UNIDIMENSIONAL PARTICULE MODEL OF CONDUCTION IN N TYPE GaAs AT LOW TEMPERATURE : CONTACTS INFLUENCE

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UNIDIMENSIONAL PARTICULE MODEL OF CONDUCTION IN N TYPE GaAs AT LOW TEMPERATURE : CONTACTS INFLUENCE

P. Hesto, J. Pone, R. Castagné

To cite this version:

P. Hesto, J. Pone, R. Castagné. UNIDIMENSIONAL PARTICULE MODEL OF CONDUCTION

IN N TYPE GaAs AT LOW TEMPERATURE : CONTACTS INFLUENCE. Journal de Physique

Colloques, 1981, 42 (C7), pp.C7-177-C7-181. �10.1051/jphyscol:1981720�. �jpa-00221656�

Colloque C7, supplément au n°10 Tome 42, octobre 1981 page C7-177

UNIDIMENSIONAL PARTICULE MODEL OF CONDUCTION IN N TYPE GaAs AT LOW TEMPERATURE : CONTACTS INFLUENCE

P. Hesto, J.F. Pone and R. Castagne

Institut d'Eleatronique Fondamentale, Batirnent 220, Universite Paris XI, 91405 Orsay, France

Résumé.- Nous décrivons un modèle particulaire unidimensionnel utilisé pour modéliser la conduction électronique dans une couche de 4000 A d'AsGa de

type N à 77 K. Nous envisageons différentes conditions d'injection, présen- tons les résultats obtenus et proposons une caractérisation possible du transport balistique.

Abstract.- We discribe a unidimensional particle model of electronic conduc- tion in a 4000 A N type layer of GaAs at 77 K. We study the contact influ- ence, show some results and propose a possible characterisation of the ballistic transport.

At cryogenic temperatures, the conduction electrons can cross a very thin layer of a semiconductor material such as AsGa within a single free flight. This is what is commonly referred to as ballistic effect 11/. This phenomenon is diffi- cult to show off by studying I-V characteristics of any sample, due to its scree- ning by the space charge effect on the potential across the layer I'll. This space charge effect depends on the carrier injection conditions, so that the theoretical studies based upon analytical analysis which do not correctly take into account these injection phenomena are still quite dependent on this ambiguity. In order to help to recognize the ballistic effect in realistic conditions, we simulate the travelling of electrons in a small thickness N type layer, varying the conditions of the electron injection. We describe the electron motion model, different elec- tron injections and show the results.

ELECTRON MOTION : The electron is moving in free flight between two interactions during a time t under the space discretized electric field.

- the electric field E^ is assumed to be constant in the i cell of thickness Ax.

E. is computed by solving Poisson's equation with fixed potential at each extremity.

The electron density is constant in a cell and calculated by a space-time average

during a time step, L

- the free flight time is given by IX (e(t)) dt = - Ln r

•'a

r : randomly generated number A : interaction probability e : total electron kinetic energy

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981720

C7- 178 JOURNAL DE PHYSIQUE

In order to compute tv, we suppose that the electron energy does not change significantly during a flight Ax. between the time t. and tf, tf - ^{t. is the}

smallest value between the transit time in a cell and the time step. So

For this approximation to be valid, it is necessary for the increase of the energy during a flight Ax. to be small, i.e. E. n Axi to be small. The cell width is fixed by this condition.

When an interaction occurs the concerned phonon is randomly determined, and we make a complete treatment of the process. In order to simplify the calculation the energy is limited to 300 meV, that is the valley transfer energy in AsGa, and we assume a non-parabolicity parameter equal to zero. So the applied potential is limited but at higher energy, there is no ballistic transport, due to the great value of the interaction probability.

INJECTION CONDITIONS : We studied the electron motion across a N type layer ,with different injection conditions :

1. A constant electron density at each extremity and an electron injected drift

*

velocity equal to - p(E) * ^E, p(E) being the stationnary mobility given in littera- ture / 3 / . This case is equivalent to a finite thickness slice in an infinite layer.

We did not see any effect due to ballistic transport (electron density and electric field remain constant). This is consistent with the ohmicity of such a device.

2 . A constant electron density at each extremity and an electron injected drift

velocity equal to 0 . This corresponds to injected electrons from a very low drift velocity layer as an N+ type layer. Shur takes very similar conditions 1 2 1 . !Je have a strong space charge effect and a dissymetry at low bias. In the N layer, the zero drift velocity electrons take the place of electrons moving with a drift velocity due to the electric field (fig.1). We have a non linear I-V characteristic due to the space charge effect.

3 . A ND = 5.10'~ ' N type layer at each extremity which corresponds to a

degenerated layer at 77 K. Ve have simulated an ohmic contact at each extremity of the N+ layers and have taken a constant total electron number. This is equivalent to a constant electron number at each end but the computation time is lower. llith these hypothesis, all the electrons flowing out at one extremity are injected at the other one.

The ohmic contact is simulated by a Schottky barrier with a transmission

coefficient D(cX) E~ being the electron kinetic energy along the x-axis. Hence the

electron flow density for E and ex energies from metal to semiconductor is / 4 / .

- f _N -

feld model.

- f

is the electron distribution function in the semiconductor with a Fermi- SC

Dirac statistic.

The electron flow density from semiconductor to metal is :

Then, for an electron having E and

energies, we know the flow probability

X

from metal to semiconductor and from semiconductor to metal. We can randomly com- pute the E and E energies for the injected electrons and we can know if an elec- tron with

and

energies flows out.

We have studied an N.; (5.10 ' I ~ m - ~ , 1000 1) , N ( 10 4cm-3, 4000 1) ~"(5.10' '~rn-~, 1000 1) device at 77 K with this model. At different applied voltages, we obtain the electron density (fig. 2), the potential and the electric field in the device.

We also determine the average energy and the drift velocity along the structure.

We modelled such a structure to show how ballistic effect occurs, to see what interest this effect presents for devices and to help us in experimental characte- risation.

The total energy for a free flight moving electron in the N layer, is constant then

= Cst

P

E is the electron potential energy, equal to the opposite of the potential P

Hence E

V .:. cst

For energies lower than the optical phonon energy the energy variations are equal to the potential ones.

For greater energies,

and V are proportionnal with a coefficient lower than 1 due to the non free flight transport of all conduction electrons (fig. 3). TJe have followed electron flights to confirm this hypothesis (fig. 4). Then we can affirm that conduction electrons have no or few interactions in the N layer. The electron velocity when only a few interactions occur is very large (10 cm/s for V 8

250 mV)

P and the transit time very low ( 5 1 ps). This time is lower than the transit time of thermal electrons across a 4000 layer ( 2 ps for Vd rr 2 . lo7 cm/s ) . This re- sult may help us to imagine very fast devices.

On account of the linearity of the I-V characteristic (fig. 5), the ballistic

effect cannot be characterized experimentally in this way. Other investigations must

be developped, for example an energy spectroscopy of the conduction electrons will

show the part of the free flight electrons in the total current.

C7- 180 JOURNAL DE PHYSIQUE

REFERENCES. -

(1) M.S. Shur, L.F. Eastman, SSE - 24 (1981) p. 1 1 (2) J.R. Barker, D.K. Ferry, H.L. Grubin, IEEE

Vol. EDL 1 (1980), p. 209

(3) E. Constant, Lille, Private Communication (4) R. ~tratton,"~unnelin~ Phenomena in Solid"

p. 105, Plenum.

0 4000

Fig.]: Space charge versus time

0 1000 xcil 5000 6000 Pig.2: Electron density versus bias

1 : V

OV, 2 : V

100 mV,

P P

Fig. 3 : Average total kinetic energy versus potential for an applied voltage V

75 mV

P

Fig. 4 : Observed Kinetic energy and velocity along x-axis for one electron.

- with no interaction

--- with one interaction

-- with two interactions

X represents an interaction