MULTIBAND TRANSPORT MODELS FOR SEMICONDUCTOR DEVICES

Giornata di lavoro Giornata di lavoro

Mathematical modeling and numerical analysis of quantum Mathematical modeling and numerical analysis of quantum

systems with applications to nanosciences systems with applications to nanosciences

Firenze, 16 dicembre 2005 Firenze, 16 dicembre 2005

Giovanni Frosali

Dipartimento di Matematica Applicata “G.Sansone”

giovanni.frosali@unifi.it

MULTIBAND TRANSPORT MODELS

FOR SEMICONDUCTOR DEVICES

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)

• 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

Energy (ev)

60 50 40 30 20 10

0 Position (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 APPROXIMATION

TWO-BAND APPROXIMATION

• 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,…

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OUR APPROACH TO THE PROBLEM

M U L T IB A N D T R A N S P O R T M U L T IB A N D T R A N S P O R T

General Multiband Models

General Multiband Models

KANE modelKANE model

MeF modelMeF model

WIGNER APPROACH WIGNER

APPROACH

SCHRÖDINGER APPROACH SCHRÖDINGER

APPROACH

HYDRODYNAMIC MODELS

HYDRODYNAMIC MODELS

QUANTUM DRIFT-DIFFUSION

MODELS QUANTUM DRIFT-DIFFUSION

MODELS

Isothermal IsothermalQDD

QDD

Chapman- Enskog

expansion

Chapman- Enskog

expansion

H   E 

n

( ) x



2 2

0

( ) ( )

2

H V x U x

   m    We filter the solution

1

( ) x



Envelope function models Envelope function models

Multiband

“KP”

system

2

( ) x ( ) x 



S chrödinger equation Hamiltonian

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..







| ( )

i

n i

R cell

n dx  R

  



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

 

i

x



MEF model: first order MEF model: first order

 

 

2 2 2

1 1

1 2

* 2

0

2 2 2

2 2

2 1

* 2

0

P 2

P 2

c

c g

v

v g

i E U U

t m x m E x

i E U U

t m x m E x

   

   

        

   

 

  

     

   



 



 



Physical meaning of the envelope function:

• intraband dynamic Zero external electric field: exact electron dynamic

MEF model: first order MEF model: first order

 

 

2 2 2

1 1

1 2

* 2

0

2 2 2

2 2

2 1

* 2

0

P 2

P 2

c

c g

v

v g

i E U U

t m x m E x

i E U U

t m x m E x

   

   

        

   

 

  

     

   



 



 



Effective mass dynamics:

• intraband dynamic

• interband dynamic

first order contribution of transition rate of Fermi Golden rule

Coupling terms:

( , )

T n  n k   k  MEF model: first order

MEF model: first order

 

 

2 2 2

1 1

1 2

* 2

0

2 2 2

2 2

2 1

* 2

0

P 2

P 2

c

c g

v

v g

i E U U

t m x m E x

i E U U

t m x m E x

   

   

        

   

 

  

     

   



 



 



Wigner function:

Phase plane representation: pseudo probability function ^{f x p}  ^, 

 ^,  ¹  ^{/ 2}   ^{/ 2} 

2

f x p x m x m e d



    

    ^  ^

Wigner equation Liouville equation

CLASSICAL LIMIT

 0



  ^x

^{n x}   ^{f x p dp}  ^, 

   

  ^{J x}   ^{p f x p dp}  ^, 

m ^       

Moments of Wigner function:

Wigner picture:

Wigner picture:

 ^H,   ^H

^H



i d

dt

     

 

   



1

,

n

n n n

x y

x y

   



   

 

 

  

 

 



  

 Density matrix

Multiband Wigner function

 ^,    ¹  ^{/ 2 , / 2} 

2

ip

ij ij

f x p x m x m e d



    

 ^W     ^  ^

Evolution equation ^{i df} _dt ^{ W}  ^H

^{ H}

 ^W

^f

WIGNER APPROACH WIGNER APPROACH

Wigner picture for Schrödinger-like models

Wigner picture:

Wigner picture: Two-band Wigner model

  ^{   }

 

p



f    V x

   / 2 m  V x

   / 2 m  

f

F F

  ^{ }

 

p



f

    V x    / 2 m  

f

F F

 

 

*

0

*

0

2

*

0

2 2

4

cc cc cc cv

g

vv vv vv cv

g

cv cv c

cc

vv

c c vv

g

v v c

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 p f i f i f f

t m m E

 

 

  





 

      

  

       

  

  

                







 





pseudo-differential operators:

  ^,  

ii ii

f x v  W 

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

Wigner picture:

Wigner picture: Two-band Wigner model

 

 

*

0

0

2

*

0

*

2 2

4

cv g

c

cc cc cc cc

vv v

g

cv cv

vv vv vv

cv cc vv

cv

g

P f

m E

P f

m E

f i P

i p f i f i f f

t

f p

f i f

t m

f p

f i f

m m E

t m











 





 

  

 

 

 

 

   

                



  



    









 



  ^,  

ii ii

f x v  W 

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

• interband dynamic: coupling like G-R via f

 x p ^,  Wigner picture:

Wigner picture: Two-band Wigner model

 

 

*

2

*

0 0

0

*

2 2

4

cv g

c

cc cc cc

vv vv vv

cv cv c

cc

vv

cv

v

v cc vv

g

g

f p

f i f

t m

f p

f i f

t m

f i P

i p f i f i f f

t

P f

m E

P f

m E

m m E





  









 

    

  

     

  

  

                

 



 



 







Mathematical setting Mathematical setting

 

 





1

: ; 1

f f L p dx dp

X    

1 1 1

X X

H    X

Hilbert space:

Weighted spaces:

1 D problem:

If the external potential

the two-band Wigner system admits a unique solution f  H

 

(0)

i d A B C dt

   

 

 



f f f

f f

 f

^, f

^, f





f

0

 D A    H f

2,

( )

ext x

U  W



2

2

* * * 2

, , 1

4

p p

diag i i p

m x m x x

A m

    

         



 e iAt

unbounded operator unitary group on H

Stone theorem

2

1 2 3

2 1

f :

) ,

( f , f f

H X

x x

A x

D     

         

Mathematical setting Mathematical setting



^,

^,



d

B  iag   

   

   

   

0

0 0 2

0 0 2

g

0 P

m E i i





 





 

  

 

 

  

 

  

 



 

 

  ^{   }

 

p



f  ^  V x

   / 2 m  V x

   / 2 m ^ 

f

F F

  ^{ }

 

p



f

    U x    / 2 m  

f

F F

C

Symmetric bounded operators

 

, H

B C  B

1, ( _x)

ij ij

f

c U

f

 



2, ( _x)

ij X ext W ij X

f c U f









Mathematical setting Mathematical setting

If the external potential U

 W

( 

) a unique solution

the two band Wigner system admits

 H f

 

(0)

i d A B C dt

   

 

 



f f f

f f

 f

^, f

^, f





f

The operator generates semigroup A B C  

The unique solution is given by f  e



 f

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 v

n      

( , ) ( , ) ( , ) exp

c c

iS x t x t n x t

 

 

  

 

( , ) ( , ) ( , ) exp ^v

v v

iS x t x t n x t

 

 

  

:

cv c v c v

n     n n e

 : S

S



 

( , ) ( , ) ( , ).

ij i j

n x t   x t  x t

We introduce the rescaled Planck constant parameter

 ^{ } 

2 R R

m l

  t

MEF model reads in the rescaled form:

with

g

m P V

K mE







R

,

l t

with the dimensional where are typical dimensional quantities

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

m

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 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 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.

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

J

The equations can be put in a more familiar form with the quantum Bohm potentials

c

,

S S

J

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 .



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

, , ,

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 , 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

,

,

, ,

t t 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.



Hydrodynamic version of the MEF MODEL

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

^

^{ 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

, , .



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

Isothermal QUANTUM DRIFT-DIFFUSION for the MEF

MODEL

Thanks for your attention !!!!!

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

, ,

  



NEXT STEPS

similar to the temperature tensor of kinetic theory.

Problems in the practical use of the Kane model:

2 2 2

1 1 2

2 1

0 0

2 2 2

2 2 1

2 2

0 0

2

2

Kane Kane Kane

Kane c

Kane Kane Kane

Kane v

i V P

t m x m x

i V P

t m x m x

   

   

       

   

 

  

    

   



 



 



Kane model Kane 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 ^{( )}

 

i

n x    x

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

k 0

k  0

k 

The 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 V an external potential.

V

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

KANE MODEL

The Kane model consists into 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

• 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)

E

^{( ) E}

^V

^{  E V}

^{, i c v  ,}

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).

E

c

( )

V 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,

( , )

( , ),

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 .