dipartimento di matematica applicata università di firenze multiband transport models for...

Dipartimento di Matematica Applicata Università di Firenze

Multiband transport models for semiconductor devices

Giornata di lavoro sulle Nanoscienze Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005Firenze 16 dicembre 2005

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Giornata di lavoroGiornata di lavoro

Mathematical modeling and numerical analysis of quantum Mathematical modeling and numerical analysis of quantum systems with applications to nanosciencessystems with applications to nanosciences

Firenze, 16 dicembre 2005Firenze, 16 dicembre 2005

Giovanni Frosali Dipartimento di Matematica Applicata “G.Sansone”

[email protected]

MULTIBAND TRANSPORT MODELS FOR SEMICONDUCTOR DEVICES




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Research group on semicoductor modeling at University of Florence

Dipartimento di Matematica Applicata “G.Sansone” Giovanni Frosali Chiara Manzini (Munster) Michele Modugno (Lens-INFN)

Dipartimento di Matematica “U.Dini” Luigi Barletti

Dipartimento di Elettronica e Telecomunicazioni Stefano Biondini Giovanni Borgioli Omar Morandi

Università di Ancona Lucio Demeio

Others: G.Alì (Napoli), C.DeFalco (Milano), A.Majorana(Catania), C.Jacoboni, P.Bordone et. al. (Modena)




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• The spectrum of the Hamiltonian of a quantum particle in a periodic potential is continuous and characterized by (allowed) "energy bands“ separated by (forbidden) “band gaps". • In the presence of additional 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

Ene

rgy

(ev)

6050403020100Position (nm)

RITD Band Diagram• Negibible coupling: single-band approximation

• This is no longer possible when the architect-ure 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.

TWO-BAND APPROXIMATIONTWO-BAND APPROXIMATION




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Schrödinger-like models (Barletti, Borgioli, Modugno, Morandi, etc.)

Wigner function approach (Bertoni, Jacoboni, Borgioli, Frosali, Zweifel, Barletti, Manzini, etc.)

Hydrodynamics multiband formalisms (Alì, Barletti, Borgioli, Frosali, Manzini, etc)

• Multiband models are needed: 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 (effective mass theory), tight-binding, pseudopotential methods,…

OUR APPROACH TO THE PROBLEM




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MU

LT

IBA

ND

TR

AN

SP

OR

TM

UL

TIB

AN

D T

RA

NS

PO

RT

General Multiband Models

General Multiband Models

KANE modelKANE model

MeF modelMeF model

WIGNER APPROACH

WIGNER APPROACH

SCHRÖDINGERAPPROACH

SCHRÖDINGERAPPROACH

HYDRODYNAMIC MODELS

HYDRODYNAMIC MODELS

QUANTUM DRIFT-DIFFUSION

MODELS

QUANTUM DRIFT-DIFFUSION

MODELS

QDDQDD

CEexpansion

CEexpansion




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H E

( )n x

( )xMultiband

“kp”system

We filter the solution

1( )x

22

0

( ) ( )2 per extH V x U xm

Envelope function models Envelope function models

2( )x




n. 7 di 30

WIGNER APPROACHWIGNER APPROACH

Lucio Demeio - collaborazioni con Paolo Bordone, Carlo Jacoboni

Luigi Barletti

Giovanni Frosali – collaborazione con Paul Zweifel

Giovanni Borgioli

[1] G. Borgioli, G. Frosali and P. Zweifel, Wigner approach to the two-band Kane model for a tunneling diode, Transp. Teor.Stat. Phys. 32 3, 347-366 (2003).




n. 8 di 30

Wigner picture for Schrödinger-like modelsWigner picture for Schrödinger-like models

x yH, H Hd

idt

1 1 1

1

,n

n n n

x y

x y

Density matrix

Multiband Wigner function

1, / 2 , / 2

2ip

ij ijf x p x m x m e d

W

Evolution equation 1x yH H

dfi fdt

W W -




n. 9 di 30

Wigner picture:Wigner picture: Two band Wigner model

1p p/ 2 / 2ij ij i j ijf V x m V x m f F F

-1p p/ 2ij ijf V x m f F F

*0

*0

2*

0

2

2

4

cccc cc cv

g

vvvv vv cv

g

cvcv c

cc

vv

c c vvg

v v c

f p Pf i f f

t m m E

f p Pf i f f

t m m E

f i Pi p f i f i f f

t m m E




n. 10 di 30

*0

0

2*

0

*

2

2

4

cvg

c

cccc cc cc

vvv

g

cvcv

vv vv vv

cv cv cc vvg

Pf

m E

Pf

m E


t

f pf i f

t m

f pf i f

m m E

t m

Wigner picture:Wigner picture: •Two band Wigner model

,ii iif x v W

intraband dynamic: zero coupling if the external potential is null




n. 11 di 30

*

2*

0

0

0*

2

2

4

cvg

c

cccc cc cc

vvvv vv vv

cvcv cv c

vg

v cc vvg

f pf i f

t m

f pf i f

t m


t

Pf

m E

Pf

m E

m m E

Wigner picture:Wigner picture: Two band Wigner model

, iii if x v W

• intraband dynamic: zero coupling if the external potential is null

• interband dynamic: coupling like G-R via ,cvf x p




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Problems in the practical use of the Kane model:Problems in the practical use of the Kane model:

2 2 21 1 2

120 0

2 2 22 2 1

220 0

2

2

Kane Kane KaneKane

c

Kane Kane KaneKane

v

i V Pt m x m x

i V Pt m x m x

Kane modelKane model

• Strong coupling between envelope function related

to different band index, even if the external field is null

• Critical choice in the cut off for the band index

• Poor physical interpretation 2( ) Kane

ii

n x x




n. 14 di 30

2| ( )

i

n i

R cell

n dx R | n

The quantity represents the mean probability density to find the electron into n-th band, in a lattice cell.

2

i x

MEF model: first order MEF model: first order

Physical meaning of the envelope function:Physical meaning of the envelope function:

2 2 21 1

1 2* 20

2 2 22 2

2 1* 20

P

2

P

2

cc g

vv g

Ui E U

t m x m E x

Ui E U

t m x m E x




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• intraband dynamic

Effective mass dynamics:Effective mass dynamics:

Zero external electric field: exact electron dynamic

2 2 21 1

1 2* 20

2 2 22 2

2 1* 20

P

2

P

2

cc g

vv g

Ui E U

t m x m E x

Ui E U

t m x m E x





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• intraband dynamic

• interband dynamic

first order contribution of transition rate of Fermi Golden rule

Coupling terms:Coupling terms:

( , )T n n k k


2 2 21 1

1 2* 20

2 2 22 2

2 1* 20

P

2

P

2

cc g

vv g

Ui E U

t m x m E x

Ui E U

t m x m E x




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Wigner function:

Phase plane representation: pseudo probability function ,f x p

1, / 2 / 2

2ipf x p x m x m e d

Wigner equation Liouville equation

CLASSICAL LIMIT

0

2,x n x f x p dp

,J x p f x p dpm

Moments of Wigner function:

Wigner picture:Wigner picture:




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MEF model characteristics:MEF model characteristics:

• Direct physical meaning of the envelope function

• Easy approximation (cut off on the index band)

• Highlight the action of the electric field in the interband transition phenomena

• Easy implementation: Wigner and quntum-hydrodynamic formalism

Hierarchy of “kp” multiband effective mass models, where the asimptotic parmeter is the “quasi-momentum” of the electron




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2 2 21

1 1 2* 20

2 2 22

1 2 1* 20

P

2

P

2

cc g

vv g

UE E U

m x m E x

UE E U

m x m E x

First order two band MEF modelFirst order two band MEF model

• The spectrum of the Hamiltonian is not bounded from below

• This prevent us to give an useful “a priori” estimates for the non linear Poisson-Schrödinger problem

Well-posedness of the problemWell-posedness of the problem

2 2

1 ''0

2( ) ( ) ( ) ( )

i

i n i n in n ngR cell

Px dx R R R

m E

2| ( )

i

n i

R cell

n dx R | n n-band Projector

Electron density




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22 21

1 1 2* 20

24 2 22

2 2 14 * 20

( )2

( )2

cc g

vv g

P VE E U

m x m E x

P VE E U

x m x m E x

• We get a bounded spectrum developing the diagonal term of the Hamiltonian to a higher order in “k”

Well-posedness of the problemWell-posedness of the problem




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Boundary conditionBoundary condition

•Outside the domain we will assume that the electron is represented by a Bloch wave

•The region of interest is bounded by two charge reservoirsReservoirReservoir

Bloch waveBloch wave




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Bloch waveBloch waveEnvelope function approximationEnvelope function approximation

Sum of travelling waves


( ) nik xn n

n

x e




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Bloch waveBloch waveEnvelope function approximationEnvelope function approximation

Sum of travelling waves


Continuity of , on the interfaceContinuity of , on the interface( )n x

(0) ' (0)n n

' ( )n x

One dimensional case: Mixed type B.C.

One dimensional case: Mixed type B.C.

Transparent boundary conditionTransparent boundary condition

( ) nik xn n

n

x e




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Existence and uniqueness of the MeF problemExistence and uniqueness of the MeF problem

2 2 21

1 1 2* 20

4 2 2 22

2 2 14 * 20

P( )

2

P( )

2

Transparent Boundary Conditions

cc g

vv g

UE q E U

m x m E x

UE q E U

x m x m E x

2, 1 21 2, ( , ) H Hq U W

• q is the liner momentum of the incident electron and it can vary from 0 to

E(q) is the energy of the incident electron




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n n-th band component

Hd

idt

1,..,t

n

General Schrödinger-like model

matrix of operator

Wigner picture:Wigner picture:

x yH, H Hd

idt

1 1 1

1

,n

n n n

x y

x y

Density matrix




n. 26 di 30

22 21

1 1 2* 20

24 2 22

2 2 14 * 20

2

2

2 2

1 2 1 20

P( )

2

P( )

2

2( )

cc g

vv g

g

UE q E U

m x m E x

UE q E U

x m x m E x

Un

x

Pn f q dq

m E

2

2,

L

W

n

U

Existence of solution of the problem:

Existence and uniqueness Existence and uniqueness Non linear problemNon linear problem




n. 27 di 30

Injective operator

jE E0 0EA

2 .H IE b MEFA E

EA

self-adjoint operator with compact resolvent

There exist a numerable sequence of eigenvaluesjE

2 .H b MeM

Existence and uniqueness Existence and uniqueness Linear problemLinear problem




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Existence and uniqueness Existence and uniqueness

22 21

1 1 2* 20

24 2 22

2 2 14 * 20

P( )

2

P( )

2

Transparent Boundary Conditions

cc g

vv g

UE q E U

m x m E x

UE q E U

x m x m E x

Linear problemLinear problem

2, 11 2, , Hq U W

The linear problem admit a solution

for almost every q

1 21 2, H H




n. 29 di 30

Theorem: the MFM-Poisson system admits a unique solution with ;

Theorem: the MFM-Poisson system admits a unique solution with ; ,n V 2,WV 1H (0,1)n

Asymptotic limit

Modified problem

E (q) E(q) i

0, ,n V n V

Fixed point theoremFixed point theorem

Existence and uniqueness Existence and uniqueness Non linear problemNon linear problem

2,

( ) : |

0,1W

V T V M

V M

Gummel map




n. 30 di 30

2*

21

* 20

( ) 02

L

c

V Em x

Energy estimate

222

2

2

* 222( )

c

V E Cxm

Asymptotic limit Asymptotic limit Non linear problemNon linear problem 0

Single band caseSingle band case




n. 31 di 30

2 2 22

2 2 12Vn C

x x x

2 2

1 2 1 20

2( ) ( ) ( ) ( )i i i i

g

Pn R R R R

m E

22 21

1 1 2* 20

24 2 22

2 2 14 * 20

P( )

2

P( )

2

cc g

vv g

UE q E U

m x m E x

UE q E U

x m x m E x

A priori estimate: Energy

Asymptotic limit Asymptotic limit Non linear problemNon linear problem 0




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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

k0k 0k

where

The Hamiltonian introduced in the Schrödinger equation is2

,2o o per

hH H V H V

m

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

Vai alla 8Vai alla 8




n. 33 di 30

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.

Kane model

Vai alla 9




n. 34 di 30

KANE MODEL

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

v x t( , )be the valence band envelope function.Let c 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 (the matrix element of the

gradient operator between the Bloch functions)

cE ( )vE, ,i iV E V i c v




n. 35 di 30

Remarks on the Kane model

• The external potential V 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).gE

( )c vV V

• The envelope functions are obtained expanding the wave function on the basis of the periodic part of the Bloch functions evaluated at k=0,

( , ) ( , ),ikxn nb x t e u k x

,c v

0 0( ) ( ) ( )c c v vx x u x u 0, ,( ) (0, )c v c vu x u xwhere .




n. 36 di 30

MMEEF MODEL F MODEL (Morandi, Modugno, Phys.Rev.B, 2005)

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 caused by the applied electric field proportional to the x-derivative of V. In the two-band case they assume the form:




n. 37 di 30

*cm• (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

*vm

c vgE

• Expansion of the wave function on the Bloch functions basis

• Introduction in the Schrödinger equation

Which are the steps to attain MEF model formulation?




n. 38 di 30

• 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

( , )nu k x

0k

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

Vai a lla 14

MEF model can be obtained as follows:




n. 39 di 30

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

• projection of the wave function on the Wannier basis which depends on

where are the atomic sites positions, i.e.

Wn

ix R iR

• 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 iR




n. 40 di 30

Comments on the MEF MODEL

• The envelope functions can be interpreted as the effective wave functions of the 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.

c v,

gE

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.( )n iR




n. 41 di 30

Physical meaning of the envelope functions

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

Mc M

v

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

This This simple picture does not apply to the Kane model.

2

0

, , , .( )

K M Mj j h

j h

Pi j h c vm E E

This fact confirms that even in absence of external potential , when no interbandThis fact confirms that even in absence of external potential , when no interbandtransition 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.




n. 42 di 30




n. 43 di 30

Hydrodynamic version of the MEF MODEL

We can derive the hydrodynamic version of the MEF 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 vn

( , )( , ) ( , ) exp c

c c

iS x tx t n x t

( , )( , ) ( , ) exp v

v v

iS x tx t n x t

: icv c v c vn n n e : c cS S

( , ) ( , ) ( , ).ij i jn x t x t x t

Vai alla 21




n. 44 di 30

We introduce the rescaled Planck constant

parameter

2R

R

m l

t

MEF model reads in the rescaled form:

withg

m P VK

mE

,R Rl t

with the dimensional

where are typical dimensional quantities

and the effective mass is assumed to be equal in the two bands

m




n. 45 di 30

Quantum hydrodynamic quantities

• Quantum electron current densities

when i=j , we recover the classical current densities

• Complex velocities given by osmotic and current velocities can be

expressed in terms of plus the phase difference

• Osmotic and current velocities

c c cJ n S

Im( )ij i jJ

v v vJ n S

, , , ,c os c el c v os v el vu u iu u u iu

, ,, , ,i ios i el i i

ii

n Ju u S i c v

nn

, , ,c v c vn n J J




n. 46 di 30

The quantum counterpart of the classical continuity equation

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

Summing the previous equations, we obtain the balance law

c v where, compared to the Kane model, the “interband density” Is missing.

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




n. 47 di 30

Next, we derive a system of coupled equations for phases , obtaining a system equivalent to the coupled Schrödinger equations. Then we obtain a system for the currents and JvThe equations can be put in a more familiar form with the quantum Bohm potentials

,c vS S

cJ

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 .




n. 48 di 30

Recalling that and are given by the hydrodynamic quantities

and , we have the HYDRODYNAMIC SYSTEM for the MEF model

, , ,c v c vn n J Jvu, ,cv cn u




n. 49 di 30

The DRIFT-DIFFUSION scaling

We rewrite the current equations, introducing a relaxation time , in order to simulate all the mechanisms which force the system towards the statistical mechanical equilibrium.

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

, , , ,c c v v

tt J J J J

Finally, after having expressed the osmotic and current velocities, in terms of the other hydrodynamic quantities, as tends to zero, we formally obtain the ZER0-TEMPERATURE QUANTUM DRIFT-DIFFUSION MODEL for the MEF system.




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Hydrodynamic version of the MEF MODEL




n. 51 di 30

NON ZERO TEMPERATURE hydrodynamic model

We consider an electron ensemble which is represented by a mixed quantum mechanical state, to obtain a nonzero temperature model for a Kane system. We rewrite the MEF system for the k-th state, with occupation probability

We use the Madelung-type transform exp / , ,k k ki i in iS i c v

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

We define , , , , , .k k k k k kc v cv c vJ J n u u

k

Performing the analogous procedure and with an appropriate closure, we get




n. 52 di 30

Isothermal QUANTUM DRIFT-DIFFUSION for the MEF MODEL




n. 53 di 30

REMARKS

We derived a set of quantum hydrodynamic equations from the two-band MEF model. This system, which is closed, can be considered as a zero-temperature quantum fluid model.

In addition to other quantities, we have the tensors and

Starting from a mixed-states condition, we derived the corresponding non zero-temperature quantum fluid model, which is not closed.

• Closure of the quantum hydrodynamic system

• Numerical treatment

• Heterogeneous materials

• Generalized MEF model

, ,c v cv vc

NEXT STEPS

similar to the temperature tensor of kinetic theory.




n. 54 di 30

Thanks for your attention !!!!!




n. 55 di 30

Non linear Schrödinger-like Poisson problem Non linear Schrödinger-like Poisson problem

• Description of the model• Description of the model

• Numerical applications• Numerical applications

• Mathematical problem• Mathematical problem

Multiband (MEF) model coupled with the Poisson eqn.

• Well-posedness and B.C.• Existence and uniqueness of the solution for the MEF-P [Ben Abdallah,Morandi]

Application to IRTD

dipartimento di matematica applicata università di firenze multiband transport models for...

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