186 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-4, NO.

4, APRIL 1968

Power Efficiency and Quantum Efficiencies of

Electron-Beam Pumped Lasers

Abstract-The object is to identify and assess factors and mech- Cp Heat capacity(unit volume)
anisms that control the conversion of electron-beam power into
coherent light through excitation of a semiconductor laser cavity.
hv Stimulatedphoton energy
First, weexamine the question of pump power losses associatedwith liw, Optical phonon energy
electron backscattering andpair creation. It is shown that power re- a Photon-absorption coefficient
tention andionization yield reflect target characteristics (atomicnum-e Radiation-ionization energy
ber and bandgapenergy) only. The external quantum efficiency, { Adhoc
conversion
efficiency
which is best expressed a s a product of quantum yield, coherence
ratio, and escape probability, involves two parameters: pumping
vin Internalquantum efficiency
ratio and output coupling. This leads to a straightforward optimiza- vex Externalquantum efficiency
tion procedure. Heating effects are analyzed in terms of a different- vd Differential quantum efficiency
ial quantumefficiency and areshown to degrade the saturation value K Thermal
conductivity
of the efficiency by a factor roughly proportional to the pulse rise .$ Overall power
efficiency
time, if adiabatic conditions hold. These considerations are illustrated
7 Pulse risetime
using power-efficiency figures reported for CdS, CdTe, and GaAs
lasers; it is demonstrated that thephoton-loss coefficient of excited w Heat-load
factor
"perfect" CdS must be less than1.5 cm-1, a t 4.2OK.
Subscript 0 refers to ambient temperature.
Subscript th refers to t'hreshold conditions.
GLOSSORY
OF SYMBOLS
Superscript 0 refers to zero-output coupling.
Depth-dose function Superscript * refers to resonator unit surface.
Bandgap energy
Photon-escape probability I. INTRODUCTION
Coherence-ratio factor
Power-retention factor
Resonator cavity length
Characteristic length
P ULSED electron beams provide an exceptionally
versatile tool for inverting the carrier population
in semiconductors and,thus, for achieving laser
action. I n this regard, electron-beam pumping hasdemon-
Total photon production rate strated much technological promise.''] With the develop-
Coherent photon emission rate ment of powerful kiloelectronvolt guns, it has become
Photon population per mode imperative to assess the factors that cont'rol the efficiency
Minority-carrier distribution of converting electrical power into coherent light by this
Incident beam power procedure; the present paper concerns that very question.
Deposited beam power Start'ing from fairly fundamental considerations, we will
Threshold beam power attemptto describe the electron-beam pumped (EBP)
Total optical power generated laser in a manner such that theparameters that determine
Coherent optical power generated the peak power output for pulsed operation canbe readily
Coherent opt'ical power emitted identified and understood. Our goal is to relate laser light
Pumping power rat'io output and beam power input by making allowances for
Radiative recombination rate the losses that occur at successive stages of the conversion
Nonradiative recombination rate process. In addition, it is hoped that this analysis will
Spontaneous contribution to rrad clarify the effect of varying some of theseparameters
Practical range and, hence, that it may lead to guidelines for designing
Bethe range optimized semiconductor structures from the standpoint
Bombarded resonator area of the emitted coherent power or t'he external quantum
Penetration depth efficiency.
Atomic number For orientation purposes the reader is referred to Fig.
Pair-generation rate 1,"' which shows a schematic of the sequence of events
leading from an electron beam to a laser beam. First,
Manuscript received January 3, 1968; revised January 19, 1968.
the transfer of beam energy to a solid target comtitutes
The author is with RaytheonResearch Division, Waltham, Mass. the end result of a complex process best described by the
KLEIX : EFFICIENCIES OF ELECTRON-BEAM PUMPED LASERS 187
1
ELECTRON
pumped GaAs lasers is discussed in Section V and shown
BEAM to accord with presently available experimental evidence.
/ 7 I n addition, wewill attemptto derive credible upper
( K) r
BACKSCATTER
limits for the optical power that can be emittedfrom
LOSSES
’r gallium arsenide units operated a t energy-deposition rates
4
f i
DEPOSITED
just below theanticipated threshold for catastrophic
ENERGY failure; the so-called “radiating-mirror” excitation
I
I ’fl---l PHONON
scheme‘31will also be considered in this context.
LOSSES EFFICIENCY
11. OVERALLPOWER
ELECTRON-
We define the overall power efficiency of an EBP laser
as the ratio of the coherent optical power emitted from
L==L--l
HOLE PAIRS

QUANTUM
both end faces of the resonator to the electrical power
carried bythe incident electron beam: E = Pout/Pin.
LOSSES
I n terms of the photon-emission rate ( X ) , the pair-
LASER
generation rate (a), and thedeposited beam energy ( P d e p ) ,
this ratio can be expressed as

Fig. 1. Block diagram of major “events”occurring inthe con-

version of a n electron beam to a laser beam. Loss mechanisms are
identifiedon the right. Quantum losses include nonradiative
recombinations, incoherent photons, and reabsorption losses. I n this manner we express the overall powerefficiency
as a product of three factors corresponding t o the three
spatial distribution of specific energy losses dE/dx, where stages of the excitation mechanism as delineated in Fig. 1.
E is the residual primary-electron energy a t a penetration These factors are as follows:
depth x . I n this context we emphasize that a substantial
1) The externalquantumeficiency qex = X / % or the
fraction of the incident electrons can be backscattered,
number of stimulated photons emitted per electron-hole
whichis of consequencebecause it reduces the power
pair;
available for laser excitation and thusdegrades the overall
2) Theionization-yield factor E G / e = %(hv)/P,,, since
efficiency. Thenextstage concerns the dissipation of
e measures the average amount of deposited beam energy
radiationenergy deposited in the target. This involves
consumed per pair’;
primarily impact ionization and phonon generation by
3) T h e power-retention factor K = P d , , / P i n which
“hot” carriers released inthelattice;interms ofeffi-
relates to backscattering losses.
ciencies, the problem reduces to estimating how much
energy is lost to phonons in the course of creating a single Backscattering losses involve all primary electrons
pair of thermalized holes and electrons. Finally, these that reemerge from thetarget subsequent to one or
nonequilibrium carriers may recombineacross the gap several elastic or inelastic collisions. Much work has been
and produce laser light, if sufficient recombinations occur done regarding electron fluxesreflected from various
viastimulatedtransitions. As inthe case of junction targets, but few investigators have actually recorded the
devices, an “external quantum efficiency” characterizes energy distributionof backscattered electrons in a context
the conversion of electron-hole pairs into emittedphotons, relevant to electron-beam pumping (10 5 E , 5 100 keV).
whichimplies consideration of the relative probability The author has considered this problem elsewhereL4’and
of radiative recombinations, of internal losses resulting has calculated backscattered energy fractions on the basis
from reabsorption, and, of course, pumping power in of a simple model that bypasses the electron-transport
relation to threshold power. equation; here, we simply wish to outline guidelines and
Adhering to this pattern, wewill first examine the results. At normal incidence, the electrons are presumed
question of pump powerlosses associated with electron to penetrate straight into the target down to the “depth
backscattering and pair creation; morespecifically, we of complete diffusion,” from whence they diffuse randomly
will determine what fraction of input powereffectively in all directions while losing energy at an exponentially
contributes to the excitat.ion(Section 11). Section I11 decreasing rate along any radius vector of the “sphere of
deals with quantum efficiencies, in other words, with the excitation.” This model, which includes a suitably defined
factors that control the light output of a semiconductor absorption coefficient p, yields
laser cavity. I n this connection it is convenient to consider
the differential quantum efficiency, since we must ex- APin/Pin= $[l-I- (pR,)z,/21 exp [ - ( p I z B ) z ~ I , (2)
amine how bombardment-induced heating may detract
from ideal performance characteristics (Section IV). The 1 W i assume that the photon energy matches the bandgap of
theoretical powerefficiency situation in electron-beam the active material.
 I88 IEEE JOURNAL OF QVANTUM ELECTROKICS: APRIL 1968
where zD = 4/(2 + 4) is the depth of complete diffusion I.o I I I
iI
,I I O

measured inunits of integratedpathlength (or Bethe

range R,). By and large, calculated backscattering losses
agree with experimental results, whichshow very little
energy dependence in the decaltiloelectronvolt range, on
assuming an effectively Z independent p R B product of
10 =t 1. Of direct relevance here is the power-retention
factor K = 1 - APin/Pi,whichis plottedagainst Z
in Fig. 2(a).Thisfactor becomes quiteimportant for
targets of moderate or high atomic number; we conclude
that the usual EBP laser materials do not retain more 0 20 40 60 80 0 I 2 3 4

than 60 to 80 percent of the incident beam power. ATOMIC

ENERGY.EG
8ANOGAP
NUMBER.2 lev1

In a semiconductor, deposited radiation energy gives Fig. 2. (a) The fraction of incident beam energy actually deposited
rise to a complex sequence of events, which makes phase 2 in the target. (b) The fraction of deposited beam energy “stored”
in the form of thermalized electron-hole pairs. (a) and (b) are
of the excitation process fairly difficult to analyze. For- basedon analytical work described previously[41 and reflect
tunately, efficiency considerations require only informa- the present experimental situation.
tion about the generation rate of the electron-hole pairs,
and we have already seen that this is best done in terms efficiency of an electron-beam pumped CdS laser cannot
of radiation-ionization energies E.’ Experimentaldeter- possibly exceed 25 or 26 percent. More important, our
minations of E yield a consistent picture in the sense that calculations confirm that Hurwitz’s measurement reflects
radiation-ionization energies do not reflect features of the “ideal” operational conditions, that is, an external
incident particle but reveal a definite correlation with quantum efficiency of one (or almost), and hence leaves
the target’s bandgap. I n effect, it has been that no room for further improvement.
the amount of beam energy dissipated per electron-hole
pair is the sum of three contributions: the intrinsic band- 111. EXTERNAL QUANTUMEFFICIENCY
gap (ZQ),optical phonon losses (ho,), and the “residual” Equation (4) relates the concept of an external quantum
kinetic energy (9/5)EG. Thus, we take it that E can be efficiency to the overall power efficiencyof an EBPlaser.
related to E , simply by writing Since vex = %/a, and thus characterizes the performance
+
‘2 E , r ( h d ,
E = (-+ (3) of an optical cavity from the point of view of stimulated
emission, we may first examine the rapport with coherent
where r is to be treatedasan adjustableparameter. and incoherent photon production, which suggests to
Within the limits of experiment,aluncertainty, the phonon mite
loss term remains roughly constant [0.5 5 r(hwR) 5
1.0 eV],whichimplies thatthe ionization-yield factor
( E G / e )is essentially a function of the bandgap andbehaves
as depicted in Fig. 2(b). We note that losses occurring I n effect, we propose to discuss the quantum efficiency
in connection with pair-creation alone may exceed 90 in a manner patterned upon the approachadoptedin
percentin the case of narrow bandgap materials; for connection with (l),that is, by expressing vex as a product
bandgaps of more than 2 eV, ionization losses approach of three factors. These factors are as follows:
the commonly assumed figure of 66 percent. 1) T h e escape-probabilityfactor F = @out/@coh which
At this point, it is tempting to assess the 26.5 percent reflects internal losses resulting from the reabsorption
powerefficiency of Hurwitz’s CdS especially
since of stimulated photons;
this disclosure has stimulatedinterest in theultimate 2 ) T h e coherence-yatio factor G = or net
capability of such devices. I n the light of the foregoing, fraction of photons generated through stimulated transi-
the power efficiency of an EBP laser is tions;
= K(EG/e)?Iex, (4) 3 ) Theinternalquantzcm efJiciency ~l~~= N / a or total
number of photons per injected electron-hole pair.4
and there is little one can do to enhance the power reten-
tion or the ionization yield.3 For CdS an appropriat’e In terms of rrad and r,,, the radiative and nonradiative
mean value of 2 is 32, which points to K = 0.75 f 0.02; recombination rates of injected pairs, we have
the ratioE G / t , on the other hand, amountsto 0.325 f0.01,
since EG = 2.5eV. Consequently, the overall power
via = I +
1 (mr/rrad)l-’. (6)
I n a fully lasing device, or more precisely, if the quasi-
Fermi levels are sufficiently far apart throughout the
2 The quantity E is often referred to as “energy required to form
a n electron-hole pair;” this is not only incorrect but highly mis-
active region, the photons are generated a t a rate T , , ~ e
leading, in view of the rapport with threshold energies for impact
ionization. Basov et al. refer to this number as “quantum yield,” which
3 This in contrast to ’lex, which collects a number of interrelated may be preferable since their terminology better identifies efficiencies
items and may indeed be optimized, as we shall see Section 111. with coherent photons.
KLEIN : EFFICIEKCIES OF ELECTROK-BEAM PUMPED LASERS

+
~ ~ ~N ) ,( where 1 rsp designates the recombination rate
via spontaneous radiation and N is the average photon
population of thecavity modes.“’ It follows thatthe
quantum yield (6) is essentially determined by the
magnit.udeof rnr/rs,,and the density of stimuhted photons.
Since N increases more or less linearly with pump power,
we expect T~~ to saturate a t some level of excitation. This
level depends critically upon the ratio T,,/T,,, in other
words, upon the nature of the recombination process a t
the threshold for laser action. The situations of interest
here are those typical of “good”laser materials, or
materials whereacross-the-gap transitions withphoton O
. ,..’.
W 1 I I I I I
emission aremuch more probable than nonradiative I 2 3 5 10 20 30 50 100

processes; accordingly, all subsequent workassumes an PUMPING POWER R A T I O ~ F & / P t ~ l

internal quantum efficiency of unity, whichwebelieve Fig. 3. Fraction of the total optical power generated incoherent
to be consistent with much of the experimental evidence form (the coherence-ratio factor) as a function of the pumping
power ratio. “Actual” distributions are t,hose considered repre-
about GaAs sentative of the minority-carrier distributionin electron-beam
The question of the degree of coherence ina semi- pumped (EBP) units and depend upon the ratio R / L of particle
range and diffusion length.lg1 Theupperinsertillustrates a
conductor laser cavity G = has been investigated conventional EBP geometry: electron beam incident upon plane
by Klein and Lavine.”’ If it is assumed that, given any (2, y), penetration along the axis z, light emission along the
dlrectlon 2.
layer dz (see Fig. 3), the photons in excess of the threshold
population all contributetoand if thedepth depend- perature, threshold occurs m-hen the peak pair-generation
ence of photon production reflects the steady-state dis- rate exceeds
tribution p ( z ) of bombardment-induced minority carriers,
t’hen
?Ith = a::’(l Lc/L), +(10)
where L J L = In ( l / d R T )(aL)-‘ is the ratio of end
G = /’*p(x)
=I
dx - (l/PR)p(z,)(z,- x , ) . (7) losses to absorption losses, or output coupling of the
resonator,[12’ whereas 3:;) relates to material properties
Here, z1 and z2 are the limits of the active region while zo only.[21On this basis we may introduce the notion of a
locates the peak of p ( z ) , which is taken to be normalized pumpingratio for zero-output coupling, (PR)’ = Pi,/Pib)),
(J: p ( x ) dx = 1 ) ; note that PR stands for “pumping and hence write
ratio,” or the ratio P i n / P t h . l l o l First consider the case
of a homogeneous distribution extending over a distanceR.
This case [ p ( z ) = 1/R and (zz - z,) = R] leads to
which will allow us to “decouple” the geometry from the
G = 1 - 1/PR (8)
other parameters. I n this context we find it most ap-
and thus to the “standard” equation of injection lasers propriate toexpress the escape probability F of stimulated
Pout= {(Pi, - P t h ) , where { represents an ad hoccon- photons, or fraction of the coherent radiationemitted
. ~ an exponential distribution (7) yields
version f a ~ t o rFor through the end faces, in the following manner:
G = 1 - (l/PR)[l + In ( P R ) ] , (9) F = [l + L/LC]-’. (12)

which fits Basov’s treatment of optically pumped GaAs I n Basov et uL1’] it is seen that for practical values of a ,
lasers.”’] With EBP devicesweconsider it a fair ap- L, and z/a
this expression does not differ much from
proximation to write p ( x ) = [exp ( - z / R ) - exp (-x/L] - Stern’s‘’’ escape probability, albeit both are approxima-
[ R - LI-’, if L designates the diffusion length of the tions. I n addition, (12) clearly demonstrates the essential
minority carriers. Fig. 3 illustrates the result of numerical role of the characteristic length LC and,bythe same
integrations performedfor a wide range of R / L ratios; token, of the optical absorption coefficient a; note that,
evidently, the nonuniform minority-carrier density of in the active region, optical losses arise from free-carrier
EBP lasers degrades theirperformance unless they are absorption as well as crystal imperfections.
operated at very high pumping power ratios. I n summary, since
The pumping ratio PR = P i , / P t h involves the power
= FqinG, (13)
P,, needed to achieve lasing under the conditions that
prevail at an input power level Pin.At any given tem- and since nonradiative recombinations are presumed
negligible a t high excitation densities, we conclude that
6 In this connection we note that Gvi, represents an internal
the external quantum efficiency of a semiconductor laser
quantum efficiency for stimulated emission. In a. fully lasing device, is primarily a function of pumping ratioandoutput
therefore, the following holds: G = [l +
l/IV-l. A confrontation coupling. Furthermore, it is clear that the cavity length
with (8) then shows that N = P R - 1, in the case of homogeneous
excitation. controls the lasing efficiency at any input-power level.
 190 IEEE J O U R N A L O F QVANTUht ELECTHOSICS, APRIL 1968
at high beam-currentdensit’ies,‘*41 the discussion may
begin by considering differential quantities such as

We call cZ(vexPin)/dPin
a “differential quantum efficiency”:

no’ 01 0 0
REDUCED CAVITY LENGTH ( L / L c I

Fig. 4. External quantum efficiency of a semiconductor laser as a Actually, since we are interest’edprimarily in the behavior
function of the reduced cavitylength, or reciprocal output-
coupling parameter. The calculations refer to input power levels at high excitation densities (vi,, = I), and since
measuredinzero-output coupling threshold units ranging from
three to infinity, and assume a coherence-ratio factor typical of
electron-beamexcitation (see Fig. 3). Kotethat t,he internal
quantum efficiency is taken equal to unity throughout, the range
of excitation.
it is a straightforward matter to establish the following:

Fig. 4 illustrates this for the case of electron-beam pump-

ing: Optimum coupling simply means optimized tradeoff
between opticalabsorption losses and threshold power which allows us t o draw some important conclusions.
losses, at a given zero-output pumping ratio.Strictly 1) Consider an ideally simple situation for which G =
speaking, an external quantum efficiency of 100 percent 1 - 1 / P R (homogeneous excitation) and dP,,,/dP,, = 0
would require an infinitely ‘‘small” cavityoperated a t (temperature-independent threshold). The dlfferential
an infinit’ely “large” input.The question then arises quantum efficiency becomes simply vd = F , which implies
whether this is compatible witha measured vex N” 1, straight watt-ampere characterist,icsthroughout the range
as seems t o be the case with Hurwitz’s‘“ CdS laser (see of e~citation.~
Table I). Since the reported power-efficiency figure refers 2) Assumenow atemperature-dependent threshold;
to a beam current I = 2001,,, it appears that, indeed, inthat case me have vd = F(l - dPth/dPi,), where
may exceed 90 percent if L / L c 5 0.1 and thus(PR)’ 2 dPth/dPi, = (clP,,/dT) (clT/dPi,) reflects the threshold-
2200; we infer that free-carrier absorption inCdS probably power increase induced by heating and may cause serious
corresponds to an (Y of 1.3 cm-’ or less.6 deviations from linearity.
3) Consider a situation typical of electron-beam pump-
IV. DIFFERENTIAL QUANTUM EFFICIENCY
ing, that is, G = exp ( - a / P R ) , where 2.5 a < <
3.0,
At a fixed input power level (PR)’, lasing can be which approximates the coherence-ratio factor for pump-
achieved only with a resonator of a certain minimum size ing ratios larger than 2.”’ The corresponding differential
(see Fig. 4); with larger cavit’iesthe output first increases, quantum efficiency is
goes through a broad maximum (optimum coupling), and
then drops off as reabsorption losses predominate. For a
given cavity, on the contrary, the efficiencyalways in-
creases vith bean1 current until saturation is reached as which will now be discussed in detail.
(PR)’ 3 m . Evidently, this will not hold if substantial Since our main interest is peak power emission, we shall
heatingtakes place, because higher temperatures may focus on heating effects associated with short-pulse mode
not only lower the internal efficiency but may raise the operations. Inthe light of materialpresented inthe
threshold power. To some extent, pulsed operation alle- Appendix it appears that, for low duty cycle operations,
viates the heating problem. Still, it is important to derive maximum emission occurs atthe end of the current
realistic upper limits for the power efficiency, by taking buildup,on the condition thatthe pulse risetime T
heating into account. obeys the following inequalities’:
Since output versus input characteristics of EBP lasers
usually reveal heating effects through slopes that decline
-_
6 Tt. i s intwest,inp
__.
___”-__ ~ ..cnmnare
t,n
.- . . ~ this
~ the cy of 2.5 em-1 obtained
to ~
by Basov et al. [I11 for optic& -excited pure GaAs. However, these
authors claim thatfree carriers per se(the nonequilibriumpopula- ’ Incidentally,judging fromour analysisone may wonder
tion) did not contribute more than 0.4 em-’. It would appear that whether the usual “efficiency” formula V d = F v i , does not tacitly
a similar situation holdsin CdS, with the added advantage that assume 9;. = 1.
“nonresonant” losses may even be smaller thanin GaAs; Nicoll’s 8 Note thatbeam voltagerepresentsacritical parameter since
latest experiments [13l indirectly confirm theseobservations. the practical range of the electrons increases as Eo1.65.[41
KLEIN : EFFICIENCIES O F ELECTRON-BEAM PUMPED LASERS 191

WithCdTe, for instance, it isseen that beam-cur-

rent pulses of 150-ns duration and 90-kV voltage, as in
V a v i l ~ v ’ s [recent
~ ~ ’ experiment, should be quite adequate
since the threshold exhibits a weak linear temperature
dependence of about 4 kW/cm2/”K in the 80” to 300°K
range. In that case (adiabatic heating), the instantaneous
temperature rise a t peak output amounts to W/R(Cp),
where W represents the integrated heatflux for an elapsed
time r :

W = w(Pin/&) 1‘
0
tdt = w(Pin/X)(7/2) (20)
( a) (b)
Fig. 5 . (a) An illustration of calculations presentedin Appendix
regarding “intrapulse” events. (b) Shows how differential quantum
efficiencies deviate from the theoretical saturation value F for
[see (29)]. It thus follows that anincrease in themagnitude finitevalues of thethermal response parameter e. Adiabatic
of the pump power raises the temperature of the resonator heating is assumed during the pulse rise period T .
at the rateof
TABLE I
(dT/dPin), = ~ ( ~ / ~ ) [ X R ( C P ) ] - ~(21)
, E B P LASEREFFICIENCIES
ANALYSISOF REPORTED

at thevery instant of maximum laser emission.

Returning now to (18), we conclude that in thecontext I Crystal

of peak power emission

GaAs
I CdS CdTe
(HurwitzLG1) (Lavine[17]) (Vavilov[Ul)
Power Efficiency
g(percent) 1 26.5 -5 1.1
where 8 = W T / R ( C p ) characterizes the thermal response
of the target; this presumes adiabatic heating, a linear
Beam Voltage E,(kV)
Temperature To( OK)
’ 60
4.2
30
77
90
82
~

function P t h ( T ) ,anda relatively weak P,, versus Pin Cavity Length L(cm) 0.05 0.05 0.052
dependence (dPth/dP,, < 4). Now consider the situation Power Retention
a t high power inputs Pi,>> Pth(To),where To is the K(percent)
Ionization Yield
75 75 67.5
specimen’s temperature before application of the pulse. E@/e(percent) 32.5 30 30
Since dPth/dPi, remains constant, the reader can easily Quantum Efficiency
Vex(Percent) 100 -22 5.5
convince himself that
Pumping Ratio PI? 200* 3.2* >8.3?
Coupling Parameter L,/L +m 1.25 0.07 i 0.015
Absorption Loss m(cm-1) <1.5 18t 42 f811

and thus that, in a first approximation, the differential * Assumes a temperature-independent threshold duringoperation.
f Derived from the differential quantum efficiency following (24).
quantum efficiency of a highly excited EBP laser is 1Obtained from the threshold current versus reciprocal length
simply’ de endence.
[ Includes, presumably, contributions from passive (noninverted)
reglons.
lim [?dl
Pin-+CC
yields an effective absorption coefficient as listed in
= F exp [- (u/2)(dP;*,/fl),B]= lim (24) Table I.
Pi n--

Fig. 5(a)illustrates (24) for threshold versus temper-

V. APPLICATION TO GaAs

ature slopes of up to 10(kW.cm-2)/”K: Given a finite Fig. 6 exhibits calculated power efficiencies for electron-
risetime (0 # 0), and assumingadiabatic heating, the beampumpedGaAs lasers with resonator dimensions
quantum efficiency falls below its “ideal” saturation ranging from 0.01 t o 10 times the characteristic length,
value FJ in accordancewith the curves drawnin that and for inputs of up to lo3 times the threshold pori-er of
plot. Take V a v i 1 0 v ’ s ~CdTe
~ ~ ~ laser, for example,which the “equivalent” zero-output cavity. The calculation as-
has a power efficiency of 1 percent a t high beam-current sumes zero-risetimepulses and involves a power-reten-
densities, or a limiting external quantum efficiency of tionfactor of 75 percent, an ionization-yield factor of
5 percent, a t an ambienttemperature of 82°K.With 30 percent, an intrinsic quantum efficiency of 100percent,“
, ‘ ~ ~ ~photon escape probability as in (12), and a coherence-
R = 17 micron^,'^' and (Cp) = 0 . 9 5 ( J . ~ m - ~ ) / ” K it a
turns out that 8 5 O.OG”K/(kW.cm-’) for pulse risetimes ratio term approximated as
of up to 150 ns, and hence that 0.05 5 F 5 0.08, which
lo An q in of 100 percent at t h e threshold for laser action requires
donor or acceptor dopings of the order of 5 X 1017 t o 3 x 10‘8 cm-3
9 Note thatif peak outputoccurs at t = 7 , lim ( P R )ie always > 2. and, presumably, operation at liquid-nitrogen temperature or below.
102 IEEE JOURNAL OF QUANTUM ELECTRONICS, APRIL 1968

Damage Threshold
For "laser-grade"GaAs(10 5 a 5 20 ern-.'; l l i n = l),
this wouldcorrespond to about 1 x loz5 pairs/s/cm3;
in other words: EBP laser action in doped GaAs requires
a minimum energy-deposition rate of 5 t,o 10 MW/cnl3.
Evident'ly, operation a t much higher excitation densities
exposes t,he material to dynamic loads t,hat may result,
in catastrophic failures. I n fact, it has been demonstrated
that semiconductors such asGe or InSb fail through
brittle fracture when exposed to pulsed high-energy elec-
w
tron environments that deliver doses of a few megarads
1 I I I , 1 , )
in time intervals of20 to 40 ns.'''' Regarding Gahs, it is
IO IO0 io00 remarkable that Basov el aZ.["' report damage induced
ZERO-OUTPUT PUMPING RATIO(PR), by incident light fluxes of20 MW/cm2 striking in pulses
Fig. 6. Calculatedoverall power efficiency of electron-beam
of 40-11s duration; from the measured absorption coefficient
pumped GaAs lasers with resonator sizes ranging from 0.01 to this corresponds to surface deposition rates of 1 GW/cm3
10 times the characteristic length, and for input power levels of and doses of 40 J/cm3, and hence shows qualitative agree-
1 to 103 times the threshold power of an equivalent, infinitely
long cavity. The experimental point refers t o data reported ment with Oswald's[181observations at room temperature.
elsewhere.['?l The damage-threshold"bracket" assumes limiting We conclude that, in the usual 10- to 100-ns regime, the
energy-deposition ratesontheorder of 1 GW/cm3; theinset
illustrates power efficiency and power output of a hypothetical threshold forpulse-induced fracture of GaAsshouldbe
but conventional GaAs structure wit,h free-carrierabsorption of the order of 1 GW/cm", which would imply that EBP
losses oi 11 cm-1.
operations at (PR)"s of 100or200 may destroy the
crystal.'*
lex' [ -
+
2.75(1 L./L']
(PR)'
for (PI?)"> 2(1 + Lc/I,), (25)
Pour .fixer les iddes, we now- consider a "system" as
specified in the inset of Pig. 6. Since the peak pumping
power densities are
G=! 0
i for (PI?)" = 1 + IJL.
Three comments are in order:
where Dm,, = 2.5 at themaximum of the depth-dose func-
1) The overall efficiencyincreasesorlevels off with t i ~ n , ' ~it' follows thatinput parameters of thatsort
increasing pump power. This is in striking contrast tojunc- demand a beam voltage of perhaps 100 kV and presume
tion devices, for which the powerefficiency a t higher a zero-coupling threshold-excitation density of about
pumping ratios quickly falls off because of the quadratic 10MW/cn13. The relevant powerefficiency and power
dependence of ohnic losses on current. output obtained witha conventional laser structure
2) The best powerefficiencyrecorded at the author's ( a = 11 c171-l) having resonator lengths ranging up Lo
laboratory amounts t o about 5 percent and refers to a 1 ern are plotted in Fig. 6 (inset); of course, we are con-
2 x 10" specimen of n-type GaAs fa,bricated into sidering here a hypothetical situation and yet, significant
a 20-mil cavity and excited a t a P R of 3.2."" Taking output isachieved only at the cost of low efficiency.
a = 18 cm-' as derived from the dependence of lasing An ingenious remedy for this state of xffairs consists of
threshold on reciprocal length, Lavine's measurement adopting a "radiating-mirror" type geometry.'"' For
reflects a [(PR)' = 7.2; L/Lc = 0.81-type situation and instance, with a GaAs deposit of 25-micron thickness (the
fits remarkably well into the pattern of Fig. 6 (see also practical range of 100-kV electrons), it turns out that
Table I). a 5 of 10 percent could be obtained a t a (PR)' level of
3) At a given input power level, there is an optimum 100, if the layer had an oe of .20 cm--l, or in other words,
cavity size 0.1 5 L/Lc 5 1.0 in accordance with the approximately 2 x l0ls d o n o r s / ~ m ~ . [ * ~ principle, ~I n
indications of Fig. 4. Since pumping ratios in excess of, lasing would set ina t a beam power level of 210 kW/cm2.
say, 200 may well be ruled out (see next paragraph), it At 1 PI/IW/cm2, andassumingproper heat sinking, a
would appear that an efficiency f of about 18 percent "mirror" of 1 cm2 maythus deliver laser pulses of 100-kW
probably represents an extremeupper limit-extreme peak power, if a suitable electron gun (100-keV, 10 A a t a
because Fig. 6 does not include the degradation normally uniformbeam density of 10 A/cm2, plus nanosecond
associated with finite pulse risetimes. pulse risetimes) can be secured.
At low temperatures, and in the absence of end losses,
lasing occurs when the pair-generation rate per unit 11 It could be, however, that this will not necessarily hold with
volume exceeds the threshold value'" "pure" GaAs (a = 2.5 ern+).
KLEIN : EFFICIENCIES OF ELECTRON-BEAM PUMPED LASERS 193

VI. CONCLUSION tions summarized in Fig. 6; it thus would appear that,

The overall powerefficiency of an EBP laser is best ultimately, EBP laser efficiencies of perhaps 10 or 20
expressed as a product of three factors: E = K(EG/€)qex, percent, in conjunction withmultikilowattpeak power
which reflects the conversion process delineated in Fig. 1. outputs, will be achieved with GaAs barring catastrophic
1) The power-retentionfactor K involves pump power failures such as pulse-induced fractures.
losses stemming fromthe backscatter of incident electrons.
APPENDIX
In the light of available data, K shows very little energy
dependence in the range of interest here, though this may For the purpose of delineating EBP laser action on an
have to be ascertained through further experimentation. “intrapulse” basis, we consider uniform energy-deposi-
2) The ionization-yieldfactor EG/e incorporates losses tion profiles and ignore minority- carrier diffusion effects;
originating from the phonon production that accompanies in other words, we assume
excitat’ionas well as thermalization of .electron-hole pairs.
pout = {(Pin - Pth), (28)
These losses are very substantial (at least 66 percent of
the deposited beam power); fortunately,theycan be where { represents a fixedconversion efficiency [c =
described in terms of radiation-ionization energies E , which K(EG/€)F, if vin = 11. Input poweris taken to evolve
vary linearly with the bandgap and are reasonably well as drawn in Fig. 5, or
under~tood.‘~’ 3) The external quantum eficiencyvex relates
the power efficiency to the performance characteristics of
the optical cavity. Weemphasize that, contrary to the
situation in junction devices, there is a direct correlation
between E and vex. that is, we assume the pulse to be made up of a linear
The external quantum efficiency, in turn,involves three edge with a risetime T followed by apeak input plateauPin.
factors. 1) The internalquantumeficiency vin which we For simplicity, we also assume that within the timeperiod
take equal to unity since we are primarily concerned with of interest, adiabatic or nearly adiabatic conditions hold,
high-excitation densities. 2) The coherence-ratiofactor G which requires that theheat-diffusion length 4 5 (~t/Cp)’”
whichis the net fraction of internalphotons that are remain much smaller than the practical range R of the
stimulated a t a given pumping power ratio, and depends electrons.[20’In that case, the temperature of the excited
on the character of the minority-carrier distribution region rises according to
(see Fig. 3). 3) The photon-escape probability F which we
propose to express by making use of the output-coupling
parameter L J L or ratio of end losses to internal losses.
We conclude that the external quantum efficiency of an
where the heat-load factor w accounts for that fraction
EBP laser is simply afunction of pumping ratio and output
of input power which is deposited in the cavity but fails
coupling; the concept of a zero-coupling pumping ratio
to be reemitted and thuseffectively contributes toheating;
(PR)’ then allows for “cavity optimization” following
since incoherent photons are not confined to the active
standard procedures. In this context we note that satura-
region w M K - {. Thermal relaxation times are likely
tion may occur a t a level below the theoretical limit F
for (PR)’ + a , if bombardment-induced heating raises to be slower than the response times involved in lasing,
so that we may write
the threshold power. This problem suggests t o consider
a differential quantum eficiency q d = d(qexPi,,)/dPin which
is seen to approach the vex value corresponding to PR =
d P i n / d P t h a t high power inputs. I n this manner we can
relatethe (‘slope” d6,,t/dPi,, whichis always propor- as long as “turn-on delays” do not enter the picture.”
tional tovd, to the thermal response of the targetincluding Considering that for normallyanticipatedtemperature
pulse risetime, on the assumption that adiabatic conditions excursions A T , the threshold-power variation isessentially
hold. linear,[15’the time evolution of the laser pulse, therefore,
Finally, it may be stated that these considerations are proceeds as follows:
consistent with present experimental evidence. I n the
case of electron-beam pumped CdS, it appears that the
overall power eEciency (26.5 percent) reported by Hur-
witz“] represents an upper limit (vex M l) indicative of
exceptionally small free-carrier losses. The 1 percent power
efficiency of V a v i l o v ’ ~ [ CdTe
~ ~ ’ laser, on the other hand,
probably involves someheatingandalmost certainly
substantial optical losses (see Table I). Lavine’s[171results fast12 Such delays may arise with beam-current pulses having very
(subnanosecond) risetimes, as inthe case of field-emission
for GaAs are in remarkable agreement with the calcula- devices.
194 IEEE JOURNAL O F QUANTUM ELECTROSICP, APRIL 1968
r41 C. A. Klein, “Backscattering of kilovolt-electron beams,”
We conclude that, upon application of an electron-beam submitted to 3rd Internat’l Conf. Electron and IonBeams in Science
pulse with a finite risetime, the shape of the coherent and Technology(Boston, illass., May 1968). For apreliminary
account, see C. A. Klein, “Furtherremarkson electron-beam
lightemittedfrom a semiconductor laser (see Fig. 5) pumping of laser materials,” A p p l . Optics, vol. 5, pp. 1922-1924,
primarily reflects the rate of change in threshold power December 1966.
[SI C. A. Klein, “Bandgap dependence and related features of
as the excited region heats up. radiation-ionization energies in semiconductors,” J . A p p l . Phys.,
1) (dP,‘Eh/dT)o= 0: The light pulse preserves the shape March 1968 (to be published).
161 C. E. Hurwitz, “High power and efficiency in CdS electron-
of the current pulse though, of course, there is no output beam pumped lasers,” A p p l . Phys. Lett., vol. 9, pp. 421423,
until the beam current passes the threshold point. December 15, 1966.
171 N. G. Basov, P. G. Eliseev, S.D. Zakharov, Y. 1 ’. Zakharov,
2) (dP,*,/dT), < R ( C p ) / w : The laser output reaches I. Pi. Oraevskii, I. 2. Pinsker, and V. P. Strakhov, ”Properties of
its maximum value right at the end of the current rise- diode quantum generators prepared from gallium arsenide,” Soviet
Phys.-Solid Stale, vol. 8, pp. 2092-2097, March 1967.
time; the output drops off over the flat portion of the 181 F. Stern, “Stimulated emission in semiconductors,” in Semi-
pulse because of further heating. conductors and Semimetals, vol. 2, It. K. Willardson and 8.C. Beer,
3) (dP,*,/dT),> n ( C p ) / ~Peak: power output occurs Eds. New York: Academic Press, 1966, pp. 371-411.
[91 C. A. Kleinand J. M . Lavine,“Effect of mhomogcneons
while the current is still rising, which shortens the dura- minority carrier d k i b u t i o n s in electron-beam pumped lasers,”
A p p l . Phys. Lett., February 15, 1968.
tion of the laser pulse to a fraction of the current-pulse I101 A. Yariv and J. P. Gordon, “The laser,” Proc. IEEE, vol.
width. 51, pp. 4-29, January 1963.
[I11 N. G. Basov, A. 2. Grasyuk, V. F. Efimkov, and V. , A .
Katulin, “GaAs semiconductor laser optically excited by r a d i a h n
havingenergy close to the forbiddenbandwidth,” Soviet 1’hys.-
ACKNOWLEDGMEKT Solid State, vol. 9, pp. 65-74, J ~ l 1967.
y
P I A. Akselrad, “Optimum design for aroom-temperalme,
Thispaper is an outgrowth of numerous stimulating pulse-operated GaAs injection laser,” A p p l . Phys. Lett., vol. 8,
pp. 260-252, May 15, 1966.
discussions (and provocative arguments) the author has 1131 F. H. Nicoll, “Far-field patterns in electron-bombardetl

had with Dr. J. M. I,avine, of t,his laboratory. Much of semiconductor lasers,’’ presented a t 1967 Semiconductor Laser Cor1E.
(Las Vegas, Nev., November 30, 1967).
the initial impetus came from H. Statz, whose suggestions [la] See, for example, C. E. Hurwitz, A. R. Calawa, and 11.. 11.

have been very fruitful. The questions and comments of Rediker, “Electron-beam pumped lasers of PbS, PbSe, and PbTe,”
IEEE J . Quantum Electronics, vol. &E-1, pp. 102-103, May 1965.
M. Pilkhun, R. Rediker, and I?. Stern at the recent IEEE 1151 V. S.Vavilov, E.L. A-olle, G. P. Golubev, V. S. Mashtakov,

Semiconductor Laser Conference turned out, quite ben- and E. I. Tsarapaeva,“Temperature dependence of stimulat.etl
emission from CdTe excited by electron-beaminjection,” Sooiet
eficial in preparing this version of the author’s contribu- Phya.-Solid State, vol. 9, pp. 657-669, September 1967.
[I61 P. Aigrain and M. Balkanski, Eds. Selected Constants Relali7ve
tion. to Semiconductors. Paris: Pergamon, 1961, p. 22.
[I71 J. M . Lavine and A. Adams, Jr., “Quantum elficiency o f
electron-beam pumped GaAs lasers,” IEEE J . Quantum Eleclron;ics
REFERENCES (Correspondence), vol. &E-3, p. 641, November 1967.
[I81 R.B. Oswald, “Fracture of silicon and germanium induced
111 For source material,see M. I. Nathan, “Semiconductor by pulsed electron irradiation,” IEEE Trans. ATuclear Scienm,
lasers,” Proc. IEEE, vol. 54, pp. 1276-1290, October 1966. vol. NS-13, pp. 63-69, December 1966.
P I For further information, see C. A.,K!ein, ‘ The excitation I191 According to data of R. Hunsperger and J. Ballant,yrle,
mechanism in electron-beam pumped lasers, m Physics of Quantum “Measurement of photon absorption loss in the active and passive
Electronics. P. Kellev. B. Lax. and P. Tannenwald,Eds.
~ New regions of a semiconductorlaser,” B p p l . Phys. Lett., vol.10, pp.
York: McGram-Hill, u1966, pp. 424-434. 130-132, February 1.5, 1967.
[31 N. G. Basov, 0. V. Bogdankevich, and A. Z. Grasyuk, [sol For a general discussion, see J. Vine and P. Einstein, “Healdn::
“Semiconductor lasers with radiating mirrors,” IEEE J . Quantum effects of an electron beam impinging on a solid surface, allowing for
Electronics, vol. QE-2, pp. 594-597, September 1966. penetration,” Proc. I E E (London), vol. 111, pp. 921-930, May 1964.