TWO-BAND MODELS FOR ELECTRON TRANSPORT IN SEMICONDUCTOR DEVICES

KINETIC EQUATIONS:

KINETIC EQUATIONS:

Direct and Inverse Problems Direct and Inverse Problems

Università degli Studi di Pavia (sede di Mantova) Università degli Studi di Pavia (sede di Mantova)

Mantova Mantova

May 15-17, 2005 May 15-17, 2005

Giovanni Frosali

TWO-BAND MODELS FOR ELECTRON TRANSPORT

IN SEMICONDUCTOR DEVICES

Two-band models for electron transport in semiconductor devices n. 2 di 42

University of Florence research group on semicoductor modeling

Dipartimento di Matematica Applicata “G.Sansone”

 Giovanni Frosali

Dipartimento di Matematica “U.Dini”

 Luigi Barletti

Dipartimento di Elettronica e Telecomunicazioni

 Stefano Biondini

 Giovanni Borgioli

 Omar Morandi Università di Ancona

 Lucio Demeio

Scuola Normale Superiore di Pisa (Munster)

 Chiara Manzini

Others: G.Alì (Napoli), C.DeFalco (Milano), M.Modugno (LENS-INFM Firenze), A.Majorana(Catania), C.Jacoboni, P.Bordone et. al. (Modena)

In the standard semiconductor devices, like the Resonant Tunneling Diode, the single-band approximation, valid if most of the current is carried by the charged particles of a single band, is usually satisfactory. Together with the single-band approximation, the parabolic-band approximation is also

usually made. This approximation is satisfactory as long as the carriers populate the region near the minimum of the band.

SINGLE-BAND APPROXIMATION SINGLE-BAND APPROXIMATION

Also in the most bipolar electrons-holes models, there is no coupling mechanism between energy bands which are always decoupled in the effective-mass approximation for each band and the coupling is heuristically inserted by a "generation-recombination" term.

Most of the literature is devoted to single-band problems, both from the

modeling and physical point of view and from the numerical point of view.

Two-band models for electron transport in semiconductor devices n. 4 di 42

It is well known that the spectrum of the Hamiltonian of a quantum particle moving in a periodic potential is a continuous spectrum which can be decomposed into intervals called "energy bands". In the presence of external potentials, the projections of the wave function on the energy eigenspaces (Floquet subspaces) are coupled by the Schrödinger equation, which allows interband transitions to occur.

-2 -1 0 1 2

Energy (ev)

60 50

40 30

20 10

0 Position (nm)

RITD Band Diagram The single-band approximation is no longer valid when the

architecture of the device is

such that other bands are

accessible to the carriers. In

some nanometric semiconductor

device like Interband Resonant

Tunneling Diode, transport due

to valence electrons becomes

important.

Various mathematical tools are employed to exploit the multiband quantum dynamics underlying the previous models:

• the Schrödinger-like models

• the nonequilibrium Green’s function

• the Wigner function approach

It is necessary to use more sophisticated models, in which the charge carriers can be found in a super-position of quantum states belonging to different bands.

Different methods are currently employed for characterizing the band structures and the optical properties of heterostructures, such as

• envelope functions methods based on the effective mass theory

• tight-binding (

• pseudopotential methods

Two-band models for electron transport in semiconductor devices n. 6 di 42

MULTI-BAND, NON-PARABOLIC ELECTRON TRANSPORT MULTI-BAND, NON-PARABOLIC ELECTRON TRANSPORT

• Wigner-function approach

• Formulation of general models for multi-band non-parabolic electron transport

• Use of Bloch-state decomposition (Demeio, Bordone, Jacoboni)

• Envelope functions approach (Barletti)

• Wigner formulation of the two-band Kane model (Borgioli, Frosali, Zweifel)

• Numerical applications

• The Wigner function for thermal equilibrium of a two-band

• Multiband envelope function models (MEF models)

• Two-band hydrodynamic models (Two-band QDD equations)

QUANTUM MECHANICS LEVEL QUANTUM MECHANICS LEVEL

In this talk we present different Schrödinger-like models.

The first one is well-known in literature as the Kane model.

The second, based on the Luttinger-Kohn approach, disregards the interband tunneling effect.

The third, recently derived within the usual Bloch-Wannier formalism, is

formulated in terms of a set of coupled equations for the electron envelope

functions by an expansion in terms of the crystal wave vector k (MEF model).

Two-band models for electron transport in semiconductor devices n. 8 di 42

Electromagnetic and spin effects are disregarded, just like the field generated by the charge carriers themselves. Dissipative phenomena like electron-

phonon collisions are not taken into account.

The dynamics of charge carriers is considered as confined in the two highest energy bands of the semiconductor, i.e. the conduction and the (non-

degenerate) valence band, around the point is the "crystal"

wave vector. The point is assumed to be a minimum for the conduction band and a maximum for the valence band.

The physical environment

0 k k  0

k  ^where

The Hamiltonian Hamiltonian introduced in the Schrödinger equation is

2

, 2

o o per

H H V H h V

    m  

where is the periodic potential of the crystal and an external potential. V

V

KANE MODEL KANE MODEL

The Kane model consists in a couple of Schrödinger-like equations for the conduction and the valence band envelope functions.



( , ) x t

be the valence band envelope function

Let 

( , ) x t be the conduction band electron envelope function and

• m is the bare mass of the carriers,

• is the minimum (maximum) of the conduction (valence) band energy

• P is the coupling coefficient between the two bands

E ( ) E

, ,

i i

V   E V i c v 

c c

Two-band models for electron transport in semiconductor devices n. 10 di 42

Interband Tunneling: PHYSICAL PICTURE

Interband transition in the 3-d dispersion diagram.

The transition is from the bottom of the conduction band to the top of the val- ence band, with the wave number becoming imaginary.

Then the electron continues propagating into the valence band.

Remarks on the Kane model Remarks on the Kane model

• The envelope functions are obtained expanding the wave function on the basis of the periodic part of the Bloch functions evaluated at ,

( , )

( , ),

n n

b x t  e u k x

,



0 0

( ) x

( ) x u

( ) x u

    

0

,

( )

(0, )

c v c v

u x  u x

where .

0 k 

• The external potential affects the band energy terms , but it does not appear in the coupling coefficient P .

• There is an interband coupling even in absence of an external potential.

• The interband coefficient P increases when the energy gap between the two bands increases (the opposite of physical evidence). E

c

( )

V V V

Two-band models for electron transport in semiconductor devices n. 12 di 42

This model is a model, i.e. the crystal momentum is used as a perturbation parameter of the Hamiltonian. The wave function is expanded on a different basis with respect to Kane model:

LUTTINGER-KOHN model LUTTINGER-KOHN model

where n, n' are the band index and the bare electron mass.

As a result, if we limit ourselves to the two-band case, we have:

m

k P  k

where are envelope functions in the conduction and valence bands,

respectively and and are, respectively, the isotropic effective masses in the conduction and valence bands.

As it is manifest, disregarding the off-diagonal terms implies the achievement of two uncoupled equations for the envelope functions in the two bands. This

means that the model, at this stage of approximation, is not able to describe an interband tunneling dynamics.

,



*

m

m

Two-band models for electron transport in semiconductor devices n. 14 di 42

M M EF MODEL E F MODEL

The MEF model consists in a couple of Schrödinger-like equations as follows.

A different procedure of approximation leads to equations describing the

intraband dynamics in the effective mass approximation as in the Luttinger-

Kohn model, which also contain an interband coupling, proportional to the

momentum matrix element P. This is responsible for tunneling between

different bands induced by the applied electric field proportional to the x-

derivative of V. In the two-band case they assume the form:

*

m

• (and ) is the isotropic effective mass

• and are the conduction and valence envelope functions

• is the energy gap

• P is the coupling coefficient between the two bands

*

m





E

• Expansion of the wave function on the Bloch functions basis

• Insert in the Schrödinger equation

Which are the steps to attain MEF model formulation?

Two-band models for electron transport in semiconductor devices n. 16 di 42

• Approximation

• Simplify the interband term in

• Introduce the effective mass approximation

• Develope the periodic part of the Bloch functions to the first order

• The equation for envelope functions in x-space is obtained by inverse Fourier transform

( , ) u k x

0 k 

See: Morandi, Modugno, Phys.Rev.B, 2005

Vai

Vai a a llalla 19 19

For more rigorous details:

For more rigorous details:

More rigorously MEF model can be obtained as follows:

• projection of the wave function on the Wannier basis which depends on where are the atomic sites positions, i.e. x R 

R

W



where the Wannier basis functions can be expressed in terms of Bloch

functions as

Two-band models for electron transport in semiconductor devices n. 18 di 42

The use of the Wannier basis has some advantages.

As a matter of fact the amplitudes that play the role of envelope functions on the new basis, can be obtained from the Bloch coefficients by a simple Fourier transform

n

( ) R



Moreover they can be interpreted as the actual wave function of an electron in the n-band. In fact, ''macroscopic'' properties of the system, like charge density and current, can be expressed in term of averaging on a scale of the order of the lattice cell.

n

( ) R



Performing the limit to the continuum to the whole space and by using standard properties of the Fourier transform, equations for the

coefficients are achieved. 

( ) x

• The envelope functions can be interpreted as the effective wave functions of the electron in the conduction (valence) band

• The coupling between the two bands appears only in presence of an external (not constant) potential

• The presence of the effective masses (generally different in the two bands) implies a different mobility in the two bands.

• The interband coupling term reduces as the energy gap increases, and vanishes in the absence of the external field V.



E

Comments on the MEF MODEL

Comments on the MEF MODEL

Two-band models for electron transport in semiconductor devices n. 20 di 42

QUANTUM HYDRODYNAMICS LEVEL QUANTUM HYDRODYNAMICS LEVEL

From the point of view of practical applications, the approaches based on microscopic models are not completely satisfactory.

Hence, it is useful to formulate semi-classical models in terms of macroscopic variable. Using the WKB method, we obtain a system for densities and currents in the two bands.

In this context zero and nonzero temperatures quantum drift quantum drift diffusion models

diffusion models are derived, for the Kane and MEF systems.

Hydrodynamic version of the KANE MODEL Hydrodynamic version of the KANE MODEL

We can derive the hydrodynamic version of the Kane model using the WKB method (quantum system at zero temperature).

Look for solutions in the form

we introduce the particle densities

Then is the electron density in conduction and valence bands.

We write the coupling terms in a more manageable way, introducing the complex quantity

with

c c v v

n      

( , ) ( , ) ( , ) exp

c c

iS x t x t n x t

 

 

    

( , ) ( , ) ( , ) exp

v v

iS x t x t n x t

 

 

    

:

n     n n e

 ^:  ^S

^{ S}

( , ) ( , ) ( , ).

ij i j

n x t   x t  x t

Two-band models for electron transport in semiconductor devices n. 22 di 42

Quantum hydrodynamic quantities

• Quantum electron current densities

when i=j , we recover the classical current densities

• Complex velocities given by osmotic and current velocities, which can be expressed in terms of plus the phase difference 

• Osmotic and current velocities

c c c

J   n S

Im( )

ij i j

J     

v v v

J   n S

, , , ,

c os c el c v os v el v

u  u  iu u  u  iu

, ⁱ

,

, ,

os i el i i

i i

n J

u u S i c v

n n

 

    

, , ,

c v c v

n n J J

The quantum counterpart of the classical continuity equation

Taking account of the wave form, the Kane system gives rise to

Summing the previous equations, we obtain the balance law

where we have used the “interband density” and the complex velocities

(cos sin )( )

n u n n i u iu

        

Two-band models for electron transport in semiconductor devices n. 24 di 42

The previous balance law is just the quantum counterpart of the classical continuity equation.

Next, we derive a system of coupled equations for phases , obtaining an equivalent system to the coupled Schrödinger equations. Then we obtain a system for the currents

(similar equation for ). J

The left-hand sides can be put in a more familiar form with the quantum Bohm potentials

c

,

S S

We express the right-hand sides of the previous equations in terms of the hydrodynamic quantities

It is important to notice that, differently from the uncoupled model, equations

for densities and currents are not equivalent to the original equations, due to

the presence of . 

Two-band models for electron transport in semiconductor devices n. 26 di 42

Recalling that and are given by the hydrodynamic quantities and , we have the HYDRODYNAMIC SYSTEM

, , ,

c v c v

n n J J u

cv

, ,

 n u

The DRIFT-DIFFUSION scaling

• We rewrite the current equations, introducing a relaxation time term in order to simulate all the mechanisms which force the system towards the statistical mechanical equilibrium characterized by the relaxation time 

• In analogy with the classical diffusive limit for a one-band system, we introduce the scaling

,

,

, .

t t J  J J  J  

    

• We express the osmotic and current velocities, in terms of the other

hydrodynamic quantities.

Two-band models for electron transport in semiconductor devices n. 28 di 42

ZER0-TEMPERATURE QUANTUM DRIFT-DIFFUSION MODEL

for the Kane system.

NON ZERO TEMPERATURE hydrodynamic model

We consider an electron ensemble which is represented by a mixed quantum mechanical state, with a view to obtaining a nonzero temperature model for a Kane system.

We rewrite the Kane system for the k-th state

We use the Madelung-type transform ^

^{ n}

^exp  ^iS

^{/ }  ^{, i c v  ,}

We define the densities and the currents corresponding to the two mixed states

We define J J

, ,



, n u u

, , .

Two-band models for electron transport in semiconductor devices n. 30 di 42

HYDRODINAMIC SYSTEM for the KANE MODEL

QUANTUM DRIFT-DIFFUSION for the KANE MODEL

Two-band models for electron transport in semiconductor devices n. 32 di 42

HYDRODINAMIC SYSTEM for the MEF MODEL

QUANTUM DRIFT-DIFFUSION for the MEF MODEL

Two-band models for electron transport in semiconductor devices n. 34 di 42

Physical meaning of the envelope functions

A more direct physical meaning can be ascribed to the hydrodynamical variables derived from the second approach.

The envelope functions and are the projections of on the Wannier basis, and therefore the corresponding multi-band densities represent the (cell- averaged) probability amplitude of finding an electron on the conduction or valence bands, respectively. .

The Wannier basis element arises from applying the Fourier transform to the Bloch functions related to the same band index .

M







The Kane envelope functions and the MEF envelope functions The Kane envelope functions and the MEF envelope functions are linked by the relation

are linked by the relation This

This simple picture does not apply to the Kane model.

2

0

, , , .

( )

K M M

j j h

j h

i P j h c v

m E E

      





n

This fact confirms that even in absence of external potential , when no interband This fact confirms that even in absence of external potential , when no interband

transition can occur, the Kane model shows a coupling of all the envelope transition can occur, the Kane model shows a coupling of all the envelope

functions.

functions.

We consider

We consider a a heterostructure heterostructure which consists of two

which consists of two

homogeneous regions separated homogeneous regions separated by a

by a potential barrier potential barrier and which and which realizes a single

realizes a single quantum well quantum well in in valence band.

valence band.

NUMERICAL SIMULATION

See: Alì, F.,Morandi,

Two-band models for electron transport in semiconductor devices n. 36 di 42

The incident (from the left) conduction electron beam is

The incident (from the left) conduction electron beam is mainly mainly reflected

reflected by the barrier and the valence states are almost unexcited by the barrier and the valence states are almost unexcited . .

The incident (from the left) conduction electron beam is

The incident (from the left) conduction electron beam is partially partially reflected

reflected by the barrier and partially captured by the barrier and partially captured by the well by the well

Kane Kane MEF MEF

Two-band models for electron transport in semiconductor devices n. 38 di 42

When the electron energy approaches the resonant level, the electron

When the electron energy approaches the resonant level, the electron can can travel

travel from the left to the right, using the bounded valence resonant state as a from the left to the right, using the bounded valence resonant state as a bridge state.

bridge state.

QUANTUM KINETICS LEVEL

QUANTUM KINETICS LEVEL

Two-band models for electron transport in semiconductor devices n. 40 di 42

Wigner function :

MEF-Wigner Model

 ^,  ¹

 ^{/ 2}   ^{/ 2}  ^'

2

iv

ij i j

f x v x m x m e dv d



    

    ^  ^

 

 

*

0

*

0

2

*

0

2 Im

2 Im

4

M

M M M

c

c cc c cv

g M

M M M

v

v vv v cv

g M

M M M M M

cv cv cv cv cv c v

g

f p P

f i f f

t m m E

f p P

f i f f

t m m E

f i P

i f p f i f i f f

t m m E

 

 

  





 

     

  

      

  

             







 



   

'

/ 2 / 2

1 '

2

i v v

i j

ij ij

v

V x m V x m

f e

dv d



 

 



  

   ^ _ ^

Vai

Vai allaalla 42 42

Numerical results: MEF-WIGNER model

Valence band

Conduction band

Two-band models for electron transport in semiconductor devices n. 44 di 42

Thanks for your attention !!!!!

Wigner picture Wigner picture

   

   

   

0

0

0

Im 2 Re

Im 2 Re

2

Kane

Kane Kane Kane Kane

cc

cc cc cc cv cv

Kane Kane Kane Kane Kane

vv

vv vv vv cv cv

Kane Kane Kane Kane Kane Kane Kane

cv

cv cv cv cc vv cc vv

f P

v f i f f Pv f

t m

f P

v f i f f Pv f

t m

f P

v f i f i f f Pv f f

t m







       

  

       

 

         











 

 

 ^,  ¹

 ^{/ 2}   ^{/ 2}  ^'

2

iv

ij i j

f x v x m x m e dv d



    

    ^  ^

Wigner function :

Kane-Wigner Model

 / 2   / 2 

1 V x

 m V x

 m

   ^{    }

 

Two-band models for electron transport in semiconductor devices n. 46 di 42

REMARKS

We have derived a set of quantum hydrodynamic equations from the two- band Kane model, and from the MEF model. These systems, which can be considered as a nonzero-temperature quantum fluid models, are not closed.

In addition to other quantities, we have the tensors and , similar to the temperature tensor of kinetic theory.

• Closure of the quantum hydrodynamic system

• Numerical treatment

• Heterogeneous materials

c

, ,

   

Numerical validation for the quantum drift-diffusion equations (Kane and

MEF models) are work in progress.