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Carrier temperature effects and energy transfers in non-radiative recombination
Solid-Sfmr Elccrmnrcs.1976.Vol. 19. pp. 2-S-233. PergamonPress Prinwd m Gm1 Britain
CARRIER TEMPERATURE EFFECTS AND ENERGY TRANSFERS IN NON-RADIATIVE RECOMBINATION J. E. PARROTT Department of Applied Physics and Electronics, UWIST, King Edward VII Avenue, Cardiff. CFl 3NU, Wales (Received 12May 1975:in revised form 13 August 1975)
Abstract-The effects of carrier temperature changes on Hall-Shockley-Read and Auger recombination are calculated and it is shown that these are small in the former type of recombination but may be very large in the latter. Energy transfer calculations show that for Hall-Shockley-Read recombination the electron system loses energy kTo for every recombination event, but in the Auger event where there are initially two electrons and one hole and finally one electron. the electron system gains energy
whilst the hole system loses energy
where p is the effective mass ratio tn./m,.
1.
INTRODIJCTlON
There are two ways in which the phenomena of recombination and carrier heating interact. Firstly, the recombination rates may be affected by the carrier temperatures. It will be shown that these effects are relatively slight except for the case of Auger recombination, or to be precise the reverse process of Auger generation which is identical to that of impact ionisation. Secondly, the carrier temperatures may be altered by the energy transferred in recombination processes. Again it is the Auger effect which provides the most significant changes. Calculations of carrier temperatures in p-n junctions[l] have already suggested that it may be possible to discuss avalanche breakdown in terms of Auger effect in a high temperature electron hole plasma. The work referred to excluded consideration of generation/recombination in the depletion layer and clearly to discuss avalanche breakdown in this way it must be included. In addition the energy loss caused by the Auger generation must be taken into account in the energy conservation equations. The results presented here should assist in carrying out this task.
2. THE EFFECTS
OF CARRIER ON RECOMBINATION
by radiative recombination. For this reason it will be only the latter two processes which will be discussed here. Recombination through the intermediary action of an energy level due to a foreign atom (referred to henceforward as a “trap”) was first discussed by Hall[2] and Shockley and Read[31. The physical model used here will be thosame as that used by them, but the treatment will begin at the level of the Boltzmann equation rather than at that of the electron or hole continuity equation. This is necessary if one wishes to deal with either carrier temperature or energy transfer effects. However, no account will be attempted of the means by which the trap exchanges energy with the lattice, nor will exicted states of the trap be considered. The rate of change of the conduction band electron distribution function f’ as a result of the presence of a trap concentration N, can be written as (
5
>,
= ep’N,J - u’u;N,(l-f,)f
(1)
where e,’ is the probability of emission from the trap of an electron of momentum p ; q,’ is the corresponding captive cross section and or the electron velocity. f, is the probability that the trap is occupied by an electron. In thermal equilibrium at the lattice temperature TO,microscopic reversibility leads to a relation between e,,
TJM’ERATURE RATES
t = g~c; e(f,-r)ikrO ep
Leaving surface recombination aside there are three principal means by which electrons and holes can recombine or be spontaneously generated. These are direct radiative recombination, recombination through an impurity centre and Auger recombination. There are other more complicted processes often combining aspects of the three main processes. If one excludes light emitting diodes, junction lasers and photodiodes, the remaining semiconductor devices are not normally much influenced
(2)
where E, is the energy of the trapping level and e that of the electron. Equation (1) then becomes 8 = N&!eO;Lfie(5-W-~(j ( at, >
-ft)]*
A similar equation can be derived for holes 229
(3)
J. E. PARROT
230
as” =N,uhuph[(l (at, >
-ft)e”+‘“~-fhfr]
(4)
where f” is the hole distribution and E the hole energy. The hole and electron equations are coupled together by their dependence on fi. Conservation of charge requires that
where &To) is the intrinsic carrier concentration at the lattice temperature. Thus eqn (8) tells us that the recombination depends on the carrier temperature through the quantities (u’u’) and (~“a”). In fact, the generation term depends primarily on the lattice temperature through n?( To); the condition PN = ni’ for vanishing R also no longer applies. To determine (~‘a’) we refer to the work of Pratt and Ridley[4]. Using a simple model they find
1
Ap’
where J d’p = 2K’ J dp, $ dp,, J dp, and Rh and R, are the net rate of disappearance of holes and electrons into the traps. In a steady state these are, of course, equal. Integrating equation (3) to obtain R. gives R.=N,[(I-+%,‘f’d’p
de. T) = 7
E- [
eSm*“z - 1 -’ i 2V%ee,e “I>
(9)
where A is a constant and the remaining quantities have their usual meanings. For moderately high energy the argument of the exponential is small leading to an approximate result
1 (6)
_ fi e(e,-,likT, V’a,’ e-” /kr,,d3p I
where E’ is the electron kinetic energy measured from the bottom of the conduction band, ec. At this point the carrier temperature assumption is introduced. This states that the electron distribution retains the MaxwellBoltzmann form except that the electron temperature T, may differ from To. Then
exp
A’T!,’ cr(e, T)=-
E
.
This can be used to evaluate (L”(T’),: 112
and similarly for holes. This leads finally to a recombination rate given by
R, = N,[N(u’cr’),(l-fr)-N,(To)e’“~-‘~”’TO(p’cr’)of,]
N,B.Bh[ NP(&“*
- d(%)]
(11) R = NB.($y
+ PB!?(.g’
+ B.N, (To) e”~-‘~1’kTo + B,,N, ( T,) e”L-Fr)‘xTo
where
(c’u’), =
I
reup’ e -r”kTed’p I
e -“lPTed’p
and (v’u’)o is the same quantity with T,, in place of T,. N,(To) is the usual effective density of states function evaluated at the lattice temperature
A similar equation can be found for R,, and if the steady state condition applies R, = R,, = R
then
Two points emerge: (i) for electrons and holes cooled by passing a forward current the recombination rate is somewhat increased, (ii) the generation rate in strongly depleted semiconductor will not be much affected by changes in T. and 7’,,.Equation (11) would correspond in n-type semiconductor to a lifetime sh-’ = N,Bh and in p-type semiconductor to a lifetime t-’ = N,B, with the carriers in temperature equilibrium (T, = T,, = To). Overall the effects of carrier temperature on recombination through trapping centres is likely to be relatively small. The position turns out to be different for Auger recombination. This can be classified depending on whether it involves two electrons and a hole or two holes and an electron. In the former case the recombination energy is largely carried away by an electron, in the latter by a hole. The theory of Auger recombination has been worked out by Beattie and Landsberg[S], Hearn[6] and Landsberg[7]. Let us consider the first case described
N(uCue). + N,(T,)( u “uhh e’60-cfr’rT, h =N(u”u’),
+P(vhuh),,
+N,(To)(~‘~‘)oe”~-“~“Xr~+N,(To)(vhuh)
0 e(ev-ec”kG
(’
and R = N’N($u’),
NP(v’u’),(vhuh )h - ni’(roKo’u’)~~Uhuh)O +P(vhuh)h +N,(~o)(~‘a’)oe”‘-‘~“Lr”+N,(~o)(uhuh)~e”’~r”’k70
(8)
231
Carrier temperature effects in recombination
above and write pl for the initial hole momentum p ; and pi for the initial electron momenta and p2 for the final electron momentum. Using Beattie and Landsberg’s calculations we can write
2-2p2.g+i2/(1+/L)+(Yg~ xs Pg 2m,
(
(14)
-4 )
g
x If”(Pl)f’(PIlf’(pD-f’@*)l
(12)
where (FIF2/ is a product of overlap integrals, g= P; - PI, I= p: + BP], p,2/2m, = Ed, p = mrlmh, a = (1+2cL)lU+p). The S-function prescribes the necessary energy conservation and momentum conservation requires that
the appropriate
where as usual E’ is the electron kinetic energy. We will now evaluate this for the two recombination processes discussed in Section 2. For recombination through traps we obtain (af’/at), from eqn (1) and hence
d3p . (15) J Using a similar notation to that employed earlier and obtaining f from eqn (7) we can write -(l-J)
p’+p;=pt+p2.
We can now introduce
There are also energy continuity or conservation equations whose solutions give the carrier temperatures. To complete these equations we need expressions for the rate of change of the internal energy of the electron and hole systems. This can be written as
Maxwell-
J
E ’ v ‘q’f
(16)
Boltzmann functions for the distribution functions. The second part of eqn (12) can then be integrated without further approximation giving?
where
E’vo, e-~“‘T&p (&‘vu)r =
I e -*“kTad3p
R2 = _ 32~“e”(F,F$m,(l+ CL)“* I 3 7r ‘2h C2E02rt
and so on. There will be a similar expression for the change in the internal energy of the holes. It will be seen The magnitude of this expression is the generation rate for that eqn (16) is much more complicated than eqn (8) for R. In particular there is a third term in the numerator. electron-hole pairs in an electron gas: it clearly depends The signi8cance of this can be seen by the fact that it strongly on the electron temperature T.. It is not possible to integrate the first term in (11) with the same degree of vanishes if T. = To but still remains if recombination is suppressed by setting u h = 0. In fact it corresponds to the accuracy unless the carrier temperatures are equal energy transfers which arise when an electron is trapped (T. = Th). Then it becomes and then re-emitted at the lattice temperature without R, _ 32a’e’jF,F2~m,(l+ w)“~ recombination. a’Rh3c-Eo2a?l;(T,) Some idea of the orders of magnitude involved can be (13b) obtained by employing the Pratt-Ridley model [41given by Thus like the generation rate, the recombination also eqn (9). Using this model increases rapidly with carrier temperature. It is to be noted that the lattice temperature, To, appears nowhere in (17) these equations. The reason is that the Auger mechanism does not involve any interactions with lattice being and the resulting expression for rate of energy change is entirely an electronic process.
(134
(18)
3. ENERGYTRANSFERS IN RECOMBINATION PROCESSES
The recombination rates calculated in Section 2 are needed to fill a place in the carrier continuity equations. tDetails of this calculation are given in an Appendix.
It will be seen that if the carrier temperatures are both equal to the lattice temperature then = -RkTo.
J. E. PARROR
232
Thus the electron system only loses kTo for every electron recombined, which is approximately what would have been expected. This is in fact a very weak process under all circumstances. The energy transfers in the Auger effect can be much more significant. It will be clear that the total kinetic energy change for the combined electron and hole systems will be equal to the gap energy times the recombination rate but it is not obvious how this is divided between the two. For the process with initially two electrons and a hole and finally just one electron the change in electronic kinetic energy will be
B =~pp,2+p;L+p;2
whichcan be rewritten in terms of g and I by using the momentum conservationcondition. This leads to the complicated form B = I + pp m12+
l/AC1-6) p(P+P)fu+P)* (Itpy g.lf (ItrY
whilst the holes lose p(*/2m1, in kinetic energy. The full
tk=
which it will be seen considerably simplifies if /3 = 1. The next step is to integrateover d21making use of the energy conserving d-function. The result for the recombinative first term in eqn (12)
is R = 64?re’~FIF,~*m.2kT. I e2e02h6 (I+CLY x ll.(l-B)“”
description
gz-2k2.g
[(
2e;
I+/+ t BEat -~a If/l
)/
kT
*I
xld’gg-‘ld3p,sinh[?:~~:;,8)~~1+p)(2n.g-ag’-k~)]
of the calculation of energy changes is given
in the Appendix. These calculations lead to the result that
au' =RAEG(~+~cL+~cL*) ( at.
>
(1+2/.r)ff+cL)
(1+2pL)u+II)
?p’(I - p,g’ C1+pL)2
t2*.&p)
(1%)
where ye*= 2m.kT.. Unfortunately it appears impossible to proceed further analytically unless )!J= I, i.e. T. = Tk. If this condition is imposed
(1%)
R, = ~4~JTl’m.2kT.(l
whilst = -RAEG-
xexp-Y* i P>‘+
t ~Yexp
ic2EFc+ sFk + Eoj,kT
EZE 0*v
which it will be seen add up to R.+FG.There will be corresponding expressions for the opposite kind of Auger process. If perfect symmetry (P = 1 etc.) existed between electrons and holes each system would gain EC/~ for every recombination event. It will be seen by comparing the two recombination processes that Auger effect will outweigh trap recombination as an energy transfer mechanism by a factor of EG/Z~TC, which is approximately 20. It has of course been realised for a long time that impact ionisation was a very efficient means of removing energy from hot electrons. What has been done here is to treat it as a quasiequilibrium process.
x )hCj-
d'p2b'2b.
g- ag*- kg’ exp - (y’p:)
Proceeding to the integration over d’g, it is necessary to evaluate
dg
V2p,.g-agz-kks2 g’
where x is the cosine of the angie between pz and g. The limits are determined by the requirement that 2pzgx-ag2-k,2~0 as illustrated in the figure. Integrating first over g the limits are
g =k%kA a
REFERENCES
1. T. Y. Stokoe and J. E. Parrott, Solid-St. Electron. lg, 811 (1975). 2. R. N. Hall, Phys. Rev. 87, 387 (1952). 3. W. Shockiey and W. T. Read, Phys. Rev. 87, 835 (1952). 4. R. G. Pratt and B. K. Ridlev, Proc. Phvs. Sot. 81.996 (1%3). 5. A. R. Beattie and P. T. L-andsberg, Proc. Roy.’ SOL’ A4Zi. 16 (1959). 6. C. J. Heam, Proc. Phys. SIX. 86, 881 (1%5). 7. P. T. Landsberg, Lectures in Theoretical Physics 8.4, 313 (1%6).
where
Carrier temperature effects and energy transfers in Auger recombination The starting point in the calculations presented here is eqn (12)
which after inserting the limits gives
Integrating by parts leads to a standard form
_ v/zp,gx - a$ - k,’ g
$pzx Y
of the main report. If we use the carrier temperature approximation where K = V&. = exp
K
2~; t &”
- (BPPfi2+P’*+P’*) 2m.
k* !/
1
<
where /3 = T./T, gives the ratio of the electron and hole temperatures. The most important quantity to consider is
- K)
Thus
[,_E’
dx(pzx -K) k, I I(@?
=-&(p:-K)’
233
Carrier temperature effects in recombination
[
2p*.g-s-$J/?m..
The term in I. g will disappear in the angular part of the integration over d’I; energy conservation then enables the elimination of I itself leaving
2r k,’ [ I+pP2.9+ l+P
32re’(FIF$m.kT.(1 e2r02h6
I
I
4
2cL2g2 2m,.
(l+r)2 I/
+ FL)“*
x exp [(2.5; + .sFh+ .sC)/kT]
P
0
Fig. I. This shows the way in which the limits in the integration over d3g are determined by the condition t
2p,gx - erg’- k,’ z=0. The area of integration is hatched.
where pz> K since otherwise the area to be integrated over vanishes as shown in the figure. The expression for R, now becomes R = 256a’e’JF,F,Pm.‘kT(l+
I
x d2p2. g- ag2- k,’ exp - (~~~22). Thus we have to evaluate three integrals, one of which is the I, considered earlier. The others are
dg
d2pzgx
-w=- h,’ 8
and
k)“*
I, = 27
dgd2p2gx - ag* - kb
c2c”2h6kI x exp j(2.9; + .s” + eo)/kT,](I,-
2KJ, + K2J,)
where the limits are as discussed in the recombination calculation. Integrating over dg gives
where I, =
J2=
dxx(ptx -K) =
LI dp2nz e-7’P,’ = c K I 2Y2
= dP2P; e-72p?z IK
and
Xe-“2+5erfcX 2 2y’
1/ ,1_2 = (X’+ 1)eP J, = K dp,p,‘eI 2Y’
l,=-$
4
Then
au'
256~‘e’(F1F2~*m,kT.
( -ii- >A= c2r,2h”a(1+p)“2(l+2p)
- VGX erfc X 2y4
x exp [(2~; + EF”+ EG x
-x2
Generally eo B kT. so this becomes = &.
)/kT. I
c~(2+3~)~,+1-4~~~~ C
-tt+2:-~2)K25-&‘J.]
Making the usual substitutions 4n3 n,‘=_siiaexp-
K2)
=g[l-3(;>‘+2e)‘].
and X’ = aso/kT.. The angular part of the integration over d’p, has already been carried out. Then J,-2KJz+K2J,=e
dxb=x’-
(
*
where >
NzL$exp(_$y
I
.r.= - dpzp;
e-72Pz2 =
-!- (x3+:X) 2yJ
[
eex’++erfc
X].
x
P=-$Jexp
(2)
e
leads to eqn (13b) in the main text. Let us now consider the question of the energy transfers in Auger recombination. We know that in total the change in the electron kinetic energy and the hole kinetic energy must equal EO, the energy gap. The change in the electron kinetic energy is
Substituting for the integrals J,, J2, J, and J, the expression involving them becomes &
[
(I + 2* - ~2)X(e-Y2- v’r erfc X) + ;I [2+3p)V/nerfcX.
Making as before the assumption that X % 1 then it becomes
(p22- p’2- pi2)/2m.. Changing to the variables I and g and using the momentum conservation condition this energy change becomes
which leads to eqn (19a) of the main text.
1
CARRIER TEMPERATURE EFFECTS AND ENERGY TRANSFERS IN NON-RADIATIVE RECOMBINATION J. E. PARROTT Department of Applied Physics and Electronics, UWIST, King Edward VII Avenue, Cardiff. CFl 3NU, Wales (Received 12May 1975:in revised form 13 August 1975)
Abstract-The effects of carrier temperature changes on Hall-Shockley-Read and Auger recombination are calculated and it is shown that these are small in the former type of recombination but may be very large in the latter. Energy transfer calculations show that for Hall-Shockley-Read recombination the electron system loses energy kTo for every recombination event, but in the Auger event where there are initially two electrons and one hole and finally one electron. the electron system gains energy
whilst the hole system loses energy
where p is the effective mass ratio tn./m,.
1.
INTRODIJCTlON
There are two ways in which the phenomena of recombination and carrier heating interact. Firstly, the recombination rates may be affected by the carrier temperatures. It will be shown that these effects are relatively slight except for the case of Auger recombination, or to be precise the reverse process of Auger generation which is identical to that of impact ionisation. Secondly, the carrier temperatures may be altered by the energy transferred in recombination processes. Again it is the Auger effect which provides the most significant changes. Calculations of carrier temperatures in p-n junctions[l] have already suggested that it may be possible to discuss avalanche breakdown in terms of Auger effect in a high temperature electron hole plasma. The work referred to excluded consideration of generation/recombination in the depletion layer and clearly to discuss avalanche breakdown in this way it must be included. In addition the energy loss caused by the Auger generation must be taken into account in the energy conservation equations. The results presented here should assist in carrying out this task.
2. THE EFFECTS
OF CARRIER ON RECOMBINATION
by radiative recombination. For this reason it will be only the latter two processes which will be discussed here. Recombination through the intermediary action of an energy level due to a foreign atom (referred to henceforward as a “trap”) was first discussed by Hall[2] and Shockley and Read[31. The physical model used here will be thosame as that used by them, but the treatment will begin at the level of the Boltzmann equation rather than at that of the electron or hole continuity equation. This is necessary if one wishes to deal with either carrier temperature or energy transfer effects. However, no account will be attempted of the means by which the trap exchanges energy with the lattice, nor will exicted states of the trap be considered. The rate of change of the conduction band electron distribution function f’ as a result of the presence of a trap concentration N, can be written as (
5
>,
= ep’N,J - u’u;N,(l-f,)f
(1)
where e,’ is the probability of emission from the trap of an electron of momentum p ; q,’ is the corresponding captive cross section and or the electron velocity. f, is the probability that the trap is occupied by an electron. In thermal equilibrium at the lattice temperature TO,microscopic reversibility leads to a relation between e,,
TJM’ERATURE RATES
t = g~c; e(f,-r)ikrO ep
Leaving surface recombination aside there are three principal means by which electrons and holes can recombine or be spontaneously generated. These are direct radiative recombination, recombination through an impurity centre and Auger recombination. There are other more complicted processes often combining aspects of the three main processes. If one excludes light emitting diodes, junction lasers and photodiodes, the remaining semiconductor devices are not normally much influenced
(2)
where E, is the energy of the trapping level and e that of the electron. Equation (1) then becomes 8 = N&!eO;Lfie(5-W-~(j ( at, >
-ft)]*
A similar equation can be derived for holes 229
(3)
J. E. PARROT
230
as” =N,uhuph[(l (at, >
-ft)e”+‘“~-fhfr]
(4)
where f” is the hole distribution and E the hole energy. The hole and electron equations are coupled together by their dependence on fi. Conservation of charge requires that
where &To) is the intrinsic carrier concentration at the lattice temperature. Thus eqn (8) tells us that the recombination depends on the carrier temperature through the quantities (u’u’) and (~“a”). In fact, the generation term depends primarily on the lattice temperature through n?( To); the condition PN = ni’ for vanishing R also no longer applies. To determine (~‘a’) we refer to the work of Pratt and Ridley[4]. Using a simple model they find
1
Ap’
where J d’p = 2K’ J dp, $ dp,, J dp, and Rh and R, are the net rate of disappearance of holes and electrons into the traps. In a steady state these are, of course, equal. Integrating equation (3) to obtain R. gives R.=N,[(I-+%,‘f’d’p
de. T) = 7
E- [
eSm*“z - 1 -’ i 2V%ee,e “I>
(9)
where A is a constant and the remaining quantities have their usual meanings. For moderately high energy the argument of the exponential is small leading to an approximate result
1 (6)
_ fi e(e,-,likT, V’a,’ e-” /kr,,d3p I
where E’ is the electron kinetic energy measured from the bottom of the conduction band, ec. At this point the carrier temperature assumption is introduced. This states that the electron distribution retains the MaxwellBoltzmann form except that the electron temperature T, may differ from To. Then
exp
A’T!,’ cr(e, T)=-
E
.
This can be used to evaluate (L”(T’),: 112
and similarly for holes. This leads finally to a recombination rate given by
R, = N,[N(u’cr’),(l-fr)-N,(To)e’“~-‘~”’TO(p’cr’)of,]
N,B.Bh[ NP(&“*
- d(%)]
(11) R = NB.($y
+ PB!?(.g’
+ B.N, (To) e”~-‘~1’kTo + B,,N, ( T,) e”L-Fr)‘xTo
where
(c’u’), =
I
reup’ e -r”kTed’p I
e -“lPTed’p
and (v’u’)o is the same quantity with T,, in place of T,. N,(To) is the usual effective density of states function evaluated at the lattice temperature
A similar equation can be found for R,, and if the steady state condition applies R, = R,, = R
then
Two points emerge: (i) for electrons and holes cooled by passing a forward current the recombination rate is somewhat increased, (ii) the generation rate in strongly depleted semiconductor will not be much affected by changes in T. and 7’,,.Equation (11) would correspond in n-type semiconductor to a lifetime sh-’ = N,Bh and in p-type semiconductor to a lifetime t-’ = N,B, with the carriers in temperature equilibrium (T, = T,, = To). Overall the effects of carrier temperature on recombination through trapping centres is likely to be relatively small. The position turns out to be different for Auger recombination. This can be classified depending on whether it involves two electrons and a hole or two holes and an electron. In the former case the recombination energy is largely carried away by an electron, in the latter by a hole. The theory of Auger recombination has been worked out by Beattie and Landsberg[S], Hearn[6] and Landsberg[7]. Let us consider the first case described
N(uCue). + N,(T,)( u “uhh e’60-cfr’rT, h =N(u”u’),
+P(vhuh),,
+N,(To)(~‘~‘)oe”~-“~“Xr~+N,(To)(vhuh)
0 e(ev-ec”kG
(’
and R = N’N($u’),
NP(v’u’),(vhuh )h - ni’(roKo’u’)~~Uhuh)O +P(vhuh)h +N,(~o)(~‘a’)oe”‘-‘~“Lr”+N,(~o)(uhuh)~e”’~r”’k70
(8)
231
Carrier temperature effects in recombination
above and write pl for the initial hole momentum p ; and pi for the initial electron momenta and p2 for the final electron momentum. Using Beattie and Landsberg’s calculations we can write
2-2p2.g+i2/(1+/L)+(Yg~ xs Pg 2m,
(
(14)
-4 )
g
x If”(Pl)f’(PIlf’(pD-f’@*)l
(12)
where (FIF2/ is a product of overlap integrals, g= P; - PI, I= p: + BP], p,2/2m, = Ed, p = mrlmh, a = (1+2cL)lU+p). The S-function prescribes the necessary energy conservation and momentum conservation requires that
the appropriate
where as usual E’ is the electron kinetic energy. We will now evaluate this for the two recombination processes discussed in Section 2. For recombination through traps we obtain (af’/at), from eqn (1) and hence
d3p . (15) J Using a similar notation to that employed earlier and obtaining f from eqn (7) we can write -(l-J)
p’+p;=pt+p2.
We can now introduce
There are also energy continuity or conservation equations whose solutions give the carrier temperatures. To complete these equations we need expressions for the rate of change of the internal energy of the electron and hole systems. This can be written as
Maxwell-
J
E ’ v ‘q’f
(16)
Boltzmann functions for the distribution functions. The second part of eqn (12) can then be integrated without further approximation giving?
where
E’vo, e-~“‘T&p (&‘vu)r =
I e -*“kTad3p
R2 = _ 32~“e”(F,F$m,(l+ CL)“* I 3 7r ‘2h C2E02rt
and so on. There will be a similar expression for the change in the internal energy of the holes. It will be seen The magnitude of this expression is the generation rate for that eqn (16) is much more complicated than eqn (8) for R. In particular there is a third term in the numerator. electron-hole pairs in an electron gas: it clearly depends The signi8cance of this can be seen by the fact that it strongly on the electron temperature T.. It is not possible to integrate the first term in (11) with the same degree of vanishes if T. = To but still remains if recombination is suppressed by setting u h = 0. In fact it corresponds to the accuracy unless the carrier temperatures are equal energy transfers which arise when an electron is trapped (T. = Th). Then it becomes and then re-emitted at the lattice temperature without R, _ 32a’e’jF,F2~m,(l+ w)“~ recombination. a’Rh3c-Eo2a?l;(T,) Some idea of the orders of magnitude involved can be (13b) obtained by employing the Pratt-Ridley model [41given by Thus like the generation rate, the recombination also eqn (9). Using this model increases rapidly with carrier temperature. It is to be noted that the lattice temperature, To, appears nowhere in (17) these equations. The reason is that the Auger mechanism does not involve any interactions with lattice being and the resulting expression for rate of energy change is entirely an electronic process.
(134
(18)
3. ENERGYTRANSFERS IN RECOMBINATION PROCESSES
The recombination rates calculated in Section 2 are needed to fill a place in the carrier continuity equations. tDetails of this calculation are given in an Appendix.
It will be seen that if the carrier temperatures are both equal to the lattice temperature then = -RkTo.
J. E. PARROR
232
Thus the electron system only loses kTo for every electron recombined, which is approximately what would have been expected. This is in fact a very weak process under all circumstances. The energy transfers in the Auger effect can be much more significant. It will be clear that the total kinetic energy change for the combined electron and hole systems will be equal to the gap energy times the recombination rate but it is not obvious how this is divided between the two. For the process with initially two electrons and a hole and finally just one electron the change in electronic kinetic energy will be
B =~pp,2+p;L+p;2
whichcan be rewritten in terms of g and I by using the momentum conservationcondition. This leads to the complicated form B = I + pp m12+
l/AC1-6) p(P+P)fu+P)* (Itpy g.lf (ItrY
whilst the holes lose p(*/2m1, in kinetic energy. The full
tk=
which it will be seen considerably simplifies if /3 = 1. The next step is to integrateover d21making use of the energy conserving d-function. The result for the recombinative first term in eqn (12)
is R = 64?re’~FIF,~*m.2kT. I e2e02h6 (I+CLY x ll.(l-B)“”
description
gz-2k2.g
[(
2e;
I+/+ t BEat -~a If/l
)/
kT
*I
xld’gg-‘ld3p,sinh[?:~~:;,8)~~1+p)(2n.g-ag’-k~)]
of the calculation of energy changes is given
in the Appendix. These calculations lead to the result that
au' =RAEG(~+~cL+~cL*) ( at.
>
(1+2/.r)ff+cL)
(1+2pL)u+II)
?p’(I - p,g’ C1+pL)2
t2*.&p)
(1%)
where ye*= 2m.kT.. Unfortunately it appears impossible to proceed further analytically unless )!J= I, i.e. T. = Tk. If this condition is imposed
(1%)
R, = ~4~JTl’m.2kT.(l
whilst = -RAEG-
xexp-Y* i P>‘+
t ~Yexp
ic2EFc+ sFk + Eoj,kT
EZE 0*v
which it will be seen add up to R.+FG.There will be corresponding expressions for the opposite kind of Auger process. If perfect symmetry (P = 1 etc.) existed between electrons and holes each system would gain EC/~ for every recombination event. It will be seen by comparing the two recombination processes that Auger effect will outweigh trap recombination as an energy transfer mechanism by a factor of EG/Z~TC, which is approximately 20. It has of course been realised for a long time that impact ionisation was a very efficient means of removing energy from hot electrons. What has been done here is to treat it as a quasiequilibrium process.
x )hCj-
d'p2b'2b.
g- ag*- kg’ exp - (y’p:)
Proceeding to the integration over d’g, it is necessary to evaluate
dg
V2p,.g-agz-kks2 g’
where x is the cosine of the angie between pz and g. The limits are determined by the requirement that 2pzgx-ag2-k,2~0 as illustrated in the figure. Integrating first over g the limits are
g =k%kA a
REFERENCES
1. T. Y. Stokoe and J. E. Parrott, Solid-St. Electron. lg, 811 (1975). 2. R. N. Hall, Phys. Rev. 87, 387 (1952). 3. W. Shockiey and W. T. Read, Phys. Rev. 87, 835 (1952). 4. R. G. Pratt and B. K. Ridlev, Proc. Phvs. Sot. 81.996 (1%3). 5. A. R. Beattie and P. T. L-andsberg, Proc. Roy.’ SOL’ A4Zi. 16 (1959). 6. C. J. Heam, Proc. Phys. SIX. 86, 881 (1%5). 7. P. T. Landsberg, Lectures in Theoretical Physics 8.4, 313 (1%6).
where
Carrier temperature effects and energy transfers in Auger recombination The starting point in the calculations presented here is eqn (12)
which after inserting the limits gives
Integrating by parts leads to a standard form
_ v/zp,gx - a$ - k,’ g
$pzx Y
of the main report. If we use the carrier temperature approximation where K = V&. = exp
K
2~; t &”
- (BPPfi2+P’*+P’*) 2m.
k* !/
1
<
where /3 = T./T, gives the ratio of the electron and hole temperatures. The most important quantity to consider is
- K)
Thus
[,_E’
dx(pzx -K) k, I I(@?
=-&(p:-K)’
233
Carrier temperature effects in recombination
[
2p*.g-s-$J/?m..
The term in I. g will disappear in the angular part of the integration over d’I; energy conservation then enables the elimination of I itself leaving
2r k,’ [ I+pP2.9+ l+P
32re’(FIF$m.kT.(1 e2r02h6
I
I
4
2cL2g2 2m,.
(l+r)2 I/
+ FL)“*
x exp [(2.5; + .sFh+ .sC)/kT]
P
0
Fig. I. This shows the way in which the limits in the integration over d3g are determined by the condition t
2p,gx - erg’- k,’ z=0. The area of integration is hatched.
where pz> K since otherwise the area to be integrated over vanishes as shown in the figure. The expression for R, now becomes R = 256a’e’JF,F,Pm.‘kT(l+
I
x d2p2. g- ag2- k,’ exp - (~~~22). Thus we have to evaluate three integrals, one of which is the I, considered earlier. The others are
dg
d2pzgx
-w=- h,’ 8
and
k)“*
I, = 27
dgd2p2gx - ag* - kb
c2c”2h6kI x exp j(2.9; + .s” + eo)/kT,](I,-
2KJ, + K2J,)
where the limits are as discussed in the recombination calculation. Integrating over dg gives
where I, =
J2=
dxx(ptx -K) =
LI dp2nz e-7’P,’ = c K I 2Y2
= dP2P; e-72p?z IK
and
Xe-“2+5erfcX 2 2y’
1/ ,1_2 = (X’+ 1)eP J, = K dp,p,‘eI 2Y’
l,=-$
4
Then
au'
256~‘e’(F1F2~*m,kT.
( -ii- >A= c2r,2h”a(1+p)“2(l+2p)
- VGX erfc X 2y4
x exp [(2~; + EF”+ EG x
-x2
Generally eo B kT. so this becomes = &.
)/kT. I
c~(2+3~)~,+1-4~~~~ C
-tt+2:-~2)K25-&‘J.]
Making the usual substitutions 4n3 n,‘=_siiaexp-
K2)
=g[l-3(;>‘+2e)‘].
and X’ = aso/kT.. The angular part of the integration over d’p, has already been carried out. Then J,-2KJz+K2J,=e
dxb=x’-
(
*
where >
NzL$exp(_$y
I
.r.= - dpzp;
e-72Pz2 =
-!- (x3+:X) 2yJ
[
eex’++erfc
X].
x
P=-$Jexp
(2)
e
leads to eqn (13b) in the main text. Let us now consider the question of the energy transfers in Auger recombination. We know that in total the change in the electron kinetic energy and the hole kinetic energy must equal EO, the energy gap. The change in the electron kinetic energy is
Substituting for the integrals J,, J2, J, and J, the expression involving them becomes &
[
(I + 2* - ~2)X(e-Y2- v’r erfc X) + ;I [2+3p)V/nerfcX.
Making as before the assumption that X % 1 then it becomes
(p22- p’2- pi2)/2m.. Changing to the variables I and g and using the momentum conservation condition this energy change becomes
which leads to eqn (19a) of the main text.
1