Non-iso thermal analysis of carrier waves in a semiconductor

Citation for published version (APA):

Roer, van de, T. G. (1971). Non-iso thermal analysis of carrier waves in a semiconductor. (EUT report. E, Fac. of Electrical Engineering; Vol. 71-E-21). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1971

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NON-ISO THERMAL ANALYSIS OF

CARRIER WAVES IN A SEMICONDUCTOR

by

Th.G. van de Roer

•

...

DEPARTMENT OF ELECTRICAL ENGINEERING

Non-iso thermal analysis of carrier waves in a semiconductor

by

Th. G. van de Roer

T.H. Report 71-E-21 augustus 1971

Abstract

Propagation of plane longitudinal carrier waves in a homogeneous isotropic extrinsic semiconductor is considered. Three transport equations for density, momentum and temperature are used with simple collision terms describing annihilation of momentum and

transfer of energy from the carriers to the lattice in terms of momentum and energy relaxation times. These time constants are assumed to be temperature dependent to include effects of velocity saturation at high electric fields.

Table of contents

I - Introduction

II - The transport equation III - Isothermal theories

111-1- Zero temperature analysis 111-2- Finite temperature analysis IV - Non-isothermal theory

IV-I- Derivation of the dispersion equation IV-2- Approximate analyses

V - Comparison with isothermal theory VI - Conclusion

I - Introduction

The collective behaviour of charge carriers in semiconductors is receiving increasing attention in recent years, mostly due to the growing importance of avalanche and Gunn-effect devices. Although the carrier gas has a finite temperature which under high electric fields may rise considerably above the lattice temperature it has been customary to neglect variations of this temperature and even the temperature itself [1,2,3]. The reasons for this are partly historical: existing theories of electron beams in vacuo, where the ordered energy is much larger than the random energy, were adapted to semiconductors. In the latter however, the situation is quite different, the random energy typically is much greater than the drift energy and its effects should not be neglected.

It is well documented that under high electric fields the carrier temperature rises considerably above the lattice temperature [4,5] and recent experiments

have shown that energy relaxation plays an important role in high frequency phenomena [5,6].

Therefore it seems worthwhile to make a systematic study of carrier wave

propagation under non-isothermal conditions. A number of authors have already made non-isothermal calculations of carrier waves. Most of them use the

higher moments of the Boltzmann-equation, which provide transport equations for density, average momentum and temperature of the carriers [7,8,9,10]. Also it has been attempted to integrate the Boltzmann-equation directly [11).

In this report the transport equation approach is followed. Carrier waves in the regions of 'high d.c.fields are studied with special attention paid to the non-linear effects that give rise to current saturation.

II - The transport equations

The transport properties of a gas consisting of one species of charged particles can be described in terms of the density distribution function feE, ~, t) which gives the particle density in phase space as a function of the space coordinates E, the velocity coordinates wand time t. This

functions obeys the Boltmann-equation:

af + 'V f at r

F

+="1 f

m w (1)

Here F is the external force exerted upon a particle, in the present case electric and magnetic forces. m is the effective mass which in the following will be assumed constant. The limits of validity of this

equation are discussed in [12].

Density n, average or drift velocity ~ and temperature T are defined as integrals over velocity space:

neE. t)

=Jf

J

f d3w (2) ~(E' t) = -1 _{fJf~ f} d3w (3) n

lk

2

T(E,

t) =

J J J

t

m w 2 f d3w

(4)

By mUltiplying (1) succeaavely by I,

~

and

i

m w2 and integrating over velocity space we obtain after a little manipulating transport equations for

::e above qua(n::t)ies:

- + 'V.n v = -at - dt . c av - + (v. "I)v - .9. (E + ~ x

l!.)

+ at - - m -k

(dV)

"InT= -m n dt c (5) (6) (7)

Here k is Boltzmann's constant, q is the particle charge and ~ and

l!.

are the electric and magnetic fields. The derivation of these equations is given in another report and can also be found in the literature [7,13].

The right hand sides of eqs. (5), (6) and (7) represent symbolically the rate of change of the quantities by collisions. To be able to make calculations we have to choose some appropriate function for these terms. Let us take the following expressions:

(~~)

= 0 c

(~:)

= -

:p

c (8) (9)

t

k

(:n

2 m v = -T P 3 _T-TL k -2 T (10) e c

These expressions have been derived in the literature by making certain assumptions about the distribution function [7,11]. They also have a clear physical interpretation:

(5) means that generation and recombination of carriers is considered unimportant because we are looking for phenomena on a time scale that is short compared to the carrier lifetime and at fields where carrier

multiplication is still small.

(6) can be interpreted such that a velocity disturbance after leaving the system to itself would decay to zero with a time constant T • In

p

general it says that the directed momentum is destroyed at a rate ~. Tp means that This directed momentum is converted to random momentum which

2

h ' f d d h' ' h ' f h mv

t ere 1S a source 0 ran om energy an t 1S 1S t e mean1ng 0 t e term

-Tp in (7). The last term of (7) finally represents transfer of thermal

energy to the crystal lattice, so the carrier temperature relaxes to the lattice temperature and not to zero.

To include current saturation T and T are assumed to be functions of T.

p e

I t would be more correct to make them functions of the total energy

0

m v 2 +

2'

3 k T) but as the thermal energy usually is much larger than the drift energy only a small error will be introduced.

Before now proceeding with calculations based on the complete set (5) •• (7) we first give a short account of isothermal theories .

III - Isothermal theories

111-1- Zero-temperature analysis

a

In the steady state (at - V - 0) (6) + (9) reduce to:

v

q T

...:....J.

E = ll!

m (1 I)

So the concept of mobility follows naturally from the concept of a momentum relaxation time. The simplest assumption we can,make now is that (II) also holds for a.c. phenomena. Non-linearity can be included by assuming \l to be

a function of E. For small variations vI' EI superimposed on large d.c. quantities v_o' Eo' all of them in the z-direction, we then have:

Consider now plane waves in the z-direction with (t.z)-dependence exp j(wt - ez). Then the continuity equation (5) gives for the a.c. quantities:

This combined with (12) gives:

On the other hand Poisson's equation demands that:

The system (14)-(15) allows two solutions:

a. e=O nl=O which describes COmmon a.c. conduction where the current density J

I is given by J_{I = q novl}

=

0

1 EI where a,

=

noll, is the a.c. conductivity.

( 12)

(13)

(14)

b. /3 = w v

o

This describes a wave that is damped by dielectric relaxation. Note

( 16)

that in complete velocity saturation

(~

=

0) it becomes undamped and in a negative differential conductance

medi~m

(ddVO < 0) it will be a growing

Eo

wave. The phase velocity of this wave is equal to the drift velocity.

111-2. Finite-temperature analysis

As a next step consider the carrier temperature to be finite but constant. A density gradient will then set up a pressure gradient which turn will give rise to a diffusion force. We can describe this phenomenologically by:

D anI vI

=

~I EI -

n-

a;-o

Comparison with (6) shows that the diffusion constant D is equal to which is known as Einstein's relation.

Assuming plane waves as before we nOw have

n = ---,S,--I w-Sv _o ( 17) T kT ....l'-.- = ~.!Q: m q (18)

Combined with (15) this again allows the sol~tion S of (16) we now have a second degree equation:

n

l = 0, but instead

"1 + j(w-Sv ) + - - = 0

o E

For small D approximate "I

solutions are:

( 19)

(20)

(21 ) There now is an extra wave travelling against the drift and heavily damped. As it is obviously connected with the introduction of diffusion we will call it a diffusion wave.

As a further refinement we may now include the inertia terms from (6) in (17) so that we have: . D J S - n n 1 o

The dispersion equation (19) then changes into: 2

D S2 - T (,.,-sv) + j (w-av )

p 0 0

It will be helpfull to define a parameter A by: k T A = - - 2 mv o D = - - 2 T v p 0 (22) (23) (24)

So A is a measure for the ratio between thermal energy and drift energy. For low frequencies such that A W T «I approximate solutions are:

p -Q fA

a

l ,2 = (A-I)Q where ~+ V o . I+Q J 2().-1)T v p 0

On the other hand, for high frequencies where WT

p we find: I±

.n:

(

~

_v

_

L-)

T V o P 0 (25) (26) 2 W £T > > 1 and p > > 1 °1 (27)

In semiconductors A usually is greater than one and in most cases even much

greater. Then Q is greater than one and it can be seen that the phase velocity of the wave traveling with the drift (the ordinary carrier wave) has increased above the drift velocity for low as well as high frequencies. This is in contradiction with the assumption commonly made that for low

frequencies (WT < < I) the inertia terms may be neglected. One then overlooks p

the fact however, that the imaginary part of

a

can be large and is in first approximation independent of frequency so that ImSv T is in the order of one·

IV - Non-isothermal analysis

IV-I. Derivation of the dispersion equation

In the preceding chapter we have found better approximations and even a new wave by including more terms from the transport equations. We now

take the last step and use the complete set. The following assumptions will be made:

There is no static magnetic field. Plane waves will be considered with all vector quantities directed along the z-axis. All quantities consist of a large d.c. part denoted by index 0 and a small a.c. part with index

It e.g.:

n

=

no + Re {n

l exp j(wt - az)} (28)

For the current density J we then have:

(29)

The relaxation times are a function of carrier temperature. Now define:

T dT

=-2....~

Yp TpO dT_o

then:

For the d.c. quantities we find from (6), (7), (9) and (10): v o T po qT po

=

_m E 0 (30) (31 ) (32) (33) (34) (35)

It will be interesting to calculate from this the slope of the static v-E-characteristic: dv _o av ₀ av IdE _{0 0} - - = - - + - - - -_{dE aE aT dT} o 0 0 0 (36)

which gives the

dvo

=

qT po

dE m (37)

o

In general the relaxation times decrease with increasing T so that Y e and yare negative [3) and the denominator is positive. Let us now define

p

a slope parameter g by: g = I + T -T o L T (Yp-Y_{e )} o (38)

It will turn out that g is an important parameter in determining the behaviour of carrier waves.

The first order a.c. parts of the transport equations read:

2mv 0 = _{- - v - Yp} T I po mv vI - - + 2 T po v o Y P -T --':;;T'- T I po 0 1- To-TL 3 To 'Ye ~T

_-2'

T I T po e (39) (40) kTI (41)

To complete the set we use the condition that the total a.c. current must vanish:

(42)

which can be derived either from Maxwell's equations or from Poisson's equation together with the continuity equation.

At this stage it is convenient to introduce reduced quantities by deviding the a.c. quantities by their d.c. counterparts, e.g.:

Besides the parameters A and g already defined in (24) and (38) we introduce the plasmafrequency

2 2 w p q no C1 0 = = -8m 8T po as a new parameter: (43)

With the help of (29) and (42) vI and EI are eliminated so that a set

'"

'"

of three linear homogeneous equations in ~, J and T is obtained (the zero subscripts on TpO and Teo are omitted from now on):

'"

wn - Sv

'"

J = 0 o 2 w . (w-Sv o _ ....£. _ L ) w T

'"

_{J - {w +} P

l(w-sv -

₂ 0 - (Sv o (A-I) Sv -

L}

o T P

'"

J + (Sv o

'"

n - (ASV -o (44) (45) (46)

This set of equations can be written in the form of an eigenvalue problem:

(A - Sv I) k = 0

o

The columnvector k is defined as

'"

'"

'"

k = (n, J, T)

(47)

(48)

I is the unity matrix and A is a matrix given by: 2 7' w 7' _{3' (~ + Yp)} (ZA-6)w + .!l. 3w-}-£. - ..1. -3Aw + T w T J T T P P e P I A= 5A-3 (5A-3)w 0 0 .2A-4 2w2 .2A-4 (3A-3)w _ j

~(3~:3)g

_

~:p)

-(ZA+2)w - _J~ 2w- --E. + J--rr w p p (49)

The dispersion equation f(a,w)

=

0 is obtained by equating the determinant of (47) to zero which gives an equation of third degree in Sv :

o with: a = - 5A

+

3

(3~:3)g)

b (51--9)w + _J

.(7

+

_T 2Ye P 2 3w2

+~-

..1.L

_{- jw}

C

O_T;2Ye

+~)

c = 9w -p _AT2 T T P P e 3 2 d = - 3w + 3ww P +

~(3g

_ 4Y

e)

+ T T AT p e p . 2 JW JW • 2 P 3g T e (50) (51) (52 ) (53) (54)

For the eigenvector ~ belonging to an eigenvalue SVo we find after some calculations:

k.=~.V

, W,

-1 ,\1 0

(5A-3) (S.v )2 _{ (ZA-6)w

+

,p.}

a.v _ 3w2

+

3w 2

+

7j

W)

______ ~1~0~ ____________ _Lp_,~'~0~~---p~---TLe- (55)

(

~Te

+

"L

_TP e) -3AW + 3j

IV-2 Approximate analyses

In this part we will try to find approximate solutions in a few parameter ranges. Specifically we will consider the limit of very high frequencies

(~) and of very low frequencies (w+o) and finally the case the d.c. drift velocity saturates completely.

It must be noted here that the high and low frequency approximations are of little practical value because the conditions under which eqs (5) .• (10) are valid break down, at high frequencies because the wavelength becomes so small that there are too few particles in a elementary volume that is small compared to the wavelength so statistical laws do not apply any more, and at low frequencies because then the time scale is no longer short

compared to the carrier lifetime. However they give an insight in the behaviour of the solutions at intermediate frequencies.

IV-2.1 High frequency limit

F or f requences so h · h h l.g t at W T 'T 2 > > 1 and ",2 ... »

p e

2

"'p we can develop the solutions of (50) in negative powers of '" which gives for the first

two terms: 6₁ Vo ~

'"

-

Js:r

. 3g (56) e

[,

62 •3vo 1 - j --L

L

_{l ' ,."]}

(57)

~

5T T 2

VISA

± e p

Apparently there are two waves travelling with the drift and one against the drift. The wave described by (56) has a phase velocity equal to the drift velocity and a damping proportional to the slope of the static v-E characteristic. The other two are fast waves with a damping that is always fini te.

IV-2.2 Low frequency limit

Let us start with the limiting case '" ~ O. Then the coefficients a and c of (50) become real whereas band d are purely imaginary. It will then be more convenient to consider the quantity r ~ - j6 for which the equation

becomes: 3 p(rv ) o with p ~ - SA 2 + q (rv ) o + 3 + r(rv ) + s ~ 0 o q ~ 7+ 2Y_E - (3A-3)1.\ Tp Te 2 4Y_p

..1L

r ~ 3", - - + P )..T2 T T .. p e 3", 2 P s = P 1.\ T e (58) (59) (60) (61 ) (62)

It is a well known theorem of elementary algebra that eq. (58) has either three real roots or one real root and two conjugate complex ones.

The latter case is curious because it means that there can be

disturbances that even at zero frequency have a wave-like character. For g = 0 we find one root equal to zero which means there is an undamped wave as in the high frequency limit.

11-2.3 Full saturation

In the foregoing it has been shown that in the case of full velocity saturation (g=O) there exists an undamped wave at very high as well

as very low frequencies. This raises the question if such a wave exists for all frequencies. That this is indeed so can be most easily shown by writing (50) in a different way:

2· _{- jyp} 2A(w-8v)(8v - ~)(8v 2 o 0 AT 0 AT W + jg]T e) p p 3(wflv -0 = (w-8v)(w-8v - j/T ) - A 82v2 o 0 P 0

Note that when either T or A is made zero this equation reduces to

e

familiar equations of isothermal theory [I,eq 4 - 36].

Now from this representation it is immediately clear that for g = 0 there is a solution 8v = w for all frequencies.

o

v - Comparison of isothermal and non-isothermal theories

We may now ask ourselves if there is a correlation between isothermal and non-isothermal analysis in the sense that one or more of the waves found in the non-isothermal analysis can be identified with waves found in the isothermal approximation.

For a start we note that using non-isothermal theory three waves are found, viz. a slow and a fast wave in the direction of the drift and a fast wave in the opposite direction:- At first sight one would say tliat the slow wave corresponds to the slow waves depicted by eqs. (16) and (20), especially as they all have a damping proportional to the slope of the v-E characteristic. However, some objections can be made against this.

First, the form of the damping terms suggests different attenuation

mechanisms in the isothermal and the non-isothermal case. In the isothermal case the damping seems to be due to conversion of drift energy into Joule

losses. In the non-isothermal theory this process is split into two parts

and only the second part, viz. transfer of heat to the lattice seems to be the cause of the damping.

Second, let T approach zero in eq. (63), then it reduces, as already said, e

to an equation of isothermal theory. However, in doing so we have excluded the solution for which ~v ~ w - jg/T and this is precisely the slow wave.

o e

Third, it appeared from eqs. (25) ••• (27) that adding inertia terms in the isothermal approximation changed the slow waves to fast waves. It is illustrative in this connection to compare the high frequency asymptotes given by (27) with those of (57).

Finally, consider the eigenvector given by (55) for the slow non-isothermal wave in the case g

=

O. Then it assumes a relatively simple form:

(

3Aw2 + 3w2 )

~(g=O, ~v _o =w) = w, w, _-31. _w+ 3' _JY

PI

p TP

For low frequencies the density and current modulation become small and we have a wave that consists mainly of a temperature modulation. It is evident that such a wave could not be found with an isothermal analysis. All this supports the conclusion that the fast forward and backward wave correspond to the ordinary carrier wave and the diffusion wave found by isothermal analysis, whereas the slow wave is a new wave owing its

existence to variations of the carrier temperature. We may therefore call it a temperature wave.

VI Conclusion

Isothermal theory of carrier waves in semiconductors reveals two waves which have been called the ordinary wave and the diffusion wave. In the non-isothermal analysis, which includes variations of the carrier

temperature, a third wave is found which travels with the drift and has a phase velocity approximately equal to the drift velocity. This wave seems to be essentially a temperature wave. It has the property that its attenuation is proportional to the slope of the static v-E-characteristic.

VII - References

1. M.C. Steele and B.Vural - Wave Interactions in Solid State Plasmas. Mc Graw-Hill 1969.

2. G.S. Kino - Carrier Waves in Semiconductors, IEEE Trans. ED-17, no.3, 178-192 (1970).

3. T. Wessel - Berg - Electromagnetic Properties of Drifted Semiconductor Plasmas, Techn. Report no. AE-2, Norwegian Institute of Technology, Trondheim,

Norway (1966).

4. E.M. Conwell - High Field Transport in Semiconductors, Academic Press 1967. 5. A.F. Gibson, J.W. Granville and E.G.S. Paige- A Study of Energy Loss

Processes in Germanium, J. Phys. Chem. Solids, ~, no 3/4, 198-217 (1961). 6. M.T. Vlaardingerbroek, P.M. Boers and G.A. Acket - High frequency

Conductivity and Energy Relaxation of Hot Electrons in Ga As -Philips Res. Reports

3i,

379-391 (1969).

7. J.E. Carroll- Non-isothermal Waves on Charge Carrier Streams, IEEE Trans. ED~, no.1, 187-189 (1966).

8. K. Blotekjaer - High-frequency Conductivity etc. in Drifted Semiconductor Plasmas - Ericsson Technics no.2, (1966), 125 -183).

9. D.C. Hanson and J.E. Rowe - Non-Isothermal Effects in Bulk Ga As Transit-Time-Mode Oscillators- Int. J. Electronics 24, no.5, 415-427 (1968).

10. K. Blotekjaer and P. Weissglas Diffusion Instabilities in Semiconductors -J.A.P. ~, 1645 (1968).

11. A.K. Jonscher and K.S. Rodger Space Charge Waves in Solids -J. Phys. C. (Solid State Physics) ~, 194-200 (1971).

12. R. Jancel and T. Kahan - Electrody~ics of Plasmas - Wiley 1966.

DEPARTMENT OF ELECTRICAL ENGINEERING Reports:

1) Dijk, J o , Mo Jcukpn and EoJo ManIlllere

AN ANTENNA FOR A SATELLITE COMMUNICATI6N GROUND STATION

(PilO VISIONAL ELEC1'RICAL Dl!:SIGII). TH-report 68-E-01. March 1968. ISBN 90 6144 001

7

2) Veefkind, A., J.H. Blom and L.H.Th. Rietjens

THEORETICAL AND EXPERIMENTAL INVESTIGATION OF A NON-EQUILIBRIUM PLASMA IN A MHD CHANNEL. TH-report 68-E-2. March 1968. Submitted

to the Symposium on a Hagnetohydrodynamic Electrical Power Generation,. Warsaw, Poland, 24-30 July, 1968. ISBN 90 6144 002 5

3) Boom, A.J.W. van den and J.H.A.M. Melis

A COHPARISON OF SOME PROCESS PARAHETBR ESTIMATING SCHEMES. TH-report 68-E-03. September 1968. ISBN 90 6144 003 3 4) Eykhoff, p., P.J.M. Ophey, J. Severs and J.OoM. Oome

AN ELECTROLYTIC TANK FOR INSTRUCTIONAL PURPOSES REPRESENTING THE COMPLE;X-FREQUENCY PLANE. TH-report 68-E-04. September 1968. ISBN 90 6144 004 1

5) Vermij, Lo and JoE. Daalder

ENERGY BALANCE OF FUSING SILVER WIRES SURROUNDED BY AIR. TH-report 68-E-05. November 1968. ISBN 90 6144 005 X 6) Houben, J .VI.M.A. . .. and P. Massee

MHD POVIER CONVERSION EMPLOYING LHIUID METALS. TH-report 69-E-06. February 1969. ISBN 90 6144 006 8

7)

Hauvel, W.MoC. van den and W.FoJ. Kersten

VOLTAGE; MEASUREMENT IN CURRENT ZERO INVESTIGATIONS. TH-report 69-E-07. September 1969. ISBN 90 6144 007

6

8) Vermij, L.

SELECTED BIBLIOGRAPHY OF FUSES. TH-report 69-E-08. September 1969. ISBN 90 6144 008 4

9) Westenbers, J.Z.

SOME IDENTIFICATION SCHEMES FOR NON-LINEAR NOISY PROCESSES. TH-Report 69-E-09. December 1969. ISBN 90 6144 009 2

10) Koop, H.E.M o, J. Dijk and ~.J. Naanders

ON CONICAL HORN AlITENNAS. TH-report 70-E-10. February 1970. ISBN 90 6144 010 6 11) Veefkind, A.

NON-EQUILIBRIUM PHENOMENA IN A DISC-SHAPED GE;NE;RATOR. TH-report 70-E-11. Harch 1970.

HAGNETOHYDRODYNAHIC ISBN 90 6144 011 4 12) Jansen, J.K.M.! M.E.J. Jeuken and CoVi. Lambrechtse

THE SCALAR FEED. TH-report 70-E-12. December 1969. ISBN 90 6144 012 2 13) Teuling, D.J.A.

ELECTRO:HC IMAGE HOTION COMPENSATION IN A PORTABLE TELEVlSION CAME;RA. TH-report 70-E-13. 1970. ISBN 90 6144 013 0

14) Lorencin, ~I.

AUTOMATIC HETEOR REFLE:CTIONS RECOIlDING EQUIPNENT. TH-report 70-E-14. November 1970. ISBN 90 6144 014 9

15)SmetB, A.J.

THE INSTRUMENTAL VARIABLE HETHOD AND RELATED IDEtlTH'ICATION SCHEHES. TH-report 70-E-15. November 1970. ISBN 90 6144 015

7

•

Reportsl

16) White, Jr., R.C.

A SURVEY OF RANDOM METHODS FOR PARAMETER OPTIMIZATION.

TH-report 70-E-16. February 1971. ISBN 90 6144 016 5

17) Talmon, J.L.

APPROXIMATED GAUSS-MARKOV ESTIMATORS AND RELATED SCHEMES.

TH-report 71-E-17.

XaxpzaB&¥

ISBN 90 6144 017 3

February 1971.

18)

Kal£~ek,

V.

MEASUREMENT OF TIME CONSTANTS ON CASCADE D.C. ARC IN NITROGEN.

TH-report 71-E-18. February 1971. ISBN 90 6144 018 1

19) Hosselet,

L.M.L.F~

OZONBILDUNG MITTELS ELEKTRISCHER ENTLADUNGEN. TH-report 71-E-19. March 1971.

XK~~~K*~aK¥

ISBN 90 6144 019 X

20) Arts, M.G.J.

ON THE INSTANTANEOUS MEASUREMENT OF BLOODFLOW BY ULTRASONIC MEANS.