Resonant enhancement of the recombination probability associated with isoelectronictrap states in semiconductor alloys: In1−x Ga x P:N laser operation (77 °K) in the yellowgreen (λ5560 Å ,ω2.23 eV )C. B. Duke, D. L. Smith, G. G. Kleiman, H. M. Macksey, N. Holonyak Jr., R. D. Dupuis, and J. C. Campbell Citation: Journal of Applied Physics 43, 5134 (1972); doi: 10.1063/1.1661085 View online: http://dx.doi.org/10.1063/1.1661085 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/43/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Enhancing the quantum efficiency of InGaN yellow-green light-emitting diodes by growth interruption Appl. Phys. Lett. 105, 071108 (2014); 10.1063/1.4892830 Yellow–green luminescence from isoelectronic nitrogen centers in GaP grown by molecularbeam epitaxy J. Appl. Phys. 70, 1841 (1991); 10.1063/1.349501 Liquid phase epitaxial (LPE) grown junction In1−x Ga x P (x0.63) laser of wavelength λ5900 Å (2.10 eV, 77°K) Appl. Phys. Lett. 25, 352 (1974); 10.1063/1.1655505 Stimulated Emission and Laser Operation (cw, 77°K) of Direct and Indirect GaAs1−x P x on Nitrogen IsoelectronicTrap Transitions J. Appl. Phys. 43, 2368 (1972); 10.1063/1.1661505 Optically Pumped In1−x Ga x P Platelet Lasers from the Infrared to the Yellow (8900−5800 Å, 77°K) J. Appl. Phys. 43, 1019 (1972); 10.1063/1.1661211

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5134 C. D. NOMICOS AND P. C. EUTHYMIOU

'E. Fortin, Phys. Status Solidi 29, KI53 (1968). 'H. Milner-Brown and E. Fortin, Can. J. Phys. 47, 2789 (1969). 'R. Bube, Photoconductivity of Solids (Wiley, New York, 1960), p. 386. '8. Holeman and C. Hilsum, J. Phys. Chern. Solids 22, 19 (1961).

'R. Bube and H. MacDonald, Phys. Rev. 128,2062 (1962). 'So Ryvkin, Photoelectric Effects in Semiconductors (Consultants Bureau, New York, 1964), p. 108.

Resonant enhancement of the recombination probability associated with isoelectronic trap states in semicond\,ctor alloys: In,_xGaxP:N laser operation (77 OK) in the yellow-green ("- ::; 5560 A, h w ~ 2.23 eV) •

c. B. Duke 1 D. L. Smith. and G. G. Kleiman l Depart"!ent of Phr:sics, Materials Resea,rch Laboratory, and Coordinated Science Laboratory, University of IIImols at Urbana-Champaign, Urbana, Illinois 61801

H. M. Macksey. N. Holonyak Jr .• R. D. Dupuis. and J. C. Campbell DefJart"!ent of Electrical Engineering, M,aterials Research Laboratory and Coordinated Science Laboratory, University of IIImols at Urbana-Champaign, Urbana, illinOIS 61801 (Received 11 May 1972; in final form 24 August 1972)

Calculations of the recombination probability of an electron and hole in III-V ternary alloys containing short-range isoelectronic traps indicate its resonant enhancement when by varying the alloy composition, the electronic trap state is made degenerate with the direct (0 minimum in the conduction band. This result is consistent with observations of strong photostimulated laser operation of the N-trap transition in In1_xGaxP (0.6~x~0.7) in cases for which this state occurs near and above the direct band gap. These results lead to III-V semiconductor laser operation at wavelengths as short as or shorter than 5560 A (2.23 eV, 77 OK), with the ultimate limit in Inl_xGa~:N being about EN ;::::Er =2.27 eV (5470 A).

I. INTRODUCTION

In earlier work1,2 we have established that short-range isoelectronic trap potentials (e. g., due to N or N-N pairs in III-V ternary alloys) lead not only to bonafide bound states below the conduction-band minima in indirect GaAs1_xP x 1 and indirect In1.~GaxP' 2 but also to resonance states above the r minimum in direct alloys of these materials. That is, the energy Er of the conduction-band minimum at r lies below the energy EN of the isoelectronic trap state. In this paper we report a second important property of this trap state: its creation of a resonant enhancement of the electron­hole recombination probability when Er ""EN" Thus we expect to be able to construct high-efficiency light­emitting diodes and semiconductor lasers at alloy compositions x near the value at which E r = EN'

Our presentation is given in two parts. In Sec. II we indicate the results of model calculations which predict the resonant enhancement. In our models, we assume that the electron moves in a short-range attractive isoelectronic trap potential (i. e., in a potential due to N, to be specific). We calculate the absorption coeffi­cient by assuming that the electron recombines with holes of the following types: free holes, holes bound to electrons (i. e., bound exciton recombination), holes bound to acceptor impurities (e. g., Zn), and holes bound to donor isoelectronic impurities (e. g., Bi). We then present in Sec. TIl the inverse of absorption be­havior, that is, recombination data on In1.~GaXP : N that display the consequences of this enhancement. Indeed, our samples of IIlo.33Gao.67P : N exhibit laser operation on the N-trap transition at wavelengths as short as or shorter than 5560 A (2.23 eV) at 77 OK.

The fundamental concepts underlying the calculations are straightforward. In alloys of GaP the conduction-band density of states near X is about 10 times that at r.

J. App!. Phys., Vo!. 43, No. 12, December 1972

Consequently, the energy of a bound electron state associated with a short-range attractive trap potential is controlled l

-3 mainly by that of the conduction-band

minima at X, i. e., Ex' In In1.xGaxP: N, for example, as x is decreased from unity, the energy of an electron bound to an isolated N trap decreases only slightly from E N- Ex - 10 meV. 1- 3 Because Er decreases with x much more rapidly than Ex, the difference (E r - EN) decreases and reaches zero at x""O. 71. 2 Because, however, the valence-band maximum lies at r, the components of the wave function of the bound electron which permit ("direct") optical recombination (absorption) transitions with free holes are those associated with the conduction­band minimum at r. Thus, as E r - EN decreases, the trapped electron's wave function contains increaSingly larger components of the conduction-band states near r, which in turn are associated with an increasing recombi­nation probability (i. e., "resonant enhancement") until E r - E N' The resulting recombination is that of a bound electron with a free hole, i. e., with a hole localized in k space. At the crystal composition at which EN=Er (not the direct-indirect transition E r = Ex), the "bound" trap state becomes a resonance state above E r . If x is decreased further (E r < EN)' this resonance broadens and ultimately disappears. 1,2 In contrast to experimental observations, 1,2 the continuum contribution to the ab­sorption coefficient (or recombination) involving a free hole exhibits no sharp structure even when the trap state is resonant. This absence of structure in the continuum absorption is a consequence of the localization of the hole in k space. In the opposite limit, in which the hole is localized in the central (nitrogen) cell, the ab­sorption coefficient reflects the resonant form of the electron's spectral density in the central cell,2 but exhibits no resonant enhancement as EN - E r . The major point in these calculations is that, in order to yield resonant enhancement and resonant structure in the intensity as EN - E r , the hole must be neither

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RESONANT ENHANCEMENT OF RECOMBINATION PROBABILITY 5135

completely localized nor unlocalized in coordinate space. This point is illustrated by the model calculations.

When we assume that the hole is sufficiently unlocalized so that resonant enhancement occurs and presume that excitation of the crystal sample yields a narrow energy range, tlE, of excited electrons near Er and holes at the top of the valence band, the effects described above combine to create resonant enhancement of the re­combination probability for Er + l:!.E ~ EN ~ E r • As the entire resonance phenomenon occurs in direct (E r SEN < Ex) semiconductors, changes in the kinetics at the direct-indirect transition4 (i. e., at Er=Ex) need not be considered explicitly.

II. EVALUATION OF THE ABSORPTION COEFFICIENT

A. Review of model of isoelectronic impurity potential

In order to explore the consequences of the short-range nature of the attractive N impurity potential, we employ the Koster-Slater one-band one-site approximation, 5

which has been shown to provide a qualitatively correct description of the nitrogen potential in GaP. 3 In this model, the impurity potential extends over only one lattice site and, in addition, is diagonal in the band indices. Thus, the matrix element of U, the attractive isoelectronic potential, between electron Wannier states is given by5

(2. 1)

The quantities Rs and n are, respectively, the site and band indices of the Wannier states. The nitrogen is assumed to be localized at RD. Equation (2.1) describes coupling only to the lowest-energy conduction band, denoted by the index c. We neglect the short-range repulsive impurity-hole interaction,3 since the hole ultimately is bound by the long-range Coulomb attraction of the electron. The probability IBE(Ro) 12 that the electron is in the (central) cell containing the impurity is given in the Wannier representation by6

(2.2a)

where

Ac(E): lim j"" dt pc(t)/(E - t + ia). li "0+ _00

(2.2b)

The quantity p~(t) is the density of states in the lowest­energy band of the host material. The value of the bound (or resonant) state energy, EN' is determined by l,2,6

(2.2c)

In our previous papers on GaAs1_XP x 1 and In1_xGaxP, 2 we used Newns's7 form to parametrize the density of states calculated via the empirical pseudopotential method of Cohen and Bergstresser. 8 The quantity Vo is determined from the bound state energy in GaP. The result of inserting this parameterization into Eqs. (2.2) is that the value of EN is determined by the value of Ex, since the density of states at X is around 30 times larger than that at r. In other words, in In1_xGaXP' for example, when the material changes from indirect to direct, EN moves out of the gap and overlaps the continuum above r, and the state changes from bound to resonant.

Another conclusion2 is that the width of the resonant state is determined mainly by the density of states at r. Evaluation of the absorption coefficient requires that we describe the electron states explicitly. For an electron energy eigenvalue, E, in the conduction band, the wave function is specified by6

wt(r) = I/!c(k, r) + 11- Vo::[Ec(k))

6I/!c(k',r)exP[i(k-k'),Ro] (2.3a) x t' Ec(k)-Ec(k')+iy ,

E:Ec(k). (2.3b) The quantity E c(k') is the host- cry stal conduction-band energy (k' is restricted to the first Brillouin zone), k is a (non-unique) wave vector determined by Eq. (2.3b), N is the number of unit cells in the crystal, and y is a finite-width parameter due to interactions with phonons, impurity scattering, and disorder. A finite width is introduced to avoid divergences in the normalization of 'lit. The quantity I/!c is a conduction-band Bloch wave which we normalize for convenience to a Kronecker delta:

(2.3c)

The integral in Eq. (2. 3c) extends over the whole crys­tal. The normalization integral of 'lit is speCified by

(2. 3d)

In order to normalize the wave function to unity, we let wt - WIt(Wt lwJ>-1/2.

In the case that the trap state is bound, the bound state wave function (normalized to unity) is given by

W ()_ 1 (- Im[Ac(EN+iy)])-1/2I]l/!c(k,r)eXP(-ik.Ro) EN r -TN y It EN-Ec(k)+iy .

(2.4)

The spontaneous recombination probability P per unit time per unit volume can be written

P = (E5/2/37T2C2) f.: w2 dw Cl g( w), (2. 5)

in which E is the semiconductor index of refraction, c is the speed of light, and Clg(W) is the optical absorption coefficient calculated with the statistical factors inverted from the usual case9 so that the higher-energy state is occupied and lower one empty. An important point to note is that for small variations in x in the neighborhood of the direct-indirect transition, the nonradiative recombination probability is approximately constant; hence, P is approximately constant. This has the consequence that, if the contribution to the recombination probability from the N trap state becomes resonantly enhanced as E N - E r , the trap "robs" direct interband transitions of their recombination probability.

The optical absorption coefficient per unit volume is expressed in the following equation;

Clg(W) =Ao~ I(i Ip· ~ If) 12a(E, - Ei -liw), (2.6a) i"

Ao: 47T2 e2 /wEm2cV, (2.6b)

in which m is the free electron mass, ~ denotes the polarization of the light, and the quantity V is the volume of the crystal. The indices i and f denote the

1. Appl. Phys., Vol. 43, No. 12, December 1972

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5136 DUKE et at.

/

F1(w) (' 1\

I \ / \

/ \ I "-

FIG. 1. Schematic drawing of the absorption coefficient in Eqs. (2.7) and (2.8) corresponding to the recombination between electrons in bound N states of energy EN and free holes. As EN - E r, the peak position in aN 0. e., nwmJIX) approaches E r and the peak height increases, producing resonant enhancement. The continuum contribution, a r, shows no resonant structure even when EN> E r, as discussed in the text. The drawing is not to scale.

initial (electron) and final (hole) states, respectively. We omit the statistical factors in Eq. (2.6) for clarity.

B. Recombination involving free holes

In our previous papers, 1,2 we considered the localized­exciton limit in which the hole is confined to the nitrogen site. In the opposite limiting case of a free hole, the results also are sufficiently simple to be written in closed form. We normalize the valence-band Bloch waves, 1/iv(q, r), according to Eq. (2. 3c) and assume a parabolic valence band. In the case of a bound trap state, Ci g comprises contributions from the bound state, Ci N' and from the continuum, Ci r :

Cig(W) == Cir(W) + CiN(W), (2.7a)

CiN(W)~2AoL; I (1/i v(k) Ip' A I \.liEN> 126(EN- Ev(k) - nw)

k

==Fo(w)F1(w), (2.7b)

Fo(w)= (Cia/W) nNO(nw - EN)l/2()(tiw - EN) ()(Em - EN)'

(2.7c)

F 1(w) = {- 'Y /Im[Ae(EN + iy)]}

x {[(me + mv)/me](EN - Ew)2 + y2r\

Cia ~ (25 /2e2m~/2 /lfl tm2c) I (1/iv I p . A l1/ie>a 12,

Em= mintEr, Ex],

(2.7d)

(2.7e)

(2. 7f)

(2.7g)

In Eqs. (2.7) Ev is the valence-band energy, 0 is the unit-cell volume, and me (mv) is the conduction­(valence-) band effective mass. We have introduced the nitrogen concentration, nN' by making the replacement 1/ N - nNS2. This is equivalent to making the approxima­tion that the crystal is composed of nNV small crystals, each of which contains only one nitrogen atom. For the

1. App!. Phys., Vo!. 43, No. 12, December 1972

crystals in our experiments, nN-1017_101B cm-3 so that the nitrogen atoms are sufficiently far apart for this approximation to be reasonable. The momentum matrix element in Eq. (2. 7e) involves an integral over the unit cell and is assumed to be independent of k: it is associated with light absorbed at r. The continuum contribution to the absorption coefficient is given by

(2.8a)

in which, expanding to first order in nNS2, we obtain

(2.8b)

The second term in Eq. (2. 8b) is omitted in Faulkner's analysis3

; it partially cancels, however, a similar term in the matrix element in Eq. (2.8a). This partial cancel­lation results in the absence of any resonant structure in the continuum absorption, as we shall show later, and, thus, requires us to examine models of localized holes.

After lengthy but straightforward calculation we obtain the continuum absorption coefficient:

Cir(W) == (Cio/w){[m/(m e + mv))3/2

x(nw -Er)1/2(nw - E r )[1 + nNOF2(w)]

+ nNOF3 (w)B(nw - Em)},

F 2(w) = (~/y) 11 - VoAe(Ew) 1-2 Im[Ac(E)],

(2.8c)

(2.8d)

(2.8e)

The important feature of Eqs. (2.7) and (2.8) for our purposes is that as EN - Er < Ex from below, the contribution to the recombination probability from the N trap, i. e., CiN(W), diverges as nw - EN [see Eqs. (2. 7b)-(2. 7d)]. Therefore, this trap extracts the spectral density from the conduction-band states for

FIG. 2. Schematic illustration of the terms F3 and F(, whose product determines the continuum absorption coefficient, a r, for a hydrogenic hole bound to a resonant electron Ii. e., Eqs. (2.12) 1. As E N- E r• the width of F( decreases and its height increases. enhancing the peak in F 3• which does not depend on EN' Thus. the spectrum exhibits resonant enhancement for EN - Er:S E h• the hole energy. The drawing is not to scale.

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RESONANT ENHANCEMENT OF RECOMBINATION PROBABILITY 5137

E > E r and deposits it in a narrow band of frequencies at nw"'EN"'E r • We can understand this "resonant en­hancement" by observing that (l N is the product of F 0

and Fl' F 1 is a Lorentzian whose peak is at nw = EN - to ~ EN - (mcimv)(Er- EN) < EN so that the peak region of F 1 does not contribute to (l N' as illustrated by the dashed line in Fig. 1. Since Fa is monotonically in­creasing and F1 monotonically decreasing for increasing nw (nw>E N), (IN exhibits a peak at nw=nwmax>ENas schematically illustrated in Fig. 1. As EN-E r , to-O and the Lorentzian peak region approaches E w Thus, a portion of F1 with larger magnitude contributes to o!w

This has the dual consequence of moving the position of the peak in O! N towards EN and of increasing the peak value of o!w In fact, when E r- EN "'1', nwmax - EN '" (mjm)y. In the limit that y - 0, this increase in the peak value of (IN becomes a divergence at nw=EN=Er .

In the limit that Ex, EN«Er' the contribution to O!/w) in Eqs. (2.8) from F 3(w) reduces to that given by Faulkner3

for GaP when we use his model for Pe(E). It is the interference terms, Eq. (2.8d), between the "direct" and trap-assisted recombination transitions that de­scribe the extraction of the recombination probability from the direct transitions to be accumulated in the N -trap recombination transitions. From a cursory examination of Eqs. (2.8), it would seem that F2 and F3 in Eqs. (2. 8d) and (2. 8e), respectively, contribute resonant structure to O!r when EN>Er' Upon looking more closely, however, we see that, when 1 B E(Ra) 12 varies more slowly than does the denominator in Eq. (2.8e), we can make the approximation

[(mv + me)2 (E _ E )2 + y2J,l '" m e1T o(E _ E ).

me '" y(mv+mc) '"

(2.9)

Inserting Eq. (2.9) into Eqs. (2.8), we find that the terms involving F2 and F3 cancel exactly. Thus, in this case a r has the structure less form illustrated in Fig. 1. This structureless behavior of (lr results from the partial cancellation produced by inserting Eq. (2.8b) into (2. 8a). Hence, a significant feature of the spectrum would have been lost by neglectingS the detailed behavior of the normalization integral and the resonant interfer­ence terms in the matrix elements in Eq. (2.8a). The absence of structure for EN> E r is not reflected in ex­periments. 1 ,2 Its prediction by the model results from the delocalized nature of the hole wave function. There­fore we must examine more localized models of holes.

One additional comment should be made. In calculating the absorption coefficient, we neglected the effect of electron-electron interactions, 10 which might produce threshold structure in the spectrum at the electron quasi- Fermi level. The only justification for this pro­cedure is that the experiments1 ,2 do not exhibit such singularities. Therefore it seems appropriate to neglect the additional complications of final state interactions at this time.

C. Recombination with holes bound to the electron (bound exciton recombination)

In the spirit of Faulkner's3 variational treatment of the bound exciton complex, we represent the bound exciton

wave function as the product of the wave function of an electron under the influence of the attractive short­range nitrogen potential and the wave function of a hydrogenic (ls) hole. We show that, in general, the spectrum exhibits both resonant enhancement similar to that described in Sec. II B and resonant structure pro­duced by the resonant N-trap state when EN> E r . In the case that the hole is very localized (i. e., the Bohr radius ao - 0), the resonant enhancement disappears.

We assume that both the hole and electron are centered at Ro=O. The hydrogenic (1s) hole wave function is specified by

1/!1S(r) = (1/ h) a~3/2 exp( - r / ao), (2. lOa)

ao~ n2E2/lle2, (2. lOb)

Il =m;nj(mc + m), (2.10c)

and the Fourier transform of (2. lOa) is given by

1/!1s(k) = (2ao),3/2 (8/ ao1T)W + a(j2)-2. (2. 10d)

Denoting the hole binding energy by Eh (i. e., Eh~ n 2a(j2/ 21l -10 meV 3), we can insert Eq. (2. lOd) and the elec­tron wave functions for Ro=O in Eqs. (2.3) and (2.4) into the expression for the absorption coefficient in Eqs. (2.6). There are again two contributions to (lg. We omit the details of the calculations since they are lengthy but straightforward. The bound electron contribution for Er>ENis given by

(l N(W) = (a/w)nNn{- y /Im[Ae(EN+ iy)]}

x {[2m e(E r - E N)/fi2]1/2 + a~1}'4 o(E N - E h - nw),

(2.11a)

0!1 ~ 25/21T2m~ 1l-3/ 2 a~3(lo' (2.11b)

Equations (2.11) are speCific to the case that E N< Er < Ex, since we are interested in determining the possibility of resonant enhancement in this region of crystal composition x. We observe from this expression that the band spectrum in Eqs. (2.7) for nw>E N is re­placed by a line emission at nw = EN - E h' Another con­clusion we can reach is that the strength of this line undergoes resonant enhancement as Er - EN from above. Since mjmv -0.07 in the III-V alloys, Il "'me and we observe that the intensity is slowly varying for Er - EN ~ E h• Thus, for a deeply bound state (i. e., ao small), the enhancement is not observable.

The continuum absorption coefficient is given in the following equations:

O!r(w) = (l2/w)nNne(tlw + Eh - E r )F3(W)F4(w), (2. l2a)

F3(W)~ [2m e(nw + Eh - EJ/n2]1/2

x [2m c(nw + E h - E r)/n2 + aii2]-\

F4(W)~ 1 + If(nw + E h) 12 + 2 Re[t(nw + E h)],

(2. 12b)

(2. l2c)

(2. 12e)

The quantity Fs is the product of a monotonically in­creasing and a monotonically decreaSing function (for

J. Appl. Phys., Vol. 43, No. 12, December 1972

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5138 DUKE et at.

2.05 2.10

hv(pump)

<f)

c Q)

c H

c o <f) <f)

E w

OJ ;>

0 Q)

0:

In GaP

100

6.0

Energy (eV) 2.15 2.20 2.25

Eg(f)

5.8

Wavelength

Inl_xGaxP'Te ' N

x ~ 0.60

Nd (3000K) = 6 x 1017/cm3

77 oK

N

5.6

(10 3 Ji)

2.30

5.4

FIG. 3. Photoluminescence spectra (77 OJ() of Inl_xGIlxP: N (x= 0.60). At low excitation levels (100 W /cm2) band-to-band recombination dominates, and at an excitation level of 4x 103

W /cm2 modes appear on the direct gap emission. As the exci­tation level is increased further the nitrogen A line (EN> E r) rapidly increases in amplitude above the direct gap recombina­tion, and at an excitation level of 4x 104 W /cm2 laser oscilla­tions occur on the A line.

increasing liw). It is, therefore, peaked as schematically illustrated in Fig. 2. If we let ao - 0 (L e., a very local­ized hole), Fg becomes very broad and we observe no structure from Fg in a r . Since Eh=1i2a;//2J.L""1i2ar//2mc' the variation of Fg with liw is determined by the magni­tude of E h• Specifically, the position of the peak in Fg lies between Er - Eh and E r • The quantity F4 is the sum of a constant background and terms proportional to the

1. Appl. Phys., Vol. 43, No. 12, December 1972

spectral density in Eq. (2.2a). The case of the N resonance above r, Le., Er<EN<Ex, is illustrated in Fig. 2. From Eq. (2. 12c), the N resonance peak in F 4 occurs at liw = EN - E h' Two effects occur when EN - Er :: E h. First, F4 becomes sharper since the width of IBE 12 is determined by the density of states at r. The other effect is that this N resonance peak in F4 reinforces the maximum in F g , so that the line shape be­comes more intense.

From the analysis of aN and C1'r in this section, we expect that strong laser operation is most likely to occur via an N-trap recombination "line" associated with either a resonance or bound electron localized near the N impurity for those alloy compositions such that IEr - ENI :: 50 meV. As described below, we have observed such behavior in In1_xGaxP : N, 0.6:s x:s O. 7. In addition, these results have been verified for GaAs1_XP x : N, where also it has been possible to show that resonance enhancement occurs in the range 1 E r - EN 1 :: 50 meV. 11 We also have evaluated the probability of radiative recombination of the bound or resonant electron with holes bound to acceptors such

2.10 2.15

;>, hll (InGa P) <f)

c Q)

C H

c 0 <f) <f)

E w

I 5.4

Energy (eV) 2.20 2.25 2.35

Inl-xGaxP' Te' N

InGaP x~0.67

ls 77°K

400 I Eg([)

I I 5.6 5.4

Wavelength (103 Ji)

FIG. 4. Photoluminescence spectra (77 "K) of In1_xGIlxP: N (x= O. 67). At low excitation levels weak band-to-band and strong A-line emission are observed (EN>E r ). As the excita­tion level is increased, the A-line emission increases because of the resonant enhancement of its recombination probability. At an excitation level of 6x 104 W /cm2 laser oscillations occur at wavelengths as short as 5560 A (2.23 eV, yellow-green).

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RESONANT ENHANCEMENT OF RECOMBINATION PROBABILITY 5139

as Zn and isoelectronic donors such as Bi. Besides the expected result that the recombination is weaker the further apart the electron and hole, these calculations bear out our major conclusions about the effect of the hole's localization upon the spectrum. That is, a very localized hole produces no resonant enhancement of the intensity of the N recombination line, although this line exists for EN> Er as well as for EN < E r . On the other hand, a very delocalized hole produces no N resonance line if EN>Er' We state the results of these calculations in the Appendix, for reference.

IIl.lnl_xGaxP:N EXPERIMENTAL RESULTS

The In1_XGa

XP : N crystals of this work are grown at

constant temperature from In-Ga solution essentially as described earlier, 2,12 except with the small modification of Ga being added to the In solution as a replacement for the previously used GaP source crystal. As before, 2,12

InP is used as source crystal which serves as the P supply for the In

1_

XGa

XP : N. The crystals are doped with

Te and with N, the latter supplied by GaN added to the synthesis ampoule. The donor doping levels of the crystals are determined by Hall measurements and the crystal composition x by electron microprobe analysis.

The experimental samples are sliced from the as-grown crystal and are etched to a thickness of -1 iJ. or less for this work, and then are cleaved and embedded into an In heat sink. 13 The sample excitation is supplied by the 5145-A output of a mode-locked argon laser. Note from the insets of Figs. 3 and 4 that the samples are opera­ted in the edge-to-edge (l s = 10-50 iJ.) Fabry-Perot cavity configuration, 12,14 rather than as layered flat cavities. 15 (Layered flat cavities exhibit higher Q's but unfortunately are limited in optical gain in the thin dimension of the sample. ) The sample configuration (i. e., active region) employed here for laser operation is similar to that of the thin layer of a p-n junction.

The results we obtain on n-type samples (Na=2-6X1017/ cm3

) are shown in Figs. 3 and 4. For the sample of Fig. 3 (x = 0.60) a noticeable separation exists between the photoluminescent output on E r == E,,( r) and on the N -trap transition. This separation and the assignment of E,(r) and N agree with absorption measurements (cf. Fig. 1, Ref. 2), which is also the case for the x = 0.67 sample of Fig. 4. The peak position of E/r) is displaced upward in energy relative to previous results2 because of band filling with electrons supplied by donors. For the sample of Fig. 3 at lower pump levels, stimulated emission begins first on E,.(r), but as the population across the sample is more fully inverted, strong laser operation occurs at higher energy [EN>E,(r)] on the N-trap transition. We note that this transition, depending upon pumping level, is 20-50 A wide and is much narrower than what we typically observe of an In1_pa

XP platelet

laser operating on E1(r). 12

A more interesting case is that of the sample (x = 0.67) which exhibits the spectra shown in Fig. 4. According to theory, as EN approaches E,(r) the narrow N-trap transition should be enhanced at the expense of the inter­band recombination transitions. This seems to occur since E.,(r), as assigned by absorption measurements, 2

is relatively weak compared to the N-trap transition.

Also, as already mentioned, the N-trap transition is quite narrow « 50 A, -20 A for some samples). At the highest pumping level shown, the sample lases abruptly (with slight "tuning" of the excitation on the sample) from the top and lower-energy side of the N-trap transition. The strong laser modes shown agree in spacing (- 8 A) with the 1-mil width of the sample [assuming (n-A dn/ dA) -7], 12 with the mode of shortest wavelength occurring at 5560 A (2.23 eV). In fact, this mode is only 80 meV less in energy than the nitrogen A line in GaP (2.310 eV, 77 OK), and is 52 meV higher in energy than the prominent NNI transition (2. 178 eV) in GaP. Furthermore, 5560 A is not the short-wavelength laser limit of In1 _xGaxP : N, which may in fact be closer to EN=E,,(r) "'2.27 eV (5470 A) as we shall describe in detail later. 16

One final feature of our results is worth mentioning: ordinarily problems are encountered in building junc­tions at alloy compositions near x = Xc where E.,( r) =E/X). At these alloy compositions (i.e., x-xc), indirect-minima donor states cause the direct-indirect transition to occur in essence prematurely, 17 which in turn causes carrier freeze-out effects in p-n junctions. For the case of the In1_

XGa

XP : N discussed here,

EN> E.,( r) and a high-energy recombination transition is possible (on EN) in alloys where Eg(r) < E,.(X) and the indirect-minima donor levels are above E,(r). This eliminates carrier freeze out on indirect-minima donor states, as, in fact, we have verified on p-n junctionslB

fabricated in the same crystal as that providing the samples of Fig. 4.

IV. SYNPOSIS AND CONCLUSIONS

Summarizing, we have presented model calculations and photoluminescence data on In1_xGa

XP : N which illustrate

the concept that not only do N traps cause resonance states at energies EN>Ep but also that if EN-E r these states "rob" the direct interband transitions of their recombination probability, creating a resonant enhancement of the intensity of the trap-assisted re­combination. Therefore, the oscillator strength which resided in a band of (unfilled) electron states for E > Er can be compressed into a narrow spectral range around fiw = EN associated with electron states that can be photo- or injection-stimulated. Consequently, we sug­gest that N-doped III-V ternary alloys (particularly In1 _ xGaxP) whose composition is chosen to make Er - EN are uniquely favorable candidates for efficient light­emitting diodes and semiconductor lasers.

Our calculations also have shown that for these effects to be optimized the hole must be neither too localized nor too de localized (i. e., a free hole). If excessive localization occurs, 1,2 there is no resonant enhancement of the intensity of the N line, contrary to experiment. If insufficient localization occurs, no resonant structure appears when EN>Er' also contrary to the observa­tions. We conclude, therefore, that our samples are characterized by the intermediate localization of the holes required to obtain both the resonance line for EN> E r , and the enhancement of its integrated intensity as EN-Er .

J. AppJ. Phys., Vol. 43, No. 12, December 1972

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5140 DUKE et al.

In our calculations, we neglected electron-electron interactions. 10 This neglect is reasonable since there is no evidence of threshold singularities at the (quasi) Fermi level in the experimental observations.

ACKNOWLEDGMENTS

The authors are grateful to G. W. Zack for help with Hall measurements and J. B. Woodhouse for electron microprobe analysis. They thank M. G. Craford (Monsanto) for many helpful conversations on N traps in ternary semiconductors and K. A. Kuehl, B. L. Marshall, Yuri S. Moroz, S. T. Romwever, and H. Corray for technical assistance.

APPENDIX

In this appendix, we first calculate the absorption coefficient for a resonant or bound electron and a hydrogenic (1s) hole bound to a Zn acceptor. We define the hole state as in Eqs. (2. 10), except that we make the replacement Il - mv. Let us assume that the Zn is located at R = 0 and the N at R = Ro. The bound contribu­tion to optical absorption is given by

) _~ nNO ( - Y ) exp(- Ro/ao) a N( w - w R~ Im[Ac(E N + iy) 1 (J(-!N _ a~2)4

X(Roao(K~ - a~2) - 2{1 - exp[(a~1 - KN)Ro]W

x O~EN-Ezn-nw), (Ala)

K~=2mc(Er-EN)/1i2, (Alb)

a3

:; 412 'lT2(m.)2 ao-Sm~3/2ao. (Alc)

The continuum contribution is specified by

ar(w)= (a2/w)nNnK",(~ + a~2t4 e(llw + E zn - E~

x{ 1 + 1f'(nw+EznW + 2 Re[f'(llw + Ezn)]

x sinK",Ro/K",Ro}' (A2a)

~:;2mc(nw+Ezn-Er)/n2, (A2b)

'( )= 0 Vo 2mc~ f E - (2'IT)31 _ VoAc(E) 1i2 Ro

x {Roao exp( - Ro/ ao)(J(-! + a~2) + 2[ exp( - Ro/ ao)

- exp(iKRo)]}, (A2c)

J(-!:;2m c(E-Er)/ll2. (A2d)

We next calculate the absorption coefficient for a hole bound to an isoelectronic donor (e. g., Bi) at R=O. 'fhe definition of the hole wave function is analogous to that of the electron in Eqs. (2.1), (2.2), and (2.4) except that we replace conduction-band parameters by the corresponding valence-band quantities and denote the Bi potential strength by Wo (i. e., Vo - Wo). We denote valence- (conduction-) band quantities by the subscript vee).

The bound electron state part of a g is (letting y - 0), for a concentration nBI of Bi,

a (_ d \ -1 (d ~ -1 aN(w)=::;- dE Av(EB1)J - dE Ac(EN)J

1. Appl. Phys., Vol. 43, No. 12, December 1972

x [exp( - KNRO) - exp( -KBRo)]R~2[EBI

-(m/m)(Er-EN)]-2o(EN-EBI-llw), (A3a)

~:; 2mflBtlll2, (A3b)

(A3c)

The corresponding continuum contribution is given by

+ 2 Re(f"(llw + Em)] sin(RqKo)/RqKo},

K~:; 2mc(llw + EBI - E r )/ll2,

as = (2ll2 /OmC)a4,

f"(E) = (O/4'IT) Vo[l- VoAc(E)]-l (2m/ll2Ro)

x [exp( -KBRO) - exp(iKRo)].

(Ma)

(Mb)

(Mc)

(Md)

*Work supported in part by the Advanced Research Projects Agency under Contract HC 15-67-C-0221, the National Science Foundation under Grants GH-33634, GH-33771 and GK-18960, and by the loint Services Electronics Program under Contract DAAB-07-67-C-0199.

tpresent Address: Xerox Research Laboratory, Webster, N.Y. 14580. lpresent address: General Motors Technical Center, Warren, Mich. 48090.

ID. R. Scifres, N. Holonyak, lr., C. B. Duke, G. G. Kleiman, A. B. Kunz, M. G. Craford, W. O. Groves, and A. H. Herzog, Phys. Rev. Lett. 27, 191 (1971).

2D. R. Scifres, H. M. Macksey, N. Holonyak, lr., R. D. Dupuis, G. W. Zack, C. B. Duke, G. G. Kleiman, and A. B. Kunz, Phys. Rev. B 5, 2206 (1972).

3R. A. Faulkner, Phys. Rev. 175,991 (1968). 4B. W. Hakki, 1. Appl. Phys. 42, 4981 (1971). 5G. F. Koster and 1. C. Slater, Phys. Rev. 95, 1167 (1954). 6J. Callaway, 1. Math. Phys. 5, 783 (1964). 7D. M. Newns, Phys. Rev. 178, 1123 (1969). 8M. L. Cohen and T. K. Bergstresser, Phys. Rev. 141, 789 (1966). 9V. L. Bonch-Bruevich in Optical Properties of Solids, edited by J. Tauc (Academic, New York, 1966), p. 331.

IORepresentative references on threshold singularities are G. D. Mahan, Phys. Rev. 163, 612 (1967); P. Nozieres and C. T. De Dominicis, Phys. Rev. 178, 1097 (1969); and David C. Langreth, Phys. Rev. B 1,471 (1970).

liN. Holonyak, Jr., R. D. Dupuis, H. M. Macksey, G. W. Zack, M. G. Craford, and D. Finn, IEEE J. Quantum Electron. (to be published).

l2D. R. Scifres, H. M. Macksey, N. Holonyak, Jr., and R. D. Dupuis, J. Appl. Phys. 43, 1019 (1972).

13N. Holonyak, Jr. and D. R. Scifres, Rev. Sci. Instrum. 42, 1885 (1971).

14N. Holonyak, Jr., D. R. Scifres, R. D. Burnham, M. G. Craford, W. O. Groves, and A. H. Herzog, Appl. Phys. Lett. 19, 254 (1971).

"N. Holonyak, Jr., D. R. Scifres, H. M. Macksey, and R. D. Dupuis, Appl. Phys. Lett. 20, 11 (1972).

16H. M. Macksey, N. Holonyak, Jr., R. D. Dupuis, and J. C. Campbell (unpublished).

17N. Holonyak, Jr., C. J. Nuese, M. D. Sirkis, and G. E. Stillman, Appl. Phys. Lett. 8, 83 (1966); M. G. Craford, G. E. Stillman, J. A. Rossi, and N. Holonyak, Jr., Phys. Rev. 168, 867 (1968).

ISH. M. Macksey, N. Holonyak, Jr., D. R. Scifres, R. D. Dupuis, and G. W. Zack, Appl. Phys. Lett. 19, 271 (1971).

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