κ

ω π ω κ

π

π  − =



 



= 

+ rr

(2.16)

On the other hand, the energy density is equal to

(

^hωN

)

^{, where N is}

amount of photons per volume unit.

Finally,

κ ω πNh

A 2

0 = . (2.17)

2.4 Complex valence band structure and Luttinger Hamiltonian

First will be described general phenomena, which define structure and parameters of a valence band. Also the effective mass method will be modified in such a way, that it will be useful for description of behavior of charge carriers in the neighborhood of the valence-band maximum, where several subbands are overlapping.

Figure 2.3. sp³ – hybridization and overlapping. To show the interaction, it is enough to have this two dimensional figure.

So, the formatting of the maximum, where the subbands are overlapping, will be described with the strong binding method:

Formation of valence band and conduction band in diamond-like structures is due to hybridization of s and p atomic functions (sp³ −hybridization). Also it is possible to assume that formation of energy band take place due to overlapping of 3 atomic orbitals of P-symmetry. These orbitals may be considered as three basis vectors X, Y, Z. So, the wave-functions can be presented as figures of eight, which are stretched along the corresponding axes. Schematic presentation of orbitals is in Fig. 2.3: X-orbitals are green and Y-orbitals are yellow.

According to strong binding method the width of formed energy bands is proportional to overlapping integral of wave-functions of neighboring atoms.

Obviously, if the energy band is wider then the effective mass is less at the top or at the maximum. It is obvious from Fig. 2.3 that X-orbitals are overlapping well between atoms which are situated along X-axis, and overlapping is bad between atoms, which are along Y-axis. Therefore, the energy band, which is formed with X-orbital overlapping, should have smaller mass along X-axis and bigger mass for quasi-momentum along Y-axis. Similar result will be for Y-orbitals overlapping:

smaller mass along Y-axis and bigger mass along X-axis. The energies of X, Y, Z states are equal at the center of Brillouin zone, where the wave vector is equal to zero. As far as valence-band maximum is in the center of Brillouin zone the energy function of the hole in the center neighborhood is proportional to k².

The effective mass method should be generalized; one has to take into account the dependence of mass on X, Y, Z-orbital combination, which describes’

Bloch amplitude of the wave function of hole [1]. In effective mass method it is most suitable to use invariant method for direct generalization of using symmetry.

So, the aim is to construct a Hamiltonian, which will describe existence of three states with different Bloch amplitudes for each wave vector. Such Hamiltonian can be considered [1] as a 3×3 matrix. The matrix elements are functions of wave vector (or quasi-momentum of holes). Below will be taken into account only two upper subbands of heavy and light holes. Those subbands correspond to maximal value of total angular momentum I=L+s_h (I = 3/2). In this case the dependence of kinetic energy from p and I will be described with standard Luttinger Hamiltonian:

( )

( )

^{ˆ ˆ}

(

^{ˆ ˆ ˆ ˆ}

)

^{, (2.18)}

2 ˆˆ ˆ 2

2 ˆ 5 2

ˆ 1

2 3

2 2 2

2 1 0 ) (

 + 

−

−

 −

  −



 

 +

=

∑

∑

≠j i

i j j i j i i

i L

h

I I I I p p E

m p K

γ γ γ γ

γ pI

where γ γ γ₁, ₂, ₃ are Luttinger parameters and m₀ is free electron mass.

Basic functions for four degenerated states at peak of valence band are:

2 . )

; ( 2 2

) (

3 1

; 2 2

) (

3

; 1 2

) (

2 3 , 2

1 ,

2 1 , 2

3 .

− ↓

 =

 

 − ↑+ ↓

=



− + ↓+ ↑

= + ↑

−

=

−

−

iY u X

iY Z u X

iY Z u X

iY u X

v v

v v

(2.19)

where arrow refer to spin.

For spherical consideration (γ₃ =γ₂), the spin projection of the hole to direction of its quasi-momentum

( )

^Jp is a “good” quantum number. So, Ip =±3 2 corresponds to subband with bigger mass and Ip =±1 2 will correspond to smaller mass, i.e. heavy and light holes,

( ) (

1 2

)

0 2 2

1 0 2

2 2 ,

2 γ −2γ = γ + γ

= m

E p m

E_hh p _lh . (2.20)

Holes can scatter on inhomogeneities, phonons etc. and transitions can be inside one subband or with changing of the mass of the holes as well. The changing of direction of hole momentum is also the changing of direction of its spin quantization direction. That’s why only several scattering acts are enough for complete loosing of initial spin direction of a hole.

The situation differs for electron in first approximation. The kinetic energy of electron, which is near to the Г-point, can be described by following expression

∑

=

= ∗

3 , 2 , 1

ˆ2

2 ˆ 1

α pα

K m

e

e , (2.21)

where

α

α ∂

∂ i r

pˆ =−h is operator of quasi momentum projection for one of three axes [100] and

m

_e^∗ is the electron effective mass. The dispersion of electron states far from Brillouin zone center cannot be described by usual parabolic dependence. For this approximation the electron spin does not interact with the momentum and spin relaxation is absent.

It is possible to suppose that fundamental absorption edge line in GaAs-like semiconductors can be presented by sum of expressions for each subband. This is almost right. The absorption coefficient is directly proportional to the above presented normalized density of states for subband and conduction band of “heavy”

holes and subband and conduction band of “light” holes. The contribution to absorption for subband of light holes is less than for subband of heavy holes due to 4 – 5 times difference between bigger and smaller masses of holes.

The wave function of states at subbands of “heavy” and “light” holes depends on the direction of wave vector. There are four states described by functions (2.19) in the center of Brillouin zone. Obviously, due to symmetry, the only nonzero matrix elements are S p_x X = S p_y Y = S p_z z = p. So, the probability of optical transitions depends on the polarization of light. The selection rules for optical transitions are shown in the Fig. 2.4.

Figure 2.4. The selection rules for optical transitions.

The electron-hole pair appears exactly at fundamental absorption edge line when the absorbed light is linearly polarized along to Z-axis and states of the pair

are e h

2 , 1 2 1 m

± . If the absorbed light is right-circularly polarized then pairs with

state e h 2 ,3 2

−1 appear three times more than pairs with state e h 2 ,1 2

1 . In similar

way for left-circularly polarized absorbed light and e h 2 , 3 2

1 − is three times more

probable than e h 2 , 1 2

1 −

− .

3 INFLUENCE OF DEFORMATION TO OPTICAL PROPERTIES OF SEMICONDUCTORS

3.1 Modification of dielectric permeability tensor induced by deformation

The changes of optical properties of semiconductors with cubic symmetry will be described below. First qualitative results may be obtained simply with the help of symmetry rules. General optical properties are characterized by permittivity tensor εˆ or impermeability tensor ηˆ=εˆ⁻¹. Tensor ηˆ will be used below because it is used in wave equation where permittivity tensor and deformation have the same symbol εˆ. Tensors of second rank in cubic crystals are transformed into scalars due to symmetry.

( )

( ) ( )

( )

( ) ( )⁰

0 0 0

0 ,

0 0

0 0

0 0

ˆ η δ η

η η η

η ij

o ij =

















= . (3.1)

Permittivity can be presented as series of deformation if the deformation is small

αβ αβε δ

η

ηij = ⁽⁰⁾ ij + pij , (3.2)

where pˆ is elasto-optic constant tensor. Elasto-optic constant tensor is often used with piezo-optic tensor πij_αβ, which connects correction data for reciprocal permittivity and strain tensor t_αβ. The connection between deformation and strain is set through coefficients of elasticity

αβ

ε_ij =S_ijαβt , (3.3)

where S is tensor of elastic compliance of a crystal.

Tensor pˆ can be described by matrix with 3⁴ =81 elements. In crystal with cubic symmetry exist only three independent elements p₁₁,p₁₂,p₄₄. Also

44 12 11,S ,S

S will be independent.

Under hydrostatic pressure deformation tensor transforms to scalar

ij

ij εδ

ε = . (3.4)

Such kind of deformation can be obtained using compressing liquid around sample.

Due to Poisson's law in that case

ij ij t

t = δ , εij =εδij =

(

S₁₁+2S₁₂

)

tδij, ηij =ηδij =

(

p₁₁ +2p₁₂

)

εδij. (3.5)

Therefore, crystal under hydrostatic pressing maintains its cubic symmetry, because permittivity and reciprocal permittivity remain scalars, and material is isotropic from optical point of view. The refraction index is independent from direction of propagation of light and its polarization and the quantity of optic axes is infinite.

In general case an anisotropic deformation removes that degeneracy. An uniaxial deformation along symmetry axis (e.g. ε_xx =ε_yy ≠ε_zz) leads to anisotropy of a impermeability tensor

( )

) 6 . 3 ( ,

) (

) 2 (

) (

2

11 12

11 12 12

11

zz xx zz xx

xx zz xx

yy xx

p p

p p p

p

η ε ε ε

ε ε ε

η η

=

− +

+

≠

≠

− +

+

=

=

and crystal has only one optic axis (Z-axis).

In case of general deformation, when deformation tensor has main eigen axes or uniaxial deformation tensor is along to free direction, crystal becomes optically biaxial.

3.2 Evolution of nondegenerate conduction band under uniaxial deformation

Above was analyzed the influence of deformation on optical properties of crystal.

Now will be considered the influence of deformation on optical properties of crystal at microscale. The answer for this question can be found with the help of

perturbation theory. It is important to use native units of length for each crystal when the potential depends on distance. Such kind of unit for deformed and strain- free crystals is lattice distance and small deformation can be considered as small perturbation Vˆdef

( )

^ε .

For distinctness, below will be taken into consideration the bottom of conduction band of GaAs. The bottom of simple conduction band in GaAs crystals is in the center of Brillouin zone and its states are doubly degenerated.

In first approximation of perturbation theory the deformation affects the states near bottom of conduction band so, that the energy states become shifted by

c V

c

E_c ≈ _def(ε)

∆ . (3.7)

Crystal deformation leads to changing of electron potential energy, namely to deformation potential.

At the same time a weak overlapping of other energy band states appears in the Bloch amplitude states

( )

^≈

( )

⁺

∑ ( )

₋

n c

def n

c

c E E

c V r n r

r 0, 0,

, χ χ

ε

χ . (3.8)

Thus the deformation leads to changing of energy gap as well as to changing of matrix element of optical transition

( ) ( )

( )

0, ˆ

( )

^{0, ... (3.9)}

, ˆ 0 , 0

3

* 3

*

− + +

+

⇒

∑ ∫

∫

≠

r d E p

E n V c

r d p

v p c

v n

c

n c n

def v c

r r

r r

χ χ

χ χ

α α

α

The deformation potential is directly proportional to deformation ε)

in first order approximation. The energy change of bottom of conduction band is a scalar.

There is only one combination of elements of tensor of rank 2 in cubic symmetry case. It is spur of matrix, which transforms as scalar

( )

⁼

( )

⁼

∑

α εαα

ε εˆ Tr ˆ

Sp . (3.10)

Shift of the conduction band bottom is directly proportional to spur of deformation tensor

ε aSp

E= . (3.11)

Now the evolution of matrix element of optical transition will be analyzed. Due to symmetry, the overlapping of other energy band states will make a correction data for matrix element of optical transition

αβ αβ

α β δ ε

ε p p A

S, ≈ + , (3.12)

which is the reason for appearance of the linear dichroism.

3.3 Shifting and splitting of subbands in valence energy band in the diamond-like semiconductors

Anisotropic deformation reduces the crystal symmetry and leads to removing of the fourfold degeneracy of the top of the valence band. Due to symmetry the general expression of Hamiltonian of a hole in deformed crystal is

{ }

{ } { } (

^{ˆ ˆ ; ˆ ˆ ˆ ˆ ˆ ˆ}

)

^{2. (3.13)}

3 ˆ

4 5

ˆ ˆ 3 ˆ

4 5

2 2 2 2

α β β α β β α

α α β αβ

α α αα

β β α

α α β

α α α

ε ε

ε d J J J J J J J J

J b Sp b a

k k J D J

k J B k B A H

+

=

−

 −



 

 + +

+

−

 −



 

 +

=

∑

∑

∑

∑

≠

≠

Additionally in the case of uniaxial deformation the deformation tensor can be written as ε_αβ =εn_αn_β, where

ε

is deformation value and nr

is unit vector directed along the deformation axis.

In the case of spherical approximation (b=d 3, B=D 3) in term of tension of crystal along the n axis, (Jn) = ±3/2 corresponds to minimal hole energy with the spin states with maximal spin projection of a hole J onto n. Such simple

expression will be kept for real crystals with cubic symmetry (b≠d 3, 3

D

B≠ ) instead of spherical symmetry only when n is directed along the high symmetry axis (for example [100], [111]). If the direction is random then the value of spin projection of a hole does not correspond to general state.

Energy spectrum in this case is

[ ] ( )

( ) ( ) ^{( )}

[ ] ( )

( )

[

^{x xx y yy z zz}

] [

^{x y xy z y zy x z zy}

]

k

zx yz xy zz

xx yy

zz yy

xx

z x y z y x k

k k

k k k

k k

k Dd Sp

k k

k k

Bb E

b d E

k k k k k k B D k B E E E E aSp Ak

E

ε ε

ε ε

ε ε

ε

ε ε ε ε

ε ε

ε ε

ε

ε

ε ε

ε ε

+ +

+

− +

+

=

+ + +

− +

− +

−

=

+ +

− +

= +

+

± +

=

2 3

) 14 . 3 2 (

; )

3 ( ,

2 2

2 2

,

2 2 2 2 2

2 2

2 2 2 2 2 2 2 2 4 2 2

/ 1 ,

2 2 , 1

So, the degeneracy of top of the valence band disappears, masses are anisotropic (they depend on the direction of motion) and the wave functions can be found with the help of energy [1].

3.4 Polarized photoluminescence in strained diamond-like semiconductors

According to the selection rules presented in chapter 2 above in the case of uniaxial stressing, bottom of the conduction band depends on states with ±12 spin projection onto the deformation axis. Absorption of light with polarization along the deformation axis is four-times higher than the absorption of light with normal linear polarization. In case of uniaxial tension the states of bottom of conduction band are characterized by ±3/2 spin projection. So, the light, which is linearly polarized perpendicularly to deformation axis, is absorbed in general way.

4 LOW DIMENSION QUANTIZATION OF ELECTRON STATES IN QUANTUM WELLS

According to the definition of type I heterojunction the energy gap E_g of one adjoining compound (A) is inside the energy gap of another adjoining compound (B). In this case the potential wells for electrons and holes are in the same border layer. Doubled heterojunction of type-I will be a structure with a single quantum well if E_g^A <E_g^B. Generally, quantum well is a system where the free charge carrier (electron or hole) has limited motion along the one direction. Hence a spatial quantization appears and energy spectrum becomes discrete with one quantum number.

To obtain clear analytical results it is useful to use method of effective mass approximation. In this method the solution for each layer of multilayer structure is written as a linear combination of independent volume solutions. Boundary conditions are used for envelopment of a wave function and its normal coordinate derivative [3].

At first steps a quantum well model will be considered with infinite energy barriers. In case of infinite barriers an envelope of electron wave function will be

) 1 (

)

( e^{( )} z

S

r ^{iqxx qyy}ϕ

ψ = ⁺ , (4.1)

where q=(q_x,q_y) is a 2D wave vector characterizing the electron motion in the interface plane. For B/A/B structure ϕ(z) function satisfies the 1D Schrödinger equation:

) ( )

2 ² (

2 2

z E dz z

d

m_A ϕ = ^zϕ

− h

, (4.2)

where mA is an electron effective mass for А material and Z axis is perpendicular to the interface plane. The ϕ(z) function is equal to zero outside the А layer. Electron energy Е consists of dimensional quantization energy Еz and kinetic energy

A

xy m

E q

2

2

h2

= (values E are counted from bottom of conduction band of A

material). Origin of Z axis is in the center of A layer. Thus the condition for infinite barriers approximation will be ) 0

(±2a =

ϕ , where а is a layer thickness and correspondingly

2

±a are the interface coordinates. Such system has reflection symmetry z→−z. So, the combination of eigensolutions of a Schrödinger equation can be divided into two groups of even and odd solutions. Even solutions has form

kz

Ccos and odd solutions has form Csinkz. Here k=(2m_AE_z/h²)^1/2 is the length of the wave vector and C is a normalization factor. Taking into account the boundary conditions the result will be

2 2

, 2 



 



= 

= E m a

k a

A z

νπ

νπ h

, (4.3)

where ν =1, 3, … for even and ν =2, 4, … for odd solutions. Electron and hole dimensional quantization states will be designated as eν and hν correspondingly.

An energy spectrum consists of following lines

Eeνq=











  +



 



 2

2 2

2 q

a m_A

νπ

h . (4.4)

They are called as subbands of dimensional quantization, or simply subbands.

Heavy holes (hh) and light holes (lh) states are quantizing independently at point q = 0 in GaAs/Al_xGa_1-xAs heterostructures, which are grown along the [001]

direction. This is why two series of a hole states hhν and lhν are formatting in QW. Those hhν and lhν states are characterized by projection of angular momentum Jz = ±3/2 and Jz = ±1/2, respectively. When the lateral wave vector is nonzero, states of heavy and light holes are mixing up. So, the valence subbands become nonparabolic.

Complicated valence band structure leads to polarization dependence of band-to-band absorption in QW. For example, for heterostructures, which are

grown with zinc blende lattice along to direction [001], a generation rate of an electron-hole pairs for optical transition at kx = ky = 0 is described by

3 . ) 4 2 / 1 ,

; 2 / 1 , (

3 , ) 1 2 / 1 ,

; 2 / 1 , (

, 0 ) 2 / 3 ,

; 2 / 1 , (

, )

2 / 3 ,

; 2 / 1 , (

2 2

2 2 2 2 2

z y x

y x

e lh

e M

ie e lh

e M

hh e

M

ie e hh

e M

∝

±

∝

±

±

=

±

±

±

∝

±

m

m m

ν ν

ν ν

ν ν

ν ν

(4.5)

5 POLARIZATION RELATION OF PHOTOLUMINESCENCE IN QUANTUM WELLS AS FUNCTION OF STRAIN

The optical transitions between general level of size quantization of electron and first two levels of size quantization of holes will be discussed in this chapter.

General hole states in strain-free QW are heavy hole states with momentum projection

2

±3

z =

J , and first excited level is a light hole states

2

±1

z =

J . For

simplification, only the transitions with zero longitudinal quasi-momentum of charge carriers will be analyzed. Phenomenological Hamiltonian describes the effect of size quantization. It can be written as



 



 −

−∆

∆ =

4 5 2

2

Jz

H , (5.1)

where J_z is a momentum projection operator J = 3/2 to Z-axis, which is the QW growth direction [001]. Canonical basis [3] for states with full momentum J = 3/2 will be used for estimations. Also the energy E in valence band is positive and is counted down. So, the energy of the general state will be E_hh = -Δ/2 for heavy holes with momentum projection ± 3/2 on Z axis, and E_lh = Δ/2 > 0 for light holes.

Interaction between holes and external deformation will be described according to Bir-Picus Hamiltonian of deformation. The states of investigated holes come from Г8-type state of valence band of GaAs. Uniaxial deformation is along the crystallographic direction [110] and the Hamiltonian is





















−

−

−

−

=

1 2

1 2

2 1

2 1

0 0

0 0

0 0

0 0

ε ε

ε ε

ε ε

ε ε

ε

i i

i i

H . (5.2)

Here

) (

2 _{11 12}

1 C C

bP

= −

ε ,

44

2 4C

= dP

ε , where b and d are deformation potential constants, P is load intensity and C_ij are elastic constants of GaAs. Uniaxial load

along the x-direction in plane of QW leads both to the energy changing Δ and to mixing of states ±3/2 and ±1/2 . Spectrum and wave functions of full Hamiltonian H =H_ε +H_∆ are

, 2 / 1 2 / 3

, 2 / 1 2 / 3 2 ,

1 2 2

1 1 2

2 2 2 1 2

, 1



 − + +

= +



 + −

+

= −

 +



 



 + ∆

=

E C i

E C i

E

E E

ε ψ ε

ε ψ ε

ε ε

m

(5.3)

where 1, 2 are indexes numbering degenerated states and C is a normalizing constant.

With the help of wave functions (5.3) and supposing that the electron states belong to s-type it is possible to estimate polarization ratio r, which describes emission transitions to a general sublevel of heavy holes:

2

1 2 2

1 2 2

1 2 2

1 2 2

) 2 ( 3 2 ) 2 ( 4

) 2 ( 3 2 ) 2 ( 4













∆ +

−

−

∆ + +

∆ +

− +

∆ +

= +

ε ε ε

ε

ε ε ε

r ε . (5.4)

Or

1 1

B

r= A (5.5)

where

2 1

2 2

2 2 1

1 ( )

) 3 ) ( (

λ ε

ε

ε ε

λ

−

∆ + +

+

∆ +

= −

A and ₂

1 2 2

2 2 1

1 ( )

) 3 ) ( (

λ ε

ε

ε ε

λ

−

∆ + +

−

∆ +

= −

B , (5.6)

2 1 2

2 +( +∆)

= ε ε

λ ,

12 11

1 C C

bP

= −

ε ,

44

2 2C

= dP

ε . (5.7)

Here b and d are deformation potential constants [1,4] and their values for GaAs are b = 1.96 eV, d = 5.4 eV. The elastic inflexibility constants have values

C11= 12.26·10^{11 dyne}/сm2

, C₁₂= 5.71·10^{11 dyne}/сm2

, C₄₄= 6·10^11dyne/сm2, Δ is a splitting of heavy and light holes sublevels due to the size quantization.

Polarization ratio for transitions to “light” holes’ sublevel can be found similarly with corresponding replacement in (5.6) of λ by

2 1 2

2 +( +∆)

−

= ε ε

λ . (5.8)

6 THERMOMETRY AT LOW TEMPERATURES

6.1 Contact phenomena

In circuits with two or more different types of conductors a temperature gradient leads to appearance of electromotive force (e.m.f.) [7].

Theory of thermo electrical phenomenon can be described by quantum mechanics. The phenomenal description with the help of classic mechanics is just qualitative and gives explanation of appearing thermo e.m.f.

In the border of two conductors A and B appear electric currents due to thermal motion. If the electron density is different (e.g. nA > nB) then a current from A to B appears and electron density in B increases. So, the potential of conductor B decreases and potential difference appears (see Fig. 6.1).

Below will be estimated the potential difference UA – UB in terms of equilibrium. The electron potential energy of conductors A and B is –eUA and –eUB

respectively, where –e is the elementary charge. Junction of conductors A and B can be considered as potential barrier: in this layer the potential energy of electrons W_contact is higher than in conductors. Moving electron from A into B can surmount the energy barrier only if its kinetic energy is higher than difference of potential energies W + eU_A between A metal and contact layer. According to Maxwell- Boltzmann distribution the electron density of such electrons is proportional to

kT eU W A

e A

n ^{−( + )/} , where k is Boltzmann constant and T is temperature. Similarly, the electron current is proportional to n_Be^{−(W+eUB)/kT} for electrons moving from B into A. The equation of equilibrium of electron currents is

kT eU W B kT eU W A

B

A n e

e

n ^{−( + )/} = ^{−( + )/} , (6.1)

or

B A B

A n

n e U kT

U − = ln . (6.2)