Modeling Nanostructures

Modeling of Nanostructures and Materials

Summer Semester 2014 Lecture

Jacek A. Majewski Faculty of Physics, University of Warsaw

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

Jacek A. Majewski

Modeling of Nanostructures and Materials

Lecture 12 – May 19, 2014

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

Continuous Methods for Modeling!

! Electronic Structure of Nanostructures!

!!

Examples of nanostructures!

!!

From atomistic to continuum methods!

!!

k.p methods!

!!

Effective mass approximation!

!!

Envelope Function Theory!

Modeling Nanostructures

Nanotechnology –

Low Dimensional Structures

Quantum

Wells Quantum

Wires Quantum

Dots A B

Simple

heterostructure

What about realistic nanostructures ?

2D (quantum wells): 10-100 atoms in the unit cell 1D (quantum wires): 1 K-10 K atoms in the unit cell 0D (quantum dots): 100K-1000 K atoms in the unit cell

Organics

Nanotubes, DNA: 100-1000 atoms (or more)

Inorganics

3D (bulks) : 1-10 atoms in the unit cell

TEM image of a InAs/GaAs dot

Si(111)7!7 Surface

GaN

InGaN GaN

HRTEM image:

segregation of Indium in GaN/InGaN Quantum Well Examples of Nanostructures

Synthesis of colloidal nanocrystals Injection of organometallic precursors

Mixture of surfactants Heating mantle

Nanostructures: colloidal crystals!

- !Crystal from sub-!m spheres of PMMA (perpex) suspended in organic solvent;

- ! self-assembly when spheres density high enough;

illuminated with white light

Bragg‘s law different

crystals different orientation

different !

Hot topic (to come) –

The curious world of nanowires

SEM of ZnTe nanowires grown by MBE on GaAs with Au nanocatalyst

Nanowire site control and branched NW structures:

nanotrees and nanoforests

(A) Nanowires can be accurately positioned using lithographic methods such as EBL and NIL. (B) Subsequent seeding by aerosol deposition produces nanowire branches on an array as in panel (A). Shown here is a top view of such a ‘nanoforest’ where the branches grow in the <111>B crystal directions out from the stems. (C) Dark field STEM image and EDX line scan of an individual nanotree. An optically active heterosegment of GaAsP in GaP has been incorporated into the branches.

http://www.nano.ftf.lth.se/

Nanowire nanolasers

Room temperature lasing action from chemically synthesized ZnO nanowires on sapphire substrate

Huang, M., Mao, S.S., et al., Science 292,

Schematic illustration A SEM image

One end of the nanowire is the epitaxial interface between the sapphire

and ZnO, whereas the other end is the crystalline ZnO (0001)

plane

Detection of single viruses with NW-FET

Patolsky & Lieber, Materials Today, April 2005, p. 20 Patolsky et al., Proc. Natl. Acad. Sci.

USA 101 (2004) 14017

Simultaneous conductance and optical data recorded

for a Si nanowire device after the introduction of influenza A solution

.

Controlled Growth and Structures of Molecular-Scale Silicon Nanowires

(a) !TEM images of 3.8-nm SiNWs

grown along the

<110> direction (c) cross-sectional image

(b) & (d) models based on Wulff construction

Yue Wu et al., NANO LETTERS 4, 433 (2004)

High Performance Silicon Nanowire Field Effect Transistors

Yi Cui, et al. NANO LETTERS 3, 149 (2003)

Comparison of SiNW FET transport parameters with those for state-of-the-art planar MOSFETs show that

“SiNWs have the potential to exceed substantially conventional devices, and thus could be ideal building blocks for future nanoelectronics.”

Heterostructured Nanowires

coaxial

heterostructured nanowire

longitudinal

heterostructured

nanowire

Heterostructured Nanowires

Transmission electron microscopy images of a GaN/AlGaN

core–sheath nanowire two Si/SiGe

superlattice nanowires Methods

for

Nanostructure Modeling

Atomistic methods for modeling of nanostructures

Ab initio methods (up to few hundred atoms)

Density Functional Theory (DFT)

One particle density determines the ground state energy of the system for arbitrary external potential

E [ !! ] == d ³ _r !!

!! ^{!! ( r ) !! !! ext ( r ) ++ F [ !! !! ]}

E [ ! ] E ₀ == ₀

ground state density

ground state energy

E[!] = d!r! !

_ext

(!r)!(!r) ^{+ T}

^S

[!]+U[!]+ E

_x

[!]+ E

_c

[!]

unknown!!!

Total energy functional

External energy

Kinetic energy

Classic Coulomb energy

Exchange

energy Correlation

energy

Atomistic methods for modeling of nanostructures

Ab initio methods (up to few hundred atoms)

Semiempirical methods (up to 1M atoms)

Tight-Binding Methods

Tight-Binding Hamiltonian

† †

i! i! i! i!, j" i! j"

!i !i ,"j

H == ! ! # c c ++ ! ! t c c

creation & anihilation operators

On-site energies are not atomic eigenenergies

They include on average the effects of neighbors

Problem: Transferability

E.g., Si in diamond lattice (4 nearest neighbors) & in fcc lattice (12 nearest neighbors) Dependence of the hopping energies on the distance between atoms

0 2 4 6 8 10 12 14 1

10 100 1 000 10 000 100 000 1e+06

Number of atoms

R (nm)

Tight-Binding

Pseudo-

potential Ab initio

Atomistic vs. Continuous Methods

Microscopic approaches can be applied

to calculate properties of realistic nanostructures

Number of atoms in a spherical Si nanocrystal as a function of its radius R.

Current limits of the main techniques for calculating electronic structure.

Nanostructures commonly studied experimentally lie in the size range 2-15 nm.

Continuous methods

Atomistic methods for modeling of nanostructures

Ab initio methods (up to few hundred atoms)

Semiempirical methods (up to 1M atoms)

(Empirical Pseudopotential) Tight-Binding Methods

Continuum Methods

(e.g., effective mass approximation)

Continuum theory- Envelope Function Theory

Electron in an external field

! ˆp

²

2m ++ V ( ! _{r ) ++ U( r )} !

!!

""

## ##

$$

%% &&

&& ! ! ( _{r ) ==} ! !!" " ( _{r )} !

Periodic potential of crystal Non-periodic external potential

Strongly varying on atomic scale Slowly varying on atomic scale

0

-5 5

!1

"3

"1

"1

"1

"3

# ‘2

# ‘25

#15

#1

$ ‘2

$ ‘2

$5

$1

$5

L ‘2 L1 L ‘3 L3

L1 !4

!1

Energy [eV]

Wave vector k

$1

"1

Ge

Band structure of Germanium

!!

_n

( ! k) U( _{r ) == 0} !

Band Structure

Electron in an external field ˆ

2

( ) ( ) ( ) ( )

2

p V r U r r r

m ! ! ""! !

## $$

++ ++ ==

%% &&

%% &&

'' ((

! ! ! ! !

Periodic potential of crystal Non-periodic external potential Strongly varying on atomic scale Slowly varying on atomic scale

Which external fields ?

"! Shallow impurities, e.g., donors

"! Magnetic field

B,

"!Heterostructures, Quantum Wells, Quantum wires, Q. Dots

( ) 2

| | U r e

!! r

== ""

! !

B curlA!== !== !! ""! A!

GaAs GaAlAs cbb

GaAs

GaAlAs GaAlAs

Does equation that involves the effective mass and a slowly varying function exist ? ˆ2

( ) ( ) ( )

2 *

p U r F r F r

m !!

"" ##

++ ==

$$ %%

$$ %%

&& ''

! ! ! !

( ) ? F r == !

Envelope Function Theory – Effective Mass Equation

J. M. Luttinger & W. Kohn, Phys. Rev. B 97, 869 (1955).

[ !! (!!i !

"

" ) ++ U( !

r ) !! !! ]F _n ( ! r ) == 0

!

! ( _{r ) == F} !

n ( _{r )u} !

n0 ( _{r )} !

U(!r) == 0 ^F

n

(!r) == exp(i!k!! !r)

(EME) EME does not couple different bands

Envelope Function

Periodic Bloch Function

“True”

wavefunction

Special case of constant (or zero) external potential

!

! ( !

r )

Bloch function

( )

U z ^F

n

( !r) == exp[i(k

_x

x ++ k

_y

y)]F

_n

(z)

k.p method for bulks

k.p Method- Luttinger-Kohn basis

Bloch functions are orthogonal in the wave vector and band index

d³_r!

!!

^{""l*(k,!r )!""n(q,!r ) ==! ##nl##(k $$! q)!}

! !_n(! k,!

r ) == exp[i! k ""!

r]u_n(! r )

d³!

!!

r^{ul*(k,!r )u! n(k,!!r ) == ""}

2##

(( ))

^3$$nl

d³!

!!

r^{ul*(k,!r )u! n(q,!}r ) "" 0^! ! k !!!

for

q

!!

^j

⁽

k, ! r ) == exp[i( ! ! k "" !

k

₀

) ## _r] ! $ $

_j

( !

k

₀

, r ) == exp[i ! ! k ## ! _r]u

j

( ! k

₀

, _{r )} ! Luttinger-Kohn basis

d³!

!!

r^{""l*(k,! !r )""n(q,!r ) ==! ##nl##(k $$! q)!}

! !

ⁿ

""

^{d3k!##n*(k,!r )! ##n(k,!r ') ==! $$(r %%! r ')!}

Luttinger-Kohn basis is orthogonal and complete

!

!

_n

( !

k, !r) == A

_nj

( ! k)""

_j

( !

k, !r)

#

j

#

Expansion of the unknown Bloch function in terms of known Luttinger-Kohn functions

k.p Method - Derivation

!p²

2m++V (!

!! r )

""

## $$

%%&&''_n(! k,!

r ) ==((_n(! k)''_n(!

k,! r )

!

!_n(!

k,!

r ) == A_nj(! k)""_j(!

k,! r )

#

j

#

A_nj(!

k) ! p²

2m++V (!

!! r )

""

## $$

%%&&

'

j

'

^{exp[i(k ((! k!}0) ))! r]**_j(!

k₀,! r ) ==

==++_n(! k)exp[i(!

k ((! k₀) ))!

r] A_nj'(! k)

'

j'

'

^**j'(! k₀,!

! r ) p !! ""i"## !!exp[i(!

k ""! k₀) ##!

r]$$_j(! k₀,!

r ) == exp[i(! k ""!

k₀) ##! r]{!! ++i(!

k ""! k₀)}$$_j(!

k₀,! r )

p!²

2m++V (!

!! r )

""

## $$

%%&&exp[i(! k ''!

k₀) ((! r]))_j(!

k₀,! r ) ==

==exp[i(! k ''!

k₀) ((! r]{"²

2m(! k ''!

k₀)²++"

m(! k ''!

k₀) ((! p ++**_j(!

k₀)}))_j(! k₀,!

r )

k.p Method - Derivation

A_nj(!

k)

!

j

!

^{"_2m^2(k!2""k!02) ++"

m(! k ""!

k₀) ##! p ++$$_j(!

k₀)}u_j(! k₀,!

r ) ==

==$$_n(!

k) A_nj'(! k)

!

j'

!

^uj'(! k₀,!

r )

{!² 2m("

k !!"

k₀)²++! m("

k !!"

k₀) """

p ++##_j("

k₀)}$$_j("

k₀,"

r ) ==

=={!²

2m("

k !!"

k₀)²++! m("

k !!"

k₀) """

p ++##_j("

k₀)}exp[i"

k₀"""

r]u_j("

k₀,"

r ) ==

==exp[i"

k₀"""

r]{!² 2m("

k²!!"

k₀²) ++! m("

k !!"

k₀) """

p ++##_j("

k₀)}u_j("

k₀,"

r )

Multiply both sides by

^ul*(! k₀,!

r )

Integrate over the unit cell

d³!

r

!

!

""

! !

^{j {[""}j(! k₀) ++"²

2m(! k²##!

k₀²)]$$_lj++"

m(! k ##!

k₀) %%! p_lj++}A_nj(!

k) ==""_n(! k)A_nl(!

k)

!p_lj== (2!!)³

"

" ^d3

!r

"

"

##

^{ul*(k!0,r )! !ˆpuj(k!0,!r )}

Momentum matrix elements

k dot p

k.p Method – Main Equation

! !

^{j {[""}j(! k₀) ++"²

2m(! k²##!

k₀²)]$$_lj++"

m(! k ##!

k₀) %%! p_lj++}A_nj(!

k) ==""_n(! k)A_nl(!

k)

System of homogeneous equations for expansion coefficients A_nj(!

k)

!!

ⁿ

( ! This equations couple all bands k)

What one needs is decoupling scheme, e.g., perturbation theory

0 (1) (2)

n n n n

E ==E ++E ++E ++!

0 0 2

0 0 0

0 0

| | '| |

n n | '|

m n n m

m H n

E E n H n

E E

!!

== ++ ++ ++

""

#

#

^!

0 0

0 0

0 0

| '|

m n n m

m H n

n n m

E E

!!

== ++ ++

""

#

#

^!

0 '

H H== ++H H n₀ ⁰ ==E n_n^{0 0}

H n ==E nn

Perturbation theory for non-degenerate states

k.p Method – Non-degenerate band Let us consider non-degenerate band and look for band energies around extremum in k

₀

e.g., conduction band minimum in GaAs

k!₀==0 !!""

H ' == !

²

2m ( "

k

²

!! "

k

₀²

) ++ ! m ( "

k !! "

k

₀

) "" "p Expansion parameter: _{s ==} ! !

k !! ! k ₀

!!_n(! k) ==!!_n(!

k₀) ++"

m(! k ""!

k₀) ##! p_nn++"²

2m(! k²""!

k₀²) ++"² m²

(! s ##!

p_nj)(! s ##!

p_jn)

!!_n(! k₀) ""!!_j(!

k₀)

j$$n

%

%

⁺⁺

u_n(!

k,_{r ) == u}!

n(! k₀,!_{r ) ++}"

m

!s !!

p!_jn

""_n(! k₀) ##""_j(!

k₀)u_j(! k₀,!_{r )}

j$$n

%

%

^++…

Energy to second order in the expansion parameter

Periodic wave function to the first order in expansion parameter

k.p Method – Effective Mass Tensor

Let us introduce second rank tensor

m m *

!!

""##

$$

%%&&

_''((

== m

!

²

))

²

**

_n

( "

k) ))k

_''

))k

_{(( "}

k =="

k₀

Reciprocal Effective Mass Tensor

Band n has extremum in k

₀

!!""_n(! k)

!!! k _{k ==}^! _k^!

0

==!!""_n(! k)

!!!

s !_{s ==0}==0

!p

_nn

++" ! k

₀

== 0

!

p_nn==0 !!!

k₀==0

e.g., GaAs

!

p_nn!!0 ""!

k₀!!0

e.g., Si Taylor series

!!_n(! k) ==!!_n(!

k₀) ++ (""!!_n(! k₀)) ##(!

k $$! k₀) ++ 1

2_%%

' '

_,&& ^((2!!n(

k!₀) ((k_%%((k_&& (!

k $$! k₀)_%%(!

k $$! k₀)_&&++

!!_n(! k) ==!!_n(!

k₀) ++"

m(! k ""!

k₀) ##! p_nn++"²

2m(! k²""!

k₀²) ++"² m²

(! s ##!

p_nj)(! s ##!

p_jn)

!!_n(! k₀) ""!!_j(!

k₀)

j$$n

%

%

⁺⁺

! m

"

s !!["

p_nn++! 2("

k ++"

k₀)]

k.p Method – Effective Mass Tensor

!!²""_n(! k₀)

!!k_##!!k_$$ =="² m%%_##$$++"²

m²

p_nj^##p^$$_jn++p_nj^##p^$$_jn

""_n(! k₀) &&""_j(!

k₀)

(

j''n

(

m m_n^*

!!

""

## $$

%%&&

''((

==))_''((++ 1 m

p_nj^''p⁽⁽_jn++p_nj^''p⁽⁽_jn

**_n(! k₀) ++**_j(!

k₀)

j,,n

-

- m – free electron mass

It is always possible to diagonalize the reciprocal effective mass tensor by proper choice of the coordinate system.

If the extremum point k0 is a general point in the BZ, the choice of the axes

depends on the details of the dynamics, that is, on the crystal potential.

If the extremum occurs at a symmetry point or along an axis of symmetry, the axes may be partially determined by symmetry.

In cubic crystals, for minima along [100], [110], [111], the symmetry axis must be a principal axis.

If k=0 is the extremum, the surface of constant energy in a cubic crystal must be spherical

k.p Method – Effective Mass Tensor

m m

_n^*

!!

""

## $$

%% &&

''''

== 1 ++ 2 m

| p

_nj^''

|

²

((

_n

( !

k

₀

) )) ((

_j

( ! k

₀

)

j**n

+

+ ^!! -!refers now to one

of the principal axes The interaction with the lower lying levels tends to decrease an effective mass

^!!j(

k!₀) <<!!_n(! k₀)

The interaction with the higher states tends to increase an effective mass

^!!j(

k!₀) >>!!_n(! k₀) For cubic semiconductors with minimum of the conduction band

in point, the band energy in the neighborhood is

!!

!!_n(!

k) ==!!_n(0) ++"²! k² 2m_n^*

Physical meaning of the effective mass ?

Dynamics of particles

In the presence of external fields, the crystalline electrons behave as particles with effective mass m*.

Effective Mass - A Two-Band Model Some insight into the nature of the results

0 = v 1 = c

EGAP

0

(0) 0

!! ==

1

(0) E

_GAP

!! ==

!p₀₁==! p₁₀^* ==!

p 1

p_!!== 3p

2

*0

1 2

3 _g

m p

m == !! mE _* ²

1

1 2

3 _g

m p

m == ++ mE

2 2

g p

E <<<< m

2

*1

2

3 _g

m p

m !! mE

1* 2

3

2 ^g

m m E

m !! p

Let us assume

This occurs, for example, in GaSb, InAs, InSb

InAsGaSb InSb

GaAsInPCdTe CdSe

ZnTeAlSbCdS

ZnSeGaPAlAs

2.0

0 1.0 3.0

0 0.1 0.2

Energy gap [eV]

Effective mass m*/m

Effective mass proportional to the energy gap

k.P – Method – Band Degeneracies

k ! ₀

The state of interest in is degenerate

^k!0

The perturbation will remove degeneracy, at least in some directions

Going from a point of higher symmetry the energy band split

Valence band top of

cubic semiconductors

It is necessary to use degenerate perturbation theory

e.g., L.I. Schiff, Quantum Mechanics Schwabl, Quantenmechanik

0 0 0

0 na n na

H!! =="" !! a==1,2,3, ,! g_n

0 ' 0

a H ++H b $$E##!!""==

--- --- Matrix

abn n n

H g !!g

In our case, perturbation is

H ' == !

²

2m ( "

k

²

!! "

k

₀²

) ++ ! m ( "

k !! "

k

₀

) "" "p For simplicity

k !

₀

== 0

H ' == !

²

2m

"

k

²

++ ! m

"

k !! "p Valence band maximum

in most of semiconductors

First order perturbation

k.P – Method – Band Degeneracies

(( )) (( )) (( ))

(0) (1) (2)

0 0 0 0

0 0 0

0 0

1

| '| | '|

| '| ^m

n n n n

ab ab ab ab

g na mc mc nb

n ab na nb

m n c n m

H H H H

H H

H !! !! !! !!

"" ## !! !!

"" ""

$$ ==

== ++ ++ ==

== ++ ++

&

%%

& & &

Löwdin perturbation theory

The essential idea of this procedure is to separate the states considered in the perturbation calculation into two sets:

one set involves a small number of strongly coupled states whose interactions is treated exactly (gn),

the second set, with more states, contains those states that are well removed in energy from the first set

One can also treat in this way situation in which some bands, although not quite degenerate, approach each other closely.

Then treatment of such bands as non-degenerate does not make sense

k.P – Method – Band Degeneracies

H_abⁿ ==!!_n(!

k₀)""_ab++"²

2m""_ab k_µµ² µµ==1

#

3

#

^{++ [}

# #

^{µµkµµ( p}µµ)^nm_ac][

# #

_$$k_$$^{( p}$$)_cb^mn]

!!_n(! k₀) %%!!_m(!

k₀)

c==1 g_m

#

m&&n

# # #

H_abⁿ ==!!_n(! k₀) ++"²

2m""_ab k_µµ² µµ==1

#

3

#

^{++ k}µµ

v==1

#

3

#

^kv µµ==1

#

3

#

_!!^{( pµµ)acnm( p$$)cbmn}

n(! k₀) %%!!_m(!

k₀)

c==1 g_m

#

m&&n

# # #

1

m_µµ!!^ab ==!!_ab!!µµ""++ 2 m

( p_µµ)_ac^nm( p_!!)_cb^mn

!!_n(! k₀) !! !!_m(!

k₀)

c==1 g_m

"

"

m##n

" "

H_abⁿ ==!!_n(!

k₀)!!_ab++!² 2m

k_µµk_v m_µµ!!^ab

v==1

!

3 µµ==1

!

!

3

!

!p

_ab^nm

== u

_na⁰

| !ˆp|u

_mb⁰

H

_abⁿ

== !!

_n

( !

k

₀

)""

_ab

++ "

²

2m

k !

²

""

_ab

++ "

m

k ## ! !p

_abⁿⁿ

++ ( ! k ## !p

_ac^nm

)( !

k ## !p

_cb^mn

)

!!

_n

( !

k

₀

) $$ !!

_m

( ! k

₀

)

c==1 g_m

%

m&&n

% % %

Band minimum in

^k!0!!! p_abⁿⁿ==0

k.P – Method – Band Degeneracies Special case: valence band in !!

^{a b, !!{ , , }x y z} n v!!

, ,

vx vy vz

3 degenerated bands:

is 3 3 matrix

abn

H !!

1 1 1

2 3 3

1 1 1

1 1 1 1 1

( ) ( ) ( )

ˆˆ (0) ( ) ( ) ( )

2 ( ) ( ) ( )

xx xy xz

yx yy yz

n v

v zx zy zz

m m m

H k k m m m

m m m m

µµ!! µµ!! µµ!!

µµ µµ!! µµ!! µµ!!

µµ

µµ!! µµ!! µµ!!

""

## ## ##

## ## ##

== == ## ## ##

$$ %%

&& ''

&& ''

== ++

&& ''

&& ''

(( ))

*

* * *

!

For cubic semiconductors, there are only three different matrix elements 1

( ) ( )

1 1 1 1 2

(0) (0)

m vm mv

g z zc z cz

xx yy zz

m n c

xx yy zz n m

p p

m m m m

L

"" == !! !!

== == == == ++

$

##

$ $ $

1

( ) ( )

1 1 1 1 1 1 1 2

(0) (0)

m vm mv

g x zc x cz

yy zz xx xx yy zz

m n c

xx yy zz yy zz xx v m

p p

m m m m m m m

M

== == == == == == == ++ "" == !! !!

$

##

$ $ $

1

( ) ( )

1 1 1 1 1 1 1 2

(0) (0)

m vm mv

g x xc z cz

xy yz zx yx zy xz

m n c

xy yz zx yx zy xz n m

p p

m m m m m m m

N

"" == !! !!

== == == == == == == ++

$

##

$ $ $

L, M, N – Dresselhaus parameters

k.P – Method – Band Degeneracies

ˆ ˆ

ˆ ˆ

H D k k ==

^µµ!! _{µµ !!}

2 2 2

2 2 2 2

2 2 2

( )

ˆˆ (0) ( )

2 ( )

x y z x y x z

v x y y x z y z

x z y z z x y

Lk M k k Nk k Nk k

H I Nk k Lk M k k Nk k

m Nk k Nk k Lk M k k

!!

"" ++ ++ ##

$$ %%

$$ %%

== ++ ++ ++

$$ %%

$$ ++ ++ %%

&& ''

!

Degenerated Valence Band of Cubic Semiconductors

det( H EI ˆˆ !! ) 0 ==

These equations can be solved analytically !!

T. Manku & A. Nathan, J. Appl. Phys. 73, 1205 (1993) J. Dijkstra, J. Appl. Phys. 81, 1259 (1997)

Pretty complicated task

SIMPLE: Find solutions along a symmetry line,

e.g.,

k !^!![k_x,0,0] !! ""line

Degenerated Valence Band of Cubic Semiconductors

k !!![k_x,0,0]

!! ""line

2

2

2

(0) 0 0

ˆˆ 0 (0) 0

0 0 (0)

v x

v x

v x

Lk

H Mk

Mk

!!

!!

!!

"" ++ ##

$$ %%

==$$ ++ %%

$$ %%

$$ ++ %%

&& ''

Dispersion relations along

2 2

(0) 2

hh v

Mk

x

!! == !! ++ ! m

2 2

(0) 2

lh v

Lk

x

!! == !! ++ ! m

Heavy hole Light hole

hh

lh (2) (1)

line

!! ""

!!

!!

_v

( ! k )

Generally are dependent

on the direction of k  Warped bands



hh L, M, N – Dresselhaus parameters lh

obtained from fits to experimental

data