TWO-BAND DYNAMICS IN SEMICONDUCTOR DEVICES: PHYSICAL AND NUMERICAL VALIDATION

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Two-band dynamics in semiconductor devices: physical and numerical validation n. 1 di 30

WASCOM05 WASCOM05

XIII Int. Conf. on Waves and Stability in Continuous Media XIII Int. Conf. on Waves and Stability in Continuous Media

Acireale - Santa Tecla, June

Acireale - Santa Tecla, June 19-219-255, 2005, 2005

Giovanni Frosali

Dipartimento di Matematica Applicata “G.Sansone”

giovanni.frosali@unifi.it

TWO-BAND DYNAMICS

IN SEMICONDUCTOR DEVICES:

PHYSICAL AND NUMERICAL VALIDATION

Giovanni Borgioli

Dipartimento di Elettronica e Telecomunicazioni

borgioli@ingfi1.ing.unifi.it

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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 assumed. 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 of 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

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The spectrum of the Hamiltonian of a quantum particle moving in a periodic potential is a continuous spectrum which can be decomposed into inter- vals 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RITD Band Diagram The single-band approximation is no longer valid when the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.

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We quote here only our different approaches to the problem:

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

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

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 (Liu, Ting, McGill, Chao, Chuang, etc.)

• tight-binding (Boykin, van der Wagt, Harris, Bowen, Frensley, etc.)

• pseudopotential methods (Bachelet, Hamann, Schluter, etc.)

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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

k 0

k  0

k  ^where

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_per

Vai alla 8 Vai alla 8

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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.

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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)

Ec ^{( )}^E_v ^Vⁱ ^{ }^{E V}ⁱ ^, ^{i c v}^ ^,

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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 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,

( , ) ^ikx ( , ),

n n

b x t  e u k x

,

c v

0 0

( )x _c( )x u_c _v( )x u_v

  

0

, ( ) , (0, )

c v c v

u x  u x

where .

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MMEF MODEL EF 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:

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*

mc

• (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

*

mv

c _v Eg

• Expansion of the wave function on the Bloch functions basis

• Introduction in the Schrödinger equation

Which are the steps to attain MEF model formulation?

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• 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 xn

0 k 

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

Vai

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MEF model can be obtained as follows:

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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.

W

n

x R i R_i

• 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( )Ri



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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}_,

Eg

standard properties of the Fourier transform, equations for the coefficients are achieved._n( )R_i

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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..

M

c _v^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

    





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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 barrierpotential barrier and which and which realizes a single

realizes a single quantum wellquantum well in in valence band.

valence band.

NUMERICAL SIMULATION

See: G.Alì, G.F., O.Morandi, SCEE2005 Proceedings

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The incident (from the left) conduction electron beam is

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

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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 wellby the well

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When the electron energy approaches the resonant level, the electron

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

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Hydrodynamic version of the MEF 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 c

iS x t x t n x t

 

 

  

 

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

v v

iS x t x t n x t

 

 

  

: ⁱ

cv c v c v

n    n n e ^  : S^c S^c



 

( , ) ( , ) ( , ).

ij i j

n x t  x t  x t

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We introduce the rescaled Planck constant parameter

 ^{ }

2 R R

m l

  t^

MEF model reads in the rescaled form:

R, 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^

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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 , , 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

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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, on the contrary of the Kane model, the “interband density”

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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_v

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

c, v

S S

Jc

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 .

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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 uv

cv, ,v

 n u

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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, ,

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.



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Hydrodynamic version of the MEF MODEL

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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 MEF system for the k-th state

We use the Madelung-type transform ^ⁱ^k ^ ⁿⁱ^k ^exp



^iSⁱ^k ^/^



^, ^{i c v}^ ^,

We define the densities and the currents corresponding to the two mixed states We define J J_c^k, ,_v^k  ^k,n u u_cv^k , , ._c^k _v^k

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obtaining

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QUANTUM DRIFT-DIFFUSION for the MEF MODEL

withwith

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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

c, ,v cv

  

vc

NEXT STEPS

similar to the temperature tensor of kinetic theory.

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TWO-BAND DYNAMICS IN SEMICONDUCTOR DEVICES: PHYSICAL AND NUMERICAL VALIDATION

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Thanks for your attention !!!!!