Solid-State Electronics Pergamon Press 1969. Vol. 12, pp. 661-669. Printed in Great Britain

THEORETICAL EFFECTS OF EXPONENTIAL BAND TAILS

ON THE PROPERTIES OF THE INJECTION LASER

M. J. ADAMS

Department of Applied Mathematics and Mathematical Physics, University College, Cardiff, U.K.

(Receiwd 23 August 1968; in revised form 25 October 1968)

Abstract-Calculations have been made of various GaAs injection laser properties, assuming the high doping levels present in these devices to be responsible for the formation of band tails with density of states varying as exp( -E/E,). Simple analytical expressions are obtained for carrier con- centrations and emission rates, which are not possible for other forms of the band tail density of states. These expressions facilitate the solution of the laser kinetic equations and thus exact relations are derived for threshold currents, lasing frequencies, I-V characteristics, and light power outputs, both above and below threshold, within the scope of the initial approximation.

R&urn&-On a calcule les proprietes de divers lasers d’injection en AsGa en assumant que les niveaux ClevCs de dope presents dans ces dispositifs sont responsables de la formation des prolonge- ments de bandes et de la densite des &tats variant d’apres exp( -E/E,,). Des expressions analytiques simples sont obtenues pour les concentrations de porteurs et les taux d’emission, qui ne sont pas possibles pour d’autres formes de la densite de prolongement de bande de&tats. Ces expressions facilitent la solution des equations cinetiques du laser et ainsi des relations exactes sont derivees pour les courants limitatifs, les frequences de laser, les caracteristiques courant-tension et les puis- sances d’illumination de sortie, au-dessus et au-dessous de la limite, dans les normes de l’approxi- mation initiale.

Zusammenfassung-Fiir verschiedene Eigenschaften von GaAs-Injektions-Lasern wurden Berechnungen durchgeftihrt unter der Voraussetzung von Bandauslaufern mit einer Zustandsdichte proportional exp( -E/Eo), die durch die hohe Dotierung in den Dioden hervorgerufen werden sollen. Einfache analytische Ausdriicke wurden gewonnen fiir die Trlgerdichte und die Emissions- bande, die mit einem anderen Verlauf der Bandschwanze nicht miiglich sind. Diese Gleichungen erleichtern die LGsung der kinetischen Laser-Gleichungen und liefern daher exakte Beziehungen fur Schwellenstrom, Laserfrequenz, Strom- Spannungscharakteristik und die Lichtemission sowohl iiber als such unterhalb der Laserwelle.

NOTATION E!.4 loss coefficient due to diffraction, free carrier absorption, etc. Q(E) density of states constant for conduction band density of states constant for valence band F, Parameter containing matrix elements, etc. velocity of light in vacua F, penetration depth of electromagnetic wave electron charge F, photon energy F3 energy measured from band edge a forbidden gap I band tailing parameter for conduction band I th band tailing parameter for valence band j,s band tailing parameter for case E,, = E,, = E, K

661

photon energy corresponding to maximum of stimulated emission number of modes per unit energy per unit volume quasi-Fermi level for electrons, measured from edge of conduction band quasi-Fermi level for holes, measured from edge of valence band escape probability of stimulated emission escape probability of spontaneous emission Planck’s constant divided by 27-r current threshold current threshold current density Boltzmann’s constant

M. J. ADAMS

length of laser cavity total light output rate per unit time number of lasing modes number of spontaneous modes internal quantum efficiency of spontaneous emission concentration of electrons in conduction band number of photons in the Mth lasing mode concentration of donors in the active region concentration of acceptors in the active region density of states function for conduction band tail density of states function for valence band tail concentration of holes in the valence band total light output power stimulated emission function spontaneous emission function integrated spontaneous emission reflectivity photon lifetime in mode 34 absolute temperature refractive index applied voltage across the junction width of laser cavity

1. INTRODUCTION

TAE HIGH concentrations of shallow impurities present in the injection laser are generally assumed to form impurity bands in the vicinity of both the conduction and valence bands. The density of states function of the resulting band ‘tails’ is a question of special importance in the calculation of the properties of the device. HALPERIN and LAX(~) have calculated the density of states in the high density limit to have the form exp( - IEI”) where n varies between fr and 2, depending on such quan- tities as screening length and effective mass. This form includes the important cases of exponential”) and Gaussianc3’ bands, which have so far received most attention in the laser literature. In particular, the exponential form has been used in the band- filling n~ode1,‘4~5’ whilst the Gaussian expression due to KANE@) has been used in threshold calcu- lations by STERN.(~) Experimental evidence for the adoption of an exponential density of states, at least for the conduction band tail in GaAs, has recently come from fluorescence experiments using electron beam bombardment.(s) Thus it would appear acceptable, especially in view of the mathe- matical simplifications thus obtained, to use a density of states function, for each band tail, of the form exp( - E/E,) where E, may, in principle, be calculated from the theory of HALPERIN and L.&l’ as a function of screening length and efIective

mass. It should be noted that this form of band tail (with smaller values of E,) might apply under certain conditions even in pure materials.‘g’

It should be emphasized that the motivation for this approach is to obtain simple analytical es- pressions for carrier concentrations and emission rates, which are not possible for other forms of the band tail density of states. These expressions enable one to solve the steady state laser kinetic equations and to derive exact relations for the threshold currents, lasing frequencies, 1-V charac- teristics, and light power outputs, both above and below threshold.

2. SPONTANEOUS AND STIMULATED EMISSION RATES

Choosing the origins of energy at the nominal band edges, and measuring positive energies away from the gap, it is therefore assumed that the conduction and valence band tails have densities of states p,(E) and p,(E), respectively, where

Strictly, these density of states functions arc only appropriate in the forbidden gap and should be replaced by the familiar parabolic forms within the bands. However, to simplify the problem it will be assumed the exponential form holds also within the bands. This effectively limits the accuracy of the theory to cases where the occupied states lie fairly deep within the tails, i.e. cases for which the quasi- Fermi level for each band is, say, 2 KT from the band edge. In order to make the problem of detcr- mining the constants A, and A, of (1) tractable, it is convenient to make the somewhat crude assumption that each impurity concentration is given by the number of states m the appropriate band tail, limited by the energy value of the nominal band edge.

Hence,

ND = A,&,; NA = A$“,. (2)

It is then possible to obtain analytic expressions for the carrier densities, n and p, in these impurity

THEORETICAL EFFECTS OF EXPONENTIAL BAND TAILS 663

bands, as follows:

m

n= s 4 expW&c) dE

_ m 1 +expNE- FJIKTI

= A, exp(F,/E,,) KT~T cosec (3)

m

P= s 4 exP(EIEbJ dE

_-oo 1 +expU-FJIKTI

= A,, exp(FJE,,) KTm cosec (4)

where F,, F, are the quasi-Fermi levels for elec- trons and holes, respectively, measured from the band edges. The integrals in equations (3) and (4) converge only if KT < E,,, E,,; the subsequent theory is somewhat restricted by these conditions, since to obtain reliable experimental agreement one usually chooses values of EoC, E,, of order 15 meV or less. This probably limits the accuracy of the theory to temperatures for which say, KT < 0.9~ E,,, E,,; i.e. T N 150°K. However, it should be noted that values of E,,, E,, as high as 36 meV have been reported. Also, as will be shown later, for the stimulated emission integral above threshold, the special case of E,, = Eob, does not require this restrictive condition, since the above integrals are not necessary. Note that the form of equations (3) and (4) for n and p, and their dependence on the quasi-Fermi levels, is similar to the analogous formulae for non-degenerate parabolic bands.

Using the same density of states functions (l), together with the “no k-selection rule” model of LASHER and STERN, the nett rate of stimulated emission, r,,(E), is given by

a3

y,,(E) = B s fc(o%(E-EG--‘) -m

1 -

I dE’

1 +exp[(F,-E+EG+E’)/KT]

= BA,A,KT?r cosec[KTn($ - i)]

where E is the photon energy, E, is the forbidden gap, and B is a parameter containing the momen- tum matrix element for the transition; LASHER and STERN estimate that B N 7.5 x 10-10cm3/sec.* Using the approximate relation between F,, F,, E, and the applied voltage V in a p-n junction (assuming the recombination occurs mainly on the p-side, say) :

eV = F, +Fv +Em (6)

equation (5) may be re-written

‘r,,(E) = BA,A,KTv cosec[KTv(-& - $)]

x exp(z ‘2)

x [exp(y) -exp (y)].

(7)

The rate of spontaneous emission, r,,(E), is then given by

y,,(E) = y,,(E)

1 - exp[(E - eV)/KT]’ (8)

I 1

’ 1 +exp[(E’-F,)jKT] i

* An alternative calculation for the donor-acceptor transition (perhaps more appropriate here) using hydrogenic wave functions as given by ZEIGER(~~) also yields B N- 7 x lo-lo cms3/sec for GaAs.

664 M. J. ADAMS

and the total spontaneous emission rate, Rsp, by

m . RsD = I rsp (E)dE N Bnp

0

(9)

From equation (7) it also follows that the maxi- mum value of the stimulated emission occurs at

It is reasonable to suppose that EM is a good approximation to the photon energies of the lasing modes, the exact values being determined by the cavity dimensions. Also, from (7),

rst(E,) = BA,A,KTm cosec[KTn(&-i)]

(11)

3. KINETIC EQUATIONS

The rate equations for electron density, n, and number of photons, NM, in the Mth lasing mode may be written’14s15’

dn I &, miv,

z- e?VLd 7 -.Ysto (12)

WLd +(E)

d,V, -I%~, WLJR,, r,t(E&f) -=~+-~I_+N~.7---

dt Tlv 4(E)

(13)

where IV and L are the width and length of the device, d is the penetration depth of the electro- magnetic wave, 7 = internal quantum efficiency of spontaneous radiation (the stimulated emission is assumed to have internal quantum efficiency

unity), m is the number (assumed small) of lasing modes out of a total of M’ spontaneous modes, 4(E) is the number of modes per unit energy per unit volume, given by

1

4(E) = &, and rM is the photon lifetime in a lasing mode, which may be related to the cavity losses as follows :

d = :[K’ + ilni] (14)

Here, c is the velocity of light, CL is the refractive index, CI’ represents the losses due to diffraction, free carrier absorption, etc., and R is the reflec- tivity of the cavity mirrors.

Considering the steady state situation only, and substituting from equations (9) and (1 l), equations (12) and (13) may be re-written

I XZ mN,YZ (J = __-__-

eWLd 7 WLd ’ (15)

()=-S!+ WLdXZ

+ N&fYZ, TM M’

(16)

where the following simplifying notations have been used:

X = BA,A,(KTr)’ cosec

(17)

E x 2

ii 1 h%-E”“)

E OC

_ (E$“;:I’;OC-E”“) 1 ,

Z = exp(z +$j. (18)

(1’))

Equations (15) and (16) may now be solved for the two unknowns NM and Z; great overall simplification is achieved by ignoring the term in l/M’ (since M’ is cxpectcd to be very large) and this will be done in what follows.

THEORETICAL EFFECTS OF EXPONENTIAL BAND TAILS

The justification for this is that, for impurity concentrations of *101’, and typical GaAs parameters, the first and third terms of (16) are of order 1020 and 1O22 respectively, whilst the second term is only of order 1Ol6 even for cases close to threshold.

lth, is given by

I th edX (23)

Thus (u) below threshold, Graphs of jth as a function of temperature cal- culated from (23) for different values of the parameters E,,, E,,, (corresponding to different doping levels) are given in Fig. 1. The figure shows the characteristic steep increase of threshold current at high temperatures and the usual cross- over points for curves corresponding to different doping levels. Note also that the expression (23) gives a linear dependence of threshold current on inverse length, as is found experimentally; the form of (23) is similar to that originally derived by LASHER.

??lN, = 0, z= 771 XeWLd

(20)

and (b) above threshold,

z = _!-- 7MY'

(21)

hM XWLd mN,=------.

-ly e (22)

Note that Z is the only quantity in (15) and (16) which is explicitly dependent upon the position of the quasi-Fermi levels and hence on the applied voltage. Equation (21) indicates that Z becomes voltage-independent above threshold.

Clearly, the lasing threshold condition is achieved at the intersection of cases (a) and (b) above, i.e. when the expressions (20) and (21) are identical, so that the threshold current density,

IO'

“E 10‘ Y

i

5

‘- 16

10:

(cl

(b)

(a)

IO IO‘ I 03

T, ‘K

FIG. 1. Threshold current density, jth, as a function of temperature: (a) E,, = 34 meV, &,, = 35 meV. (b) E,, = 28 meV, Eo,, = 29 meV. (c) E,, = 22 meV E,, = 26 meV. (7 = 1, d = 10-%m, p = 3.6,~~ =

4 X 10 -12sec.)

665

4. I-V CHARACTERISTICS

To obtain the required current-voltage relations from the expressions obtained above, it is necessary to have one further relation between the quasi- Fermi levels F,, F,. The simplest assumption here is that of charge neutrality, which is likely to be approximately satisfied throughout the junction :(18)

n-p = ND-N, (24)

In particular this equation will be applied in the active region of the device, i.e. on the p-side near the junction.

Substituting for n and p from equations (3) and (4), and using (6) and (19), one obtains, after a little manipulation,

eV=E,

+E (N,-N,)+~[(N,-N,)2+4XZIBl oc

2A,KT?r cosec (KT?r/E,,)

+E

ou ln (NA-No)+~/[(ND-NA)~+~XZIBI l 2A,KTn cosec(KTrlE,,) ’

(25)

This equation gives the I-V relationship both above and below threshold when Z is replaced by the appropriate expressions (20) or (21). A typical set of curves thus obtained, at different tempera- tures, is shown in Fig. 2; the variation is

666 M. J. ADAMS

FIG. 2. I-V characteristics at different temperatures for the parameters ND = 1018cm-3, Nb = lO’%m-:‘, E,, = 15 meV, E,, = 16 meV, 7M = 2.4 x 10 -%x,

7 = 1, WL = 0.0025 cm2 (E, was calculated, for GaAs, from the formula of VARSHNI."~')

exponential up to threshold, at which point the voltage becomes stabilized, all further current leading to an increase in coherent light output.

In addition, the use of (25) in equations (7) and (S), together with the appropriate expressions for 2, yields the emission rates as a function of photon energy for given currents. Figure 3 shows a set of spontaneous emission rates calculated in this way, at the thresholds corresponding to a number of different temperatures. These curves may be compared with those calculated by Stern in Ref. 7, using a Gaussian density of states. Clearly, Stern’s theory gives narrower spontaneous emission curves than for the present theory, especially for higher temperatures, since Kane’s density of

states’@ has the conventional behaviour of tending to the parabolic form within the bands. However, at low temperatures there is reasonable agreement and it is difficult to distinguish between the two results by comparison with experimental curves. Similarly, the photon energies at which the lasing peak is expected to occur, which are shown in Fig. 3, and their characteristic shifts to lower values for higher temperatures are also practically identical with those calculated by Stern. The exact expression for the lasing energy is given by combining equations (10) and (25), together with the value of 2 given by (21). However, a simpler expression (which gives good agreement with experiment) is obtained by considering the special

THEORETICAL EFFECTS OF EXPONENTIAL BAND TAILS 667

lOOoK I2YK

~ 80°K

L meV

FIG. 3. Spontaneous emission as a,function of photon FIG. 4. Lasing photon energy, EM, as a function of energy, at lasing threshold for various temperatures. length L. Parameters used were: ND = NA = 5 x The parameters used were: ND = 101*cm-a, Na = 10’s cmm3, E, = 7 meV, GL’ = 15 cm-l, E, calculated 1018cm-3, I?,, = 15 meV, I?,, = 16 meV, 7.,., = 2.4 x from VARSHNI,“~) R = 0.33.

10 -12sec, E. calculated from VARSHNI.(~~)

case E,, = E,, = E,, say. This case has been considered separately and yields the following expression for the lasing photon energy EM:

E, = E,-E, In [N;;;.M]. (26)

The advantage of this expression is that it has been derived without the assumption E, > KT, and provides a method for experimentally determining EO,, from values of EM, for given NA, ND and TV. An example of the use of (26) is in giving the variation of EM with length, L, of the cavity. The resulting curves are shown in Fig. 4, using parameters appropriate to the experimental results of PILKUHN et d.;(16) a reasonable agreement is

.36

.42

.46

I I , \ ) 0.02 0.04 0.06 0.08 c

L, cm ). IO

found, even though the only temperature depen- dence used was that of the gap E,.(lg)

5. TOTAL LIGHT OUTPUT

The total light output rate per unit time, 9, is given by

9 = F3 WLdR,, +F,mN, g, (27)

where 1;a and F, are the ‘escape probabilities’ for the spontaneous and stimulated emission, respectively. The analytical expressions derived above may be substituted into (27) to yield the anticipated linear dependence on current both above and below threshold. However, it is perhaps more useful to calculate the total light output

668 M. J. ADAMS

power, P, defined as follows:

cc

P = F,eWLd s

Er,,(E) dE

0

~stt&t) + F,eE,mN, -.

4(E) (28)

Using equations (7) and (8) for r,,(E), one obtains:

n3 CXZ

I ET&E) dE N

I EY&E) dE

0 -72

= BA,A,(KTm)2Z cosec

E U, say. (29)

Hence, using equations (ll), (21), (22) and (29), one may obtain the following expressions for I’:

(0) Below threshold,

P = F,eWLdU

(b) Above threshold,

(30)

P = F,e~‘LdI/+F,E,(I-I,,).

(31)

Insertion of the usual expressions for Z and EM shows that these relations have the form postulated by CHEROFF et ~1.‘~~) and one may therefore use their expressions for Fl and F3. The resulting graphs of P against I for different temperatures are shown in Fig. 5 and correspond to the appro- priate experimental results of LAMORTE et ~1.~~~) In particular, the alteration of these graphs with temperature is in reasonable agreement with experiment.

6. DISCUSSION It has been shown that the adoption of the

assumption of exponential band tails seems worth- while in view of the analytic nature of the theory

J. X10’ amp/cm’

FIG. 5. T’otal light power output as a function of currwt density for various temperatures. Parameters used were:ND :- 10’8cm-3,Nj, : 1019cm-3,1?‘,, 7 22nwV,

l&‘,, = 25 meV, 7~,, 0.48 x 10 -l%C!c, ‘1 1

which results. The desirability of simple cx- pressions for emission rates, threshold currents, etc. is obvious, especially in view of the many approximate treatments’22 - 24) which have been applied to obtain such expressions. In addition, the theoretical work of HALPERI~V and LXX”) supports the exponential assumption for certain combinations of parameters. However, the use of their results to predict the parameters Ai,, E,,, and their variation with doping, temperature, etc. is limited by the difficulties associated with current theories of the screening length.(25) In view of these difficulties it seems more useful to regard the band tail spreading energies as adjust- able parameters. Fitting experimental results then yield fairly low values (2 10 me\‘) for these parameters which limits parts of the theory to temperatures less than, say, 120’K. Clearly, at higher temperatures the Fermi levels will tend to enter the parabolic parts of the bands and other approximations(22*23) will be more valid. Never- theless, the present theory does offer a more quantitative approach to previous band-filling calculations.‘4s5)

THEORETICAL EFFECTS OF EXPONENTIAL BAND TAILS 669

Acknozuledgement-I am grateful to Professor P. T. LANDSBERG for his help and encouragement during the course of this work. This work was done under a C.V.D. Contract and is published by permission of the Ministry of Defence (Navy Department).

11.

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