Recombination in narrow-gap semiconductors

RECOMBINATION IN NARROW-GAP SEMICONDUCTORS

Günter NIMTZ II. Physikalisches Institut der Universitlit zu Köln, Zülpicher Stra/3e 77, D-5000 Köln 41, Germany

I

NORTH-HOLLAND PUBLISHING COMPANY- AMSTERDAM

PHYSICS REPORTS (Review Section of Physics Letters) 63. No. 5(1980) 265-300. NORTH-HOLLAND PUBLISHING COMPANY

RECOMBINATION IN NARROW-GAP SEMICONDUCTORS Günter NIMTZ II. Physikali.sches Inst itut der Universitdt zu Köln, Zulpicher StraI3e 77. D 5(AXI KoIn 41. Germans Received February 1980

Contents: I. Introduction 2. Narrow-gap semiconductors 2.1. General remarks 2.2 Hg 1 ~Cd~Te 2.3. Pb1 ~Sn,Te 2.4. Pb1 ~Sn,Se 3. Experimental methods to study recombination mechanisms 31. Carrier lifetime 3.2. Photoconductivity (steady-state) 3 3. Time-resolved photoconductivity 3.4. Time-resolved photo-Hall effect 3.5. Impact-ionized carriers

267 269 269

3.6. Photomagnetoelectric effect (PME) 4. Recombination mechanisms 4.1 General remarks

‘77 277 ‘7’

270 271 272 272 172 277 274 27~

4.2. Shockley Read mechanism 4.3. Radiative recombination 44. Auger recombination 4 5. Single-phonon recombinatton transitions 46 Plasmon recombination 4 7 Cyclotron-resonance enhanced Auger transitions 5. Conclusions References

‘78 280 282 191

294 295 ‘98 199

175

Abstract: This review is intended to be an introduction to the recombination mechanisms important for narrow-gap semiconductors. The study of recombination mechanisms gives information on the various interactions the free carriers happen to meet in crystals. On the other hand recombination mechanisms are limiting the performance of all the modern solid state electronic devices It was recently observed that the dominant recombination processes in narrow-gap semiconductors are different from those important for the classical semiconductors. Due to the narrow band gap the intrinsic carrier density is rather large and in addition the band gap energy is of the same order of magnitude as the energy of elementary excitations as LO-phonons or plasmons, and typical cyclotron-resonance energies. Accordingly processes as the Auger effect or recombination by emitting single phonons or single plasmons become highly probable. These mechanisms have been observed recently and also a new magnetic quantum effect was discovered. The new processes have proved to be important for devices like infrared detectors and infrared solid state lasers, but revealed also some fundamental properties of semiconductors.

Single orders for this issue PHYSICS REPORTS (Review Section of Physics Letters) 63, No. 5(3980) 265 300. Copies of this issue may he obtained at the price given below. All orders should he sent directly to the Publisher. Orders must he accompanied by check Single issue price DII. 16.00, postage included

Gunter Nim&. Recombination in narrow-gap semiconductors

267

1. Introduction At the end of the forties physicists learned about the basic properties of semiconductors and the transistor was born. In the laboratories there was a feeling about the electronic revolution due to the semiconductors. It was recognized atthat time that recombination is one of the most important quantities in determining device specifications. Recombination processes establish the thermodynamical equilibrium state, which might be disturbede.g. by the injection of carriers, by impact-ionization or by photoionization in diode structures or transistors. The fundamental theories for the description of these phenomena were developed in the fifties. It was observed that the recombination mainly proceeds in two ways, one of which being the emission of photons. More often, however, the recombination is established by trapping one hole and one electron on the same impurity level in the band gap. Other recombination mechanisms seemed to be less important for a recombination in classical semiconductors. This review is intended to be an introduction to the dominant recombination processes in narrow-gap semiconductors. Research with the so-called narrow-gap semiconductors began around 1960. Narrowgap semiconductors are understood to have a forbidden band gap of the order of magnitude of 0.1 eV, thus their band gaps roughly being one order of magnitude smaller than those of the classical semiconductors as silicon for example. Like in the case of the classical semiconductors the device aspects gave the first impulse for the development of narrow-gap materials and the study of their physical properties. Narrow-gap semiconductors are thought to be useful for the generation and detection of infrared radiation, particularly for the range of wavelengths between 5 and 20 rim. This part of the electromagnetic spectrum is of importance at least for two reasons: there are atmospheric windows (3—5 ~m and 8—14 p~m),accordingly this radiation is appropriate for interatmospheric communication, and secondly the 300 K radiation has its maximum at a wavelength of 11 p~m,as shown in fig. 1. Accordingly there are many device aspects of this infrared radiation region for which detectors and emitters were expected to be produced from narrow-gap semiconductors having band gaps equivalent to the photon energy in question. In fig. 2 an infrared picture demonstrates the usefulness of such devices in ocological problems. The picture was taken 300 m above the river Elbe near Hamburg with an infrared camera being sensitive at wavelengths around 11 urn. On the upper part of the picture the river is seen to be heated by warm water of the cooling system of a power-station (left corner, lower part). The embankment of the river Elbe is seen as a dark (cooler) ribbon. There is some interest in taking such temperature pictures, since the thermal pollution of rivers and lakes, is often harmful to their ocology. The temperature picture shown in fig. 3 is concerned with the pollution of the sea. A vessel

I

L

Suol 50010

m2~pm~ ,OO~

0~02

~Ji

~rt1l:Ktr

0,5 1

2

5

10 20

50 100

)~.(pm)

Fig. 1. Blackbody radiation distribution for two temperatures.

Gunter Ni,ntz, Reconihinanon in narrow gap semiconductors

268

ILL1

Fig. 2 Thermal imaging of a power station on the river Elbe near Hamburg. DFVLR

~l]

(the white hot spot on the picture is the infrared picture of a 150 m long ship) is littering hydrocarbons. The spiral-like hydrocarbon film is caused by the vessel’s propellers. The hydrocarbon film has a reduced infrared emission compared with the non-polluted sea surface. The temperature resolution of modern infrared cameras is better than 0.2°C. Of course there are numerous other important device applications for infrared detectors and infrared emitters. Semiconductors having narrow-band gaps equivalent to the infrared photon energy are mainly ternary mixed crystals. The components are either elements of the lind and VIth group of the periodic system, or elements of the IVth and VIth group. The recombination of the free carriers plays a key role in infrared devices. The dominant recombination mechanism is limiting both the sensitivity and the rise time of a photodetector. The efficiency p ..aaaaanaaaa~a

~

p.

r I U U I S S S S II

P I P PS S S S S S PP ~

* $

~

-

‘

I

I

I

~

I

Fig. 3. Thermal imaging of a sea pollution due to hydrocarbons littered by a ship. SAT (I)

I

*

I

I

I I

I

I

*

Günter Nimtz. Recombination in narrow-gap semiconductors

269

of a light emitting diode is determined by the non-radiative recombination transitions. Investigating the recombination behaviour of narrow-gap semiconductors it was found that some mechanisms become important which were never observed with the classical semiconductors. Due to the small band gap, there is an exponentially higher intrinsic carrier density. This favours the Auger recombination, a mechanism being strongly dependent on the number of carriers. On the other hand the band gap energy is of the same order of magnitude as the energies of various elementary excitations, e.g. phonons, plasmons or cyclotron-resonance energies. Accordingly single-particle recombination channels are opened and become extremely efficient at the energy resonance between band gap and elementary excitations of the crystal. For example the plasmon recombination transition was expected to be of such an efficiency that once it is energetically allowed it should become the dominating mechanism. Analogous to the classical semiconductors most of the investigations with the narrow-gap semiconductors were carried out with respect to device applications, however, also valuable fundamental physical properties were discovered and understood in the course of the investigations. In this introductory article on recombination in narrow-gap semiconductors we begin by introducing the most important narrow-gap materials and they are discussed particularly with respect to properties different from those of the classical semiconductors. Modern experimental methods to study the recombination behaviour are sketched, and some definitions relevant for the non-equilibrium state of carriers are given. Experimental and theoretical data of the new phenomena are presented and discussed. An extensive theoretical description of the Auger effect in narrow-gap semiconductors is given. For all the other recombination mechanisms only the basic relations are given and the favourable conditions discussed. The importance of the various mechanisms for fundamental research as well as for device applications is outlined.

2. Narrow-gap semiconductors 2.1. General remarks A semiconductor having an energy gap smaller than 300 meV is usually called a narrow-gap semiconductor. Sometimes the element semiconductor tellurium with a band gap of 330 meV is put into this class. The most popular semiconductor belonging to this class is InSb, a Ill—V compound semiconductor with a band gap of 220 meV. Both InSb and tellurium do show a recombination behaviour which we are used to from classical semiconductors having energy gaps of about 1 eV. However, in semiconductors with energy gaps below approximately 200 meV the dominant recombination mechanisms become different. Most of these narrow-gap semiconductors are ternary mixed crystals composed either by Il—VI or by IV—VI compounds. As it will be shown in the following sections in these mixed crystals the band gap can be varied with composition ratio. At present there are mainly three alloy systems of importance: Hg1 ~Cd~Te, Pb1 ~Sn~Te and Pb1 ~Sn~Se.These crystals are the basis for detectors and lasers in the 3 to 30 urn infrared region of the spectrum. Other narrow-gap alloys will not be introduced in this article, since they did not become important neither for applications nor for basic research up to now. All the mixed crystals are rather difficult to grow if compared with element or binary semiconductors. The main problems are the crystal homogeneity and lattice defects. Due to the different solubiity of the three components in the mixed crystal, segregations and vacancies are very probable.

270

Günter Nimtz Recombination in narrow-gap semiconductors

2.2. Hg1 ~Cd~Te The mixed crystal Hg1 ~Cd~Teis composed of the two Il—VI compounds HgTe and CdTe [2]. Both compounds crystallize in the zincblende lattice and have very similar lattice constants of 646.2 nm and 648.2 nm respectively. HgTe is a semimetal as shown in fig. 4. Conduction band and valence band are degenerate. Often it is said, HgTe has a negative energy gap, since the f’6-band is below the F8-bands, thus being inverted to an InSb-like band structure. With increasing content of CdTe in the mixed crystal the F6-band is lifted, becomes degenerate with the 18-bands for a ratio x 0.16 (T 4.2 K) and for x > 0.16 the material becomes semiconducting, since a positive band gap is established. In good approximation the gap energy is proportional to the CdTe content in the crystal. Eventually the band gap reaches 1.6 eV in pure CdTe. The semiconducting compositons show an InSb-like bandstructure, where the 16-band and the F,,~-bandspresent extrema of the conduction and valence bands respectively. The [‘,,8-valencebands are splitted due to spin—orbit interaction. The infrared mixtures are those with x 0.16—0.25, thus being between the semimetallic state and a semiconducting state with a band gap up to 200 meV. The energy gap dependence on x-value is temperature dependent for these compositions: for a fixed x-value the band gap increases with temperature. Accordingly the band-crossing is shifted to smaller x-values with increasing temperature. The conduction band of this material is strongly non-parabolic, i.e. the effective mass has a pronounced dependence on wave number. The energy bands are spherically symmetrical near the extrema. The narrow-gap compositions of HgCdTe have an extremely small effective band A 1m for conduction x 0.17 and T mass. = 4.2 K. typical value of the effective conduction band-edge mass is ~10 Accordingly the material is very important for the investigation of magneto-transport effects in the extreme magnetic quantum limit, a region which is not reached in other semiconductors at magnetic fields available in laboratories [31.Both non-parabolicity and small effective mass are caused by the strong interaction of the F(,~and F 5-bands, which, of course, decreases with increasing energy gap. The valence band mass is about two orders of magnitude bigger than the conduction band mass, and the deviation from parabolicity is much smaller in the valence band than in the conduction band. The valence band near the band extremum of the semiconducting compositions is represented by the two-fold I’8-levels. As indicated in fig. 4 by the strongly different curvatures there is a heavy and a light hole band. The effective mass of the light hole is of the same order of magnitude as that of the conduction band electron. The heavy hole mass, however, is assumed to be 0.55m5 [21.Due to the much —

—

Eg(eV)~

Hg1

cd~Te(42K)

10

0 H9Te

~onducfor

02

04

X

06

08

10 CdTe

Fig. 4 Energy gap of the mixed crystal Hg1 ~CdTe as a function of mol fraction s (2)

Günter Nimtz, Recombination in narrow-gap semiconductors

271

higher density of states in the heavy hole band the transport properties of p-type Hg1 ~Cd~Te are governed by the heavy holes. 2.3. Pb1..-~Sn~Te This lead salt alloy is formed by the two IV—VI compounds PbTe and SnTe [4,5]. PbTe and SnTe are semiconductors with energy gaps of 0.22 eV and 0.27 eV (T = 77 K) respectively. Both binary compounds crystallize in the cubic NaCI-structure. The energy gap can be varied continuously between the values of the constituent compounds and zero by changing the crystal composition parameter x as is demonstrated in fig. 5. For a composition x = 0.4 at 77 K the crystal has a zero band gap, i.e. has become a semimetal. This property is causedthe by conduction an energy band from one compoundvalence to the 6-band performs bandinversion and the p-like L6~-bandthe other. In PbTe the s-like L band, while in SnTe it is the other way round. The exchange of the bands takes place at a composition parameter x = 0.4 causing the degeneracy of the two bands. The band structure of the lead-chalcogenides is much more complicated than that of the above introduced mercury-cadmium-telluride crystal. The band extrema of conduction and valence bands are at the L-points of the Brillouin zone as shown in fig. 6. The constant-energy surfaces in k-space are ellipsoids along the diagonals of the Brillouin zone, which is the case for both conduction and valence bands. At high carrier densities also energy states between the main ellipsoids become occupied as illustrated in fig. 6. The anisotropy of the effective masses, i.e. the ratio of the principal axes of the main

ellipsoids is about ito 10. The conduction band and the valence band effective masses are nearlyequal. For the longitudinal and the transverse masses values of m 11 = 0.25 and m1 = 0.024m0 were found, where m0is the free electron mass. The band gap of Pb1 ~Sn~Teis strongly temperature dependent. The semiconductors on the PbTe side have a positive temperature coefficient, whereas on the SnTe side the temperature coefficient of the band gap is negative. The change of gap energy with temperature is about 0.4 meV/K and can be used,

as we shall see later, for a wavelength tuning in laser devices.

x 0.3

Eq

—

0

PbTe PbSe

Pb~_5Sn~Te (77K)

02

04

X

06

08

Fig. 5. Energy gap of the mixed crystals Pbi ~Sn,Te and Pbi as a function of mol fraction x 151.

10

SriTe SnSe

u5fl~5C

L Fig. 6. Bnllouin zone for Pb1 ~Sn~Teand Pb1 xSnuSe with energy surfaces [5].

272

Gunter Nmitz, Recombination in narrow-gap semiconductors

2.4. Pb1 ~Sn~Se Also the Pb1 ~Sn~Se alloy is composed of two IV—VI compound semiconductors [4, 51. For compositions with x 0.43 Pb1 ~Sn~Se crystallizes in the cubic NaCI-structure, at higher contents of SnSe an orthorhombic structure (B29) is found. In the region of the NaCl-structure the band gap can be adjusted to any value between zero and 170 meV. Up to now investigations were carried out mainly with crystals of NaCI-structure. Similarly to the Pb1 ~Sn~Te system a band inversion causes a zero energy gap at a distinct composition, which takes place for Pb1 ~Sn~Seat x = 0.1 (T = 77 K). The band extrema are situated at the L-points in k-space and also there are ellipsoids of constant energy surfaces along the diagonals of the Brillouin zone as has been illustrated schematically in fig. 6. However, the ratio of the principal axes is smaller than in the Pb1 ~Sn~Tesystem being only about 2. An interesting feature of the Pb1 ~Sn~Secrystal is, that the application of a magnetic field causes an increase of the effective band gap for x <0.1, but a decrease for crystals with x >0.1. This property will be discussed in more detail in section 4.7.

3. Experimental methods to study recombination mechanisms 3.1. Carrier lifetime Electrons can be pushed from the valance band to the conduction band by a number of different mechanisms. The electrons may become thermally excited, in this case the lattice delivers the ionization energy. In the language of quasiparticles we say the electron is excited to the conduction band by absorption of phonons. For device applications the photoabsorption process is important. A photon having an energy equal or somewhat higher than the band gap can push an electron into the conduction band. The inverse process is called radiative recombination. An electron may also be brought to the conduction band by impact-ionization in strong electric fields. In the field free carriers are accelerated up to a kinetic energy which exceeds the band gap energy. These so-called hot carriers can push an electron from the valence band to the conduction band via electron—electron impacts. Hot carrier impact-ionization is for example the basic mechanism of IMPA1T diodes. In other electronic devices as in pn-junction diodes or in bipolar transistors excess carriers are generated by injecting minority carriers. These four mechanisms are the most important ones by which electrons can be transferred to the conduction band. In the case that the thermodynamical equilibrium state is disturbed by the generation of carriers we are talking of excess carriers. With respect to device specifications the lifetime of these non-equilibrium carriers (excess carriers) is interesting, which in turn is governed by the dominant recombination process. For an excess carrier there may be defined different lifetimes, for instance the time occupying a conduction band state, or the time neccessary to come back to the valence band. (We restrict most of our discussions to the recombination behaviour of electrons, since that of holes proceeds in an analogous way.) In the following we are going to call the lifetime r the time a carrier contributes to the electrical conductivity —

q(J.1~n+ /~p)

—

q[/-Ln(no +

fle)+

/~Lp(po +

pe)1.

(I)

Here q is the absolute value of the elementary charge, 1a,, and ~ are the carrier mobilities of electrons

GünterNimfz, Recombination in narrow-gap semiconductors

273

and holes respectively, n0 and Po the thermal equilibrium densities of the free carriers, and n~,and Pe the excess carriers. The lifetime defined in this way may not be the electron—hole recombination time. If for instance the recombination proceeds via impurity energy terms in the energy gap, electrons and holes

might have quite different lifetimes, since the capture rate of the impurity for an electron might differ from that for a hole by many orders of magnitude. Accordingly one type of carrier may be free and contribute to the electrical conductivity whereas the other is already trapped on an impurity site. In this case one has to deal with two lifetimes one for electrons and another one for holes. In general it is observed that lifetimes are strongly dependent on material and on temperature, values are usually found between 10 2 and 10_12 s. Accordingly one has to choose an appropriate experimental

method depending on the prevailing lifetime. Some of the commonly used methods are sketched in the following subsections of this section. The time dependence of the carrier density in the conduction band is given by [6—9]:

4~—g—r+~VI~.

(2)

g is the generation rate, r the decay rate, and the last term is the change in carrier density due to

diffusion. An analogous equation holds for the holes in the valence band. In general the various rates are different for holes and electrons. For the decay rate of the excess electrons, which corresponds to the recombination rate if both electrons and holes have the same lifetime, often a linear dependence on carrier density is assumed and thus (2a) r=n/r.

(3)

This Ansatz yields the well-known exponential decay of non-equilibrium carriers after the generation is switched off and neglecting diffusion. As will be pointed out later there are some important recom-

bination mechanisms found which do not show such a simple relaxation of the non-equilibrium state. In this case the lifetime r becomes a function of the number of excess carriers. 3.2. Photoconductivity (steady-state) By the absorption of a photon an electron—hole pair is generated. The photon energy must correspond to a direct electron transition from the valence band to conduction band. “Direct” means that the transition has to proceed between band extrema having the same k-value. A typical

experimental set-up to measure the photoconductivity is shown in fig. 7. The photoconductivity is measured as a change in current under short-circuit condition or as a change in voltage under open-circuit condition. The change in conductivity due to the photoionized excess carriers is = q(nep~n+pe~p)

(4)

where here and in the following equations we assume that the material is homogeneously ionized and that both electrons and holes have the same lifetimes. Under steady-state conditions the photo-current

274

Günter Nisntz, Recombination in narrow-gap semiconductors

R\

\~PHOT

Fig. 7 Sketch of an experimental photoconductivity set-up.

per sample width d (increment in current with constant applied electric field E) is given by the relation jpcqfl4)(1R)(/2n+/.Lp)TEd

i

(5)

with flepe~(1R)Td

i.

Here ~j is the quantum efficiency, ~ the photon flux, and r the carrier lifetime. In this equation it is assumed that both types of carriers have the same r and that surface recombination can be neglected. Reflexion losses at the surface are taken into account by the reflexion coefficient R. Quite often the photoconductivity voltage U~,is measured in these experiments. This signal corresponds to the voltage increment at constant current and is obtained from eq. (5)

u

(/~tfl+/.L

PC~

0)flçb(1R)Ubr j.t~n0d

(6)

with Ub the bias voltage and assuming n0 ~‘ Po. Thus the photoconductivity response is proportional to the carrier lifetime. Usually it is impossible to determine accurately the additional parameters of eqs. (5, 6). Therefore it is often much more convenient to measure directly the lifetime of the excess carriers, i.e. to measure the decay of the photoconductivity signal as it is discussed in the following section. If such a time-resolved measurement cannot be carried out, an additional steady-state experiment, namely the photomagnetoelectric effect (section 3.6) may help to determine the lifetime. 3.3.

Time-resolved photoconductivity

In these days most of the photoconductivity experiments are carried out with lasers. Many laser systems can be operated in a pulsed mode, the pulses having decay times often much shorter than the carrier lifetimes in question. So laser pulses may be used for photoionization of excess carriers and the decay of the photoconductivity L~cr(t)=

q[n~(t)ji~+pe(t)/.ipl

(7)

corresponds to the lifetime of the excess carriers. This is usually a good approximation since energy and momentum relaxation of the carriers being much faster than the recombination. The interpretation of the photoconductivity signal may become ambiguous, if fle(t) and pe(t) have different decay times. In this case additional informations are necessary for a lifetime determination, which may be obtained from the

Günter Nimtz, Recombination in narrow-gap semiconductors

275

time-resolved Hall effect (section 3.4) or from the photomagnetoelectric effect (section 3.6). An example for two strongly different lifetimes is given in section 4.2. There may arise a serious problem from the non-homogeneous photoionization rate in the sample. Usually the samples are thicker than the penetration depth of the ionizing laser radiation, which in the

case of the fundamental absorption is of the order of i07 m_t. Thus there is a gradient of carrier density in the sample and the observed decay of photoconductivity may be difficult to interpret if the recombination time is dependent on the number of excess carriers. 3.4. Time-resolved photo-Hall effect It was mentioned before (sections 3.1 and 3.3) that in general electrons and holes may have different lifetimes. In this case the interpretation of the photoconductivity decay may be ambiguous. A rather

complicated method to obtain information on both electron and hole lifetime is described in section 3.6. A comparatively simple method to get more information about lifetimes is to measure first the photoconductivity decay according to eq. (7), and in a second experiment the Hall-voltage decay which is given by the relation UH(t) =

UH(0) + AUH(t)

(8)

with H(t)

-

[~n(t)-js~p(t)JBI q(J2~n(t) + ~t 2d~ 0p(t))

Here n (t) = Tio + fle(t), p(t) = Po + Pe(t), and B, I and d are magnetic field, current and sample width respectively. If the mobilities are known and they don’t differ too much in magnitude we may calculate from the experimental data of conductivity and Hall-voltage decay both electron and hole lifetime. At least the Hall-voltage tells us, what type of carrier determines the conductivity decay and thus whether both carriers have the same lifetime or not. 3.5. Impact-ionized carriers As mentioned above gradients of carrier density are inevitable due to the exponential dependence of photoabsorption on sample depth. To investigate fairly high excess carrier densities it is often more

appropriate to generate the carriers by impact ionization in electric fields. Particularly in polar semiconductors with narrow-band gaps, rather small electric fields between 10~and i05 V/rn are sufficient for the impact ionization of carriers. At these fields the carriers are accelerated to kinetic energies high enough to push additional carriers from the valence band into the conduction band. If the material is homogeneous and a uniform electric field is applied the ionization process takes place over the whole sample. An experimental set-up to measure the lifetime of the impact-ionized carriers is

sketched in fig. 8 [ii, 12]. The samples are shaped as posts with electrical contacts at the ends to which voltage pulses are applied. To measure the change of conductivity, i.e. the decay time after the ionization is switched off (at the end of the voltage pulse), the sample is probed by microwaves. The microwave absorption in the semiconductor is a function of the conductivity namely of the number of free carriers. For this reason, the sample is placed in a rectangular waveguide, where from the

276

Günter Nimtz. Recombination in narrow-gap semiconductors

,J_[~~~se voltoge

waveguide

output

rput insutcitor

sampLe

Fig. 8. Experimental set-up to measure the lifetime of impact ioniied carriers with microwases II 121

microwave transmission by the sample the change of conductivity, particularly its decay time can be obtained. The decay time of the conductivity change corresponds to the excess carrier lifetime, since the lifetime is usually by many orders of magnitude longer than the energy relaxation time of the carriers. Microwave frequencies between 101~and lO~Hz are appropriate for such experimental studies yielding time resolution up to 10 ~s. A typical microwave transmission signal is shown in fig. 9. A voltage pulse of 80 ns is applied to the sample. Instantaneously the carriers are heated in the electric field causing a reduction of the carrier mobility and thus a decrease of electrical conductivity. Due to the conductivity decrease the microwave transmission is increased. At the same time the impact ionization is developing an avalanche of excess carriers which becomes evident about 50 ns after the voltage pulse has been applied, which is clearly seen from a break-down in microwave transmission as a result of the strong increase of conductivity in the sample. Up to now we have discussed the modern experimental techniques which have been developed during the last years in connection with the progress in laser technology and the enormous improvements in electronics. The classical methods are extensively described in some books e.g. [7—9]. In the last section we are going to discuss a classical method to study recombination effects. This method is particularly applied if both lifetimes of minority and majority carriers are looked for, and the methods of sections 3.3 to 3.5 are not appropriate.

0

100

nsec

200

300

_____________________________________

5’

~ra~

~ieraY~

time Fig. 9. (a) Applied voltage for impact ionization and (b) corresponding microwave transmission signal versus time [11. 121.

Günter Nimtz, Recombination in narrow-gap semiconductors

277

3.6. Photomagnetoelectric effect (PME)

In fig. 10 an illustration of the photomagnetoelectric effect is sketched. Assuming that the ionizing radiation is strongly absorbed in the semiconductor there is built up a gradient of ionized carrier density. The excess carrier density decreases exponentially from the surface, where the radiation is incident, to the volume of the sample. Thus both electrons and holes diffuse into the volume but become separated due to the magnetic field. Perpendicular to the directions of both incident radiation and magnetic field a potential difference UPME is produced. This is known as the PME and it may be considered as the Hall effect of the diffusion current of photoexcited carriers. The short-circuit current due to this potential is given by the relation [7] 1PME

— —

~4qB(1

—

R)1~J~T~) (r~

2 ,j,, (j~ \1/2 r /.Lpi~JJpTp)

‘i3/

where ~ is the quantum efficiency, 4i the incident radiation in quanta per unit time and unit area, B is the magnetic field, D is the diffusion constant, and 1 the length of the sample. According to eq. (5) we may express the photocurrent neglecting surface recombination ~ j~c—

10 ()

I

Thus from the experimental data of both the photomagnetoelectric effect and the photoconductivity electron and hole lifetimes may be determined, if the carrier mobilities p.~and /L 1,

respectively are known, and having in mind the Einstein relation qD

=

for electrons and holes

1zkT.

ne

Fig. 10. Photomagnetoelectric effect. The photoionization takes place near the surface. The diffusion of the ionized carriers proceeds from the surface into the sample. In the magnetic field B the diffusion currents of positive and negative carriers become separated.

4. Recombination mechanisms 4.1. General remarks In the introduction it was pointed out that with decreasing band gap recombination mechanisms become important, which do not play any major role in the classical semiconductors having band gaps

278

Günter Nimtz. Recombination in narrow -gap semiconductori

around 1 eV as silicon, germanium or the compound semiconductor GaAs where two mechanisms dominate: recombination via lattice defects, usually called Shockley—Read recombination and radiative recombination. In narrow-gap semiconductors there have been found in addition the Auger mechanism and the recombination by emission of single phonons or single plasmons. Furthermore in magnetic fields a new magnetic quantum effect was discovered: an oscillatory dependence of the Auger transition rate on magnetic field. In this section we are going to introduce all the important recombination mechanisms, which will be discussed and illustrated by experimental data. However, more space will be devoted to the very recently observed recombination effects which are coupled with a narrow energy gap. 4.2. Shockley—Read mechanism Lattice defects either caused by vacancies, by interstitials or by impurities may have electronic states with energy levels in the gap, which is schematically shown in fig. ha. The recombination via an impurity proceeds due to the capture of an electron and a hole. In general the transition rates for electrons and holes are quite different thus causing different lifetimes. The recombination energy released in this process will either be emitted as electromagnetic radiation or more often transferred to the lattice as elastic energy. The latter process is causing the following theoretical problem: the recombination energy is often some orders of magnitude higher than a typical

phonon energy. For such a transition many phonons have to be emitted. However, an instantaneous emission of many phonons is expected to have an extremely low probability. In order to explain the observed high

non-radiative capture rate of hydrogen-like impurities one assumes cascade-like transitions [13—15].The electron is assumed to move down to the ground state of the impurity step by step via excited states of the level in question as shown in the scheme in fig. lib. Each step being so small on the energy scale that the energy is equal to that of single phonons. Many recombination centers, however, are not hydrogen-like, lacking the excited states which are necessary for a cascade electron capture. In this case a finite change of the shape of the atomic potential energy during the electronic transition is assumed to take place. In the adiabatic approximation of this model a multi-phonon transition becomes possible. However, numerical estimates of the capture cross-section are rather difficult to obtain [141. The statistical theory of the recombination via recombination centers was developed first by Shockley and Read [16].This type of recombination is often called Shockley—Read recombination. The theory of the capture cross-sections of recombination centers is still under discussion e.g. [14, 15, 17, 18]. Usually CB-’~ PC

CB

~ ~RC

VB

ci)

b)

.

Fig. 11. (a) Energy level of a recombination center RC. and (b) excited states of a hydrogen like impurity.

Günter Nimtz, Recombination in narrow-gap semiconductors

279

capture cross-sections are deduced from temperature dependent experimental data, assuming the

cross-sections to be temperature independent. This assumption is by no means based on theoretical arguments, but in many cases a fairly good description of the experimental data was achieved. The average electron lifetime is determined essentially by a statistic factor and the elementary transition as r 1N

(11)

Jf(k)v(k)u(k)dk

Here ~

is the concentration of the trap levels, f~(k)is the electron distribution function, v(k) is the electron velocity at the crystal momentum k, and o-~(k)is the electron capture cross-section for the trap level in question. Capture cross-sections are observed in the very broad limits between 10 16 and 1025 m2 [14]. Among other semiconductors the Ill—V compound InSb shows a recombination behaviour below room temperature which can be explained by assuming a Shockley—Read recombination. The model is

illustrated in the insert of fig. 12. The lifetimes observed in both n- and p-type material can be described by assuming a recombination center having two levels T

1 and T2 corresponding to a singly and a doubly charged recombination center. Experimental investigations were carried out by several groups [19].It was observed using the method discussed in section 3.6 that in p-type material the electrons, being the minority carriers, have a much shorter lifetime than the holes [201.The experiments with p-type InSb were repeated some years later with the direct method introduced in section 3.5, where the decay of the ionized excess carriers can easily be watched with an oscilloscope [21].Experimental data obtained with this direct method are presented in fig. 12. The semi-logarithmic plot of the decay of the excess

conductivity shows at first a fast decay corresponding to the excess electrons’ decay and then a much slower decay corresponding to the slow capture of the holes. The ratio of hole to electron lifetime is about two orders of magnitude. In various samples of the IV—VI compound PbTe also a Shockley—Read dominated recombination was observed by Schlicht et al. [22].There are some arguments that the Auger effect may govern the

recombination of non-equilibrium carriers in this narrow-gap semiconductor, as is discussed in section 4.4. However, the low temperature (T 100 K) lifetime data from high mobility epitaxial layers could be explained only by assuming a Shockley—Read mechanism. Lifetime data of two n-type samples are

20

p-InSb

CB

10

(3’)

VB

T2j.

,-T~

5~ 2L

*S4

1

*,S3

O,E 0

0,5

1,0 1,5 tIps) Fig. 12. Transient decay of excess carriers of two p-type InSb samples at 77 K. The rapid decay of the excess conductivity Ao- is due to electron trapping, the slow decay caused by hole trapping 1211.

280

Günter Nimtz, Recombination in narrow-gap semiconductors

2

___

~

a

1L

PbTe

n

i03~~i~~

-

iO~~

10

experimental

‘~

/

\ iOo

~

-

______

theoretica

—..

~

— -

0

100

200 TIK)—’3W

Fig. 13. Experimental and calculated lifetime versus temperature for n-type PbTe with (a) 1022 and (b) 19 x 10r electrons/m~, F 77 K. The theoretical values are obtained for the Auger mechanism [221.The measured decay was strongly non-exponential at temperatures below 150 K. The shading indicates the instantaneous lifetimes observed during a decay.

presented in fig. 13. Comparing the observed lifetimes with the expected values of minority carrier sweep-out time and measuring the time resolved photo-Hall effect (section 3.4) it was found that the observed lifetimes represent the lifetime of the majority carriers. The lifetime of the minority carriers was estimated to be smaller than 10 ~s, thus being by many orders of magnitude smaller than the lifetimes of electrons. The theoretical data are calculated for an Auger dominated recombination. however, due to the fast minority carrier capture, obviously, the Auger mechanism is prevented in these crystals. It was also observed that the carrier decay was not a linear one, accordingly a quasi-differential lifetime strongly depending on the number of excess carriers was measured. This is indicated by the shaded area in fig. 12 at low temperatures. 4.3. Radiative recombination

In the last years this recombination mechanism the reverse process of photoionization has become increasingly important due to the development of light-emitting semiconductor devices. Radiative recombination may take place in any semiconductor having a direct band gap, which means that conduction band minimum and valence band maximum are situated at the same k-value in the Brillouin —

—

Günter Nimtz, Recombination in narrow-gap semiconductors

281

zone. The rate of spontaneous radiative recombination is given by the relation [6] “ ~‘ R R~hsc2Jexp(hv/kT) 8ire~..f a (hi.’) (hp)2 d’h

(12 a

—

where e~represents the high-frequency dielectric constant and a (hi.’) the absorption coefficient dependent on photon-frequency i’. The generation rate in thermal equilibrium is given by G R 8ire,~ a(h~)(hv)2 d’h h3c2 Jf exp(hv/kT)—1 t~V

12b

Thus the radiative transition rate can be calculated from experimental absorption data by using the detailed balance for emission and absorption. Theoretical and experimental data by Kinch et al. [23] for an intrinsic non-degenerate semiconductor are presented in fig. 14. According to this graph the radiative recombination becomes dominant only at low temperatures if compared with the Auger mechanism. However, even at low temperatures this will not happen in real crystals, since they are extrinsic due to doping or residual lattice imperfections. In the extrinsic range of conduction the lifetime for radiative transition in n-type material is given by TRe =

2TRI(fl

(13)

1/nO)

as follows from simple statistical arguments. ~ 3

1~Ri represents

the radiative lifetime for intrinsic material.

‘

T(K) 200

111

77

10 Hg

t~

795Cd205Te (s) -

THEORETICAL

THEORETICAL TAi

-

intrinsic

io6—

~J3

-

extrinsic

5

7

9

11 13 15 lIT (10~K~)—4.

Fig. 14. Theoretical and experimental lifetime data versus temperature for n-type Hgo 7qsCd~esTe[231. Solid lines represent theoretical values for

radiative and Auger recombination respectively.

282

Gunter Nimtz. Recombination in narrow gap semiconductors

The Auger lifetime is more strongly dependent on the extrinsic carrier density TAe =

2rA(n,/n~,)2

(14)

since two electrons and one hole (or one electron and two holes) are necessary for a transition, whereas in the radiative transition only one electron and one hole take part. (See also eqs. (15—20).) Accordingly not the spontaneous but mainly the stimulated emission was observed in narrow-gap semiconductors [4,24, 26]. In this case the radiative recombination rate is tremendously enhanced by the radiation field in the semiconductor device. The structure of a Pb~~Sn~Telaser diode is sketched in fig. 15 [25—27]. Excess carriers become injected from the n-type PbTe layer into the active Pb 1 ~Sn~Te layer by applying a voltage between the two metal films. Typical dimensions of such a device are given in fig. 15. The metal films on top and bottom are forming the electric contacts. The p-PbTe substrate is coated with two MgF-stripes. In the uncoated region the active layer forms a waveguide in stripe geometry 50 ~imwide. A typical stripe length (=resonator length) for such a double hetero-structure laser diode is between 200 and 500 ~m.Radiation 2. powers up to about 10 mW are obtained with these diodes at injection currents of the order of 108 A/m Most of the mixed crystals have a temperature dependence of band gap which may be applied for tuning the wavelength of the emitted radiation. Experimental data of the wavelength dependent on temperature are shown for two laser diodes in fig. 16 [25—271. In various lead salt laser diodes a tunability of more than one octave was obtained, which is very useful for infrared spectroscopy. In order to reduce the thermal heating of the device or to obtain a higher peak power output a diode may also be operated with pulsed emission. 4.4. Auger recombination

The Auger recombination in a semiconductor is based on the Coulomb interaction of the free carriers. The energy from an electron—hole recombination is being released to a third particle, either to 20 X

Mg F 2

075 pm~ 1 5 pm

~Opm

-

n

_____

-

Au

In

PbTe J~b1 Sn 5Te

Pb

(pm) ~

15 10

Sn 0782

Te 0218

~

lJu(so

200 pm

p Pb Te

Au

In

Fig. 15 Structure of a double hetero-structure laser diode in stripe geometry [261

5

C0~

ow

100

—

15011K]

Fig. 16. Temperature dependence of emission wavelength for PbSnTe and PbSe laser diodes [261.

Günter Nimtz, Recombination in narrow-gap semiconductors

/A~-

~V// e—e

283

/C B //~///// VB/~/~V h—h

Fig. 17. Auger recombination by electron—electron impact and by hole—hole impact.

an electron (e—e transition) or to a hole (h—h transition) as sketched in fig. 17. The free carrier taking the recombination energy is pushed to an excited state in the conduction or valence band respectively. The Auger effect may cause a direct transition between conduction and valence band, but may also take part in a Shockley—Read process, i.e. in a transition from a band state to a recombination center level in the

energy gap. The basic theory of the Auger effect in semiconductors was developed by Beattie and Landsberg in 1959 [281.At that time the Auger effect was assumed to play a minor role in the recombination behaviour of semiconductors. Only InSb being the very narrow-gap semiconductor in the

fifties seemed to represent a candidate for a dominant Auger recombination at elevated temperatures. Recently, however, with the development of the alloy narrow-gap semiconductors, it became evident that in fact the Auger effect represents the most important recombination mechanism in this class of

semiconductors. This becomes plausible from statistical arguments [29]: In an Auger recombination transition either two electrons and one hole or one electron and two holes are involved. Accordingly we expect the recombination rate to be dependent on carrier density as ~ n2p

(iSa)

or for the hole—hole impact rh h

np2.

(15b)

For the intrinsic carrier density in a semiconductor the relation holds n~x T3 exp{—Eg/kT}.

(16)

Thus one expects the recombination rate to be an exponential function of the band gap Eg. This behaviour is qualitatively found in many semiconductors as shown in table 1. The recombination coefficient given in table 1 is defined by the relation CAZ~—T 1n~.

(17)

This relation anticipates that in extrinsic material with a small deviation from thermal equilibrium the excess carrier lifetime is proportional to n ~ and p~2in n- and p-type material respectively. This proportionality follows from the Auger recombination rate r=An2p+Bnp2.

(18)

Gunter Nimtz. Recombination in narrow-gap semiconductors

284

Table 1 Auger recombination coefficients of some semiconductors (CA Semiconductor

Energy gap E~(meV)

C

Si GaSb PbS Te InSb Pb44Sn~~Te Hg~,i~Cd,1 ~Te

1121) 7(X) 37)) 33)) 18)) ((8) 80

4 xl)) 1)) 1)) 2 x 1)) 2 x 10 4x ~ 4 x 10

4 (cm~7s) U

2( 24

r n

T (K)

Ref.

3(’) 3(8) 3(8) 3(8) 3(8) 77 77

[55) hb] [57] h8] [59] [22] [2, II. 12)

With eqs. (3, 18) we obtain for the lifetime of electrons 2-~(An~, + 2BnOp,)). T

=

(19)

2An0p0 + 2Bp~,+

Assuming n 0~

p°and n0

~‘

tie

=

Pe the lifetime will be in n-type material

(20a) and in the case of p-type material 2. p0

(20b)

The recombination coefficients A, B and CA of eqs. (17, 18) can be calculated according to Beattie and Landsberg’s model from the relation [311

26(Ei,+E,, E~ E~). (21) 1)f(E2)(1 -f(E~)).eE2rnnIUifI Here the Coulomb scattering process of two electrons from an initial state i (1, 2) with both electrons in the conduction band to a final state f = (1’, 2’) with one electron in the conduction band (1’) and the other in the heavy hole valence band (2’) (fig. 19 and section 2.2) is considered. The valence band is assumed to be parabolic with an effective mass mh ~

(i + kTn’(~))~f(E

E

2k~’/2mh (22a) 2 = —h and its states are occupied according to classical statistics. The density of heavy holes in the valence

band is given by p

=

N~(T)e

OkT

(22b)

where N~(T)= 2(mhkT/21rh2)312

(22c)

Günter Nimtz, Recombination in narrow-gap semiconductors

285

and ~ denotes the chemical potential. n’(~)= dn/d~,V is the volume, f(E) the Fermi function and U1~= g(i2; 1’2’)—g(2i; 1’2’)

(22d)

is the relevant matrix element of the screened Coulomb potential with gk~i2~1’2”I

— —

jj

ki’+k~’.ki+k2

1~V~1’F(2, ftl) F(i, 2’) 2 1’) (j,. \2 .j..

I’-

e

—

~

Here e’,. is the high frequency dielectric constant (=optical dielectric constant) and F(i, 1’) is the overlap integral of the lattic-periodical parts of the Bloch functions of states 1 and i’, respectively. The screening parameter A is given by the relation A2 = 4irq2n’(~)/e;

(22f)

1/A is the Debye length in the case of non-degenerate conduction electrons and is the Thomas—Fermi length in the degenerate case.

Beattie and Landsberg calculated the lifetime using a parabolic band, classical statistics, and estimates for the overlap integrals. The theoretical values are presented in fig. 18. It is evident from the

comparison of experimental and theoretical lifetimes for InSb that the Auger effect becomes dominant at temperatures above 300 K. The theoretical lifetimes for a radiative transition presented in the same figure are much higher than the Auger lifetimes. This suggests that the radiative mechanism is less important than the Auger effect for InSb. At temperatures below 300 K a Shockley—Read process is found to govern the recombination in InSb as discussed in section 4.2.

From the Beattie—Landsberg model follows, that besides a narrow-band gap the existence of different effective masses in the conduction and in the valence band are very advantageous for an Auger

transition. As was pointed out by Beattie and Landsberg the dominant temperature dependence for the non-degenerate and intrinsic semiconductor obtained from eq. (21) is Ai

~ (Eg(TY\3”2 ex ex 1 ~ .~&l \ kT I ~l.kTJ ~1i+~i.kTJ

~23

where ~ has to be taken as m~Im~ if m~< my and the lifetime is determined by electron—electron collisions, and p. m~/m~ if my < m,, and the lifetime being determined by hole—hole collisions. The first factor results from the density of states of the valence band or the conduction band depending on whether e—e or h—h collisions are considered. The second factor represents the intrinsic carrier density. The last exponential results from momentum and energy conservation in this process. As can be seen from the sketch in fig. 19 vertical transitions in k-space are not allowed. If both valence and conduction band masses are of the same value the effective energygap E for the transition is 1.5 times the real band

gap Eg according to eq. (23). In the so-called infrared semiconductors of the Hg

1 ~Cd~Tealloy system

with x-values around 0.2 both the energy gap (about 100 meV) and the effective mass ratio (p.

10.2)

favour the Auger effect. Numerical data taking into consideration the strong non-parabolicity of the conduction band in narrow-gap Hg1 ~Cd~Tecrystals were obtained first by Petersen [30].Similar values were calculated by Buss [23]and are shown in fig. 14 for comparison with experimental data between room temperature and 65 K. There is an excellent agreement between the experimental and numerical

286

Günter Nimtz, Recombination in narrow-gap semiconductors 106

,~ I

1 L •1

/

T (s)

Radiative Auger

,

I

,

NA ~ 3)

__

~f II io7 LJ(_\ •

\

(cm

ii

0,9 1014

~g*l2~

2~4~10i2 los/TOK

18.

Fig. Lifetime versus reciprocal temperature of different InSb samples. The broken lines represent theoretical values for intrinsic InSb for radiative and Auger recombination transitions 128, 291.

a)

::~

b)

Fig. 19. Auger transition for symmetrical and unsymmetrical bands.

data. Even in the extrinsic range at temperatures below 140 K the measured lifetime seems to be governed by the Auger effect. The experimental data by Kinch et al. [23] shown in fig. 14 were measured at excess carrier densities very small compared with the thermal equilibrium density. According to eq. (18) the recombination rate and thus the lifetime become dependent on the density of excess carriers if the condition tie ~ t2~is reached. The Auger transition rate has the strongest dependence on carrier density among the known recombination mechanisms. Thus measuring the transient decay should also give information on the dominant recombination mechanism. Such an experiment may even give a more reliable interpretation than comparing theoretical data with the measured lifetime. Theoretical data may have an error up to

one order of magnitude due to the lack of accuracy of the various parameters necessary for the calculation. Decay measurements were carried out by Nimtz et al. [12]with n-type Hg 1 ~Cd~Teat 77 K. Excess carriers were generated by impact ionization in the volume of the sample by applying voltage pulses. After the ionization was switched off the carrier decay was measured with microwaves as described in section 3.5. The transient decay of the excess carriers governed by Auger transitions is expected to be [6,12] dfl~

dt

—

fle(fle+flo+Po)[(flo+ ne)+~8(po+ne)]

2n~rA~

24

Günter Nimtz, Recombination in narrow-gap semiconductors

287

with p.112(i+2p.) (2+p.)

[(iTJ~\.~&

1’L~ \i+p.)kT

For a decay dominated by radiative recombination transitions one expects from statistical arguments a

weaker dependence on excess carriers as dne_ne(no+po+ne) dt

2

)

GR,

where the generation rate for radiative transitions are presented in fig. 20 where r

GR

is given by eq. (i2b). Numerical data of eqs.

(24,25)

0

is the lifetime for n~4 n0.

The experimental values of different n-type Hgo,795Cd0,205Te samples show the strong dependence on carrier density expected for an Auger recombination process. Thus both the temperature dependent lifetime data [23] and in addition the density dependent transient decay have proved that the Auger

effect represents the leading recombination process in narrow-gap n-type Hg1 ~Cd~Te. So far experimental and theoretical lifetime data were introduced which were restricted to temperatures where classical statistics are expected to represent a good approximation. Gerhardts et al. [31] have extended the recombination studies to low temperatures. Gerhardts adapted the Beattie and Landsberg model to the Hg1 ~Cd~Temixed crystal taking band structure and overlap integrals from a k p-calculation. Calculating with non-classical statistics a carrier lifetime was obtained as plotted in fig.

21. The results show qualitatively a lifetime depending exponentially on temperature at low and at high 10 Hg795Cd205Te (Te77K)

~

\

n0

002E~.

0

____

calculated radiative ‘ Auger

~6 ~-__ _______

1

2 tI-k

‘~ _______

Fig. 20. Transient decay of excess carriers in n-type Hgi ,,Cd~Te.Solid and broken lines represent numerical values for Auger and radiative recombination. r,, is the lifetime for ne 4 n0 [121.

3/T 0<) F1O20m~3~+211~ 10 100 1O Fig. 21. Lifetime versus reciprocal temperature for n-type Hg 0 8Cd0.2Te. Solid lines represent theoretical data for two extrinsic carrier densities, dots represent experimental values [31].

288

Günter Nimtz, Recombination in narrow gap semiconductors

temperatures. At intermediate temperatures, however, the lifetime is expected to decrease with decreasing temperature. This theoretical dependence may be understood from the discussion of the relation which follows from eq. (21) considering only terms with a strong temperature dependence [311, similarly as it was done for a non-degenerate and intrinsic semiconductor with eq. (23) LCJdsk1 Jd3k2f(E1)f(E2)(1

f(E1.))~Nv(T)C

F~/kT

(26)

In this equation the exponential with 2

E~, mm —

2lflh

(k1 + k2 k1) —

corresponds to the last one eq. (23). In the semiconducting compound with sufficiently large Eg we may neglect f(E 2, thus at high temperatures in the intrinsic 1). Then the integral is proportional range the temperature dependence is dominated by r ~to nn 2 ~ exp(Eg/kT). At lower temperatures the material becomes extrinsic, the carrier density being independent of T. If T is still so large that the exponential term in eq. (26) is not yet important, we have T N~(T) T312, i.e. r decreases in the intermediate temperature interval with decreasing T. At very low temperatures the exponential becomes important and ‘r increases again rapidly with decreasing temperature, the apparent activation energy being by a factor of the order of mC/mh smaller than at high temperatures (in Hg~. 8Cd0.2Te: 10

mC/mh

2)

In fig. 21 also experimental data are presented. The agreement with the theoretical data is excellent in the intrinsic and extrinsic range at temperatures above 50 K. The experimental data are shifted to somewhat higher temperatures at lower temperatures if compared with the theoretical data. The lifetime saturation observed in the experiment at very low temperatures is caused by an experimental limitation. Here the lifetime becomes saturated due to a minority carrier sweep-out in the electric field of the bias voltage. According to various experimental and theoretical investigations it has become evident that the Auger effect represents the most important recombination mechanism in narrow-gap semiconductors of the ternary alloy system Hg1 ~Cd~Te.Theoretical values for different semiconducting and semimetallic compositions of the same mixed crystal are plotted in fig. 22. According to these data the low temperature exponential increase of lifetime, which takes place in semiconducting crystals disappears in semimetallic crystals, since for Eg < 0, we have E0 = 0. But also at high temperatures the simple exponential behaviour is missed in semimetallic compositions, since the chemical potential remains in the conduction band. Then the Fermi functions 1(E1) imposes an important restriction on the possible recombination transition. The influence of the screening of the Coulomb interaction due to free carriers is demonstrated in fig. 22. The screening radius A is defined in eq. (22f). The broken lines in fig. 22 represent unscreened data with A 0,3.whereas thethe solid represent screened data for and an extrinsic carrier with density of 10~ Obviously freelines carrier screening becomes more more important decreasing electrons/rn energy gap, particularly in semimetallic crystals. —

An interesting feature was obtained by calculating the overlap integrals of conduction and valence band Bloch-functions with functions from a k p-calculation. This calculation yields in the extrinsic low-temperature range the proportionality r n 813, whereas earlier estimates [28,32,33] yielded .

T ~

fl 2. 0

0

289

Günter Nimtz, Recombination in narrow-gap semiconductors

0xe

I

lOOK 10

hal

a,l,,ai

10K 100

N ~ l.a.”..,

3/T (K)

~

10

Fig. 22. Auger lifetime versus reciprocal temperature for various compositions of the Hgi 4Cd4Te mixed crystal [31].

Both theoretical and experimental results of the temperature dependence of lifetime in semiconducting n-type Hg1 ~Cd~Temade us3 to suggest, that a photoconductivity detector may be operated more favourably near 100 Kwith thananat extrinsic the usualcarrier 77 K. density of about i0~ electrons/rn Near 100 K the lifetime has its maximum and additionally the hole mobility is lower at 100 K than at 77 K, accordingly a higher bias voltage can be applied to the detector element without causing a minority carrier sweep-out. The detectivity of an n-type Hg 1 ~Cd~Teinfrared photoconductor may be substantially improved, if the device is cooled down to 4.2 K. However, the full gain in detectivity will be obtained only under a reduced background radiation as it may be the condition in astrophysical measurements taken at a satellite. A 300 2mHzhh’2W K-background Iradiation limits the detectivity of a HgCdTe photoconductivity detector to about D*i0l Besides Hg 1 ~Cd~Te some lead salt mixed crystals represent the most important narrow-gap semiconductors as was mentioned in the introductory sections. Does the Auger effect also play the leading part in the carrier recombination in narrow-gap lead salts? From the discussions above one would expect a less important role if compared with narrow-gap Hg1 ~Cd~Tesemiconductors, since in the lead salts in question valence and conduction band structures are nearly symmetrical (myB/mcB

1). Accordingly the effective energy gap for a recombination transition is markedly higher than the intrinsic band gap (see eq. (23)). This handicap may be canceled to some extent by taking into

consideration the many valley-structure of the energy bands as shown in fig. 6. It was pointed out by Emtage [34]that due to the interaction between electrons of non-equivalent valleys, i.e. in valleys elongated parallel to different diagonals in k-space, the mass ratio p. becomes essentially r = m~/m1. Here m1 and m1 represent the masses in the k-directions parallel and perpendicular to the principal axes of the energy ellipsoids. For Pb1_~Sn~Tecrystals this ratio is about 1/iO. An intervalley scattering

29))

Günter Nipntz, Recombination in narrow-gap semiconductors

process is sketched in fig. 23. Emtage showed that the intervalley scattering yields with p.

1 an

exponential n~exp(— TEgI2kT),

T

(27)

instead of T

~

n~exp(—E0I2kT).

Therefore r plays a similar part as p. (see eq. (23)) in the temperature dependence and magnitude of the Auger recombination rate. Many experimental investigations were carried out with various compositions of Pb1 4Sn4Te, however, most of the observed lifetime data seem to be determined by a Shockley—Read recombination [5,22, 35, 36]. Often two decay times of the excess carriers were observed and for instance Schlicht et al. [22] found a minority carrier trapping in PbTe samples at low temperatures. In Pb0 8Sn61 2Te crystals, however, having a smaller bandgap than PbTe the lifetime may be governed by the Auger effect. Experimental and theoretical data versus temperature are shown in fig. 24. The experimental values show a behaviour as expected for an Auger recombination at temperatures above 20 K. It is somewhat strange that the observed lifetime is about one order of magnitude higher than the theoretical one, which was calculated according to Emtage’s model [22].At present it is speculated that the screening of the electron-electron interaction has not been considered properly. Since the recombination energy is much higher than the relevant optical phonon energythe Coulombinteraction was assumed to be screened only by valence electrons. Therefore in eq. (22) the high frequency dielectric constant (~,. 50) was considered and not the low frequency dielectric constant. However, the low frequency dielectric constant (zr static dielectric constant) being extremely high (er,. 1000—10 000) in the lead salts a screening due to the lattice oscillations is expected to some extension [5]. Thus we conclude that in the lead salts with decreasing energy gap the Auger effect becomes more and more important, however, the agreement between theory and experiment is not satisfactory until now. The main reasons for the

~ ‘~~~“0 p~p

-

10

vaUey

10

~ey

~3~CB

5(the~

i

/

-

_

\

I ‘~ Fig. 23 Auger transition due to an e—e impact between two non equivalent energy valleys

.l.,,.i

~

10 100 T (K)— Fig. 24. Temperature dependent lifetime of n-Pb,inSn,,2Te. Theoretical data are calculated for radiative and Auger recombination [22].

Günter Nimtz, Recombination in narrow-gap semiconductors

291

discrepancies are assumed to be due to the lack of accurate band parameters of the lead salts, and due to the theoretical description of the screening in this problem. As discussed in section 4.3 the radiative recombination rate can be given in terms of optical absorption data as was shown first by Roosbroeck and Shockley. Such a calculation is based on detailed balance between radiative recombination and photoionization. Landsberg [37]calculated the analogue for Augerrecombination and impact ionization. In this investigation an expression for the Auger lifetime depending on the probability of impact ionization was obtained. However, to get the probability of impact ionization from experimental data is much more difficult than to obtain that of photoionization from optical data. This seems to be the reason that the

interesting theoretical results have not been tested in an experiment until now. 4.5. Single-phonon recombination transitions

In a semiconductor the various types of the electron—phonon interaction contribute to the scattering of free carriers. Particularly at elevated temperatures the transport properties are governed by the

electron—phonon interaction. In the recombination of carriers, however, the electron—phonon interaction is only involved in combination with lattice defects. In a semiconductor having a bandgap of the order of 1 eV a direct recombination of electron—hole pairs by the emission of phonons or other elementary excitations such as plasmons is not probable. Since the energy of these excitations is usually only of the order of 0.01 eV, a large number of these excitations have to be released. Thus other recombination mechanisms have proved to be much more likely. A direct phonon recombination, however, might become important, even dominant, in narrow-gap semiconductors having bandgaps

corresponding to the energies of such elementary excitations. Interband recombination processes by emitting single elementary excitations have been considered first by Dimmock et al. [38] with respect to phonons and by Wolff [39] in the case of plasmons. According to their discussions these recombination channels are expected to reduce substantially the detectivity of infrared photodetectors or to quench the LOiC) TOiri uescreened 51 LOiXI t I

O

ph

)o.u(

~ t

9

1.00

0

91. 23

25 meV 15

~~_—+

210.10

1.5

057

/

007

12

Pb063 Sn0

013

—.—

a

1~O 160 T (K) Fig. 25. Lifetime and photoconductivity of Pbi 4Sn0Te versus temperature (=gap energy). The arrows indicate the temperature where the gap energy equals the energy of phonons and plasmons [40]. 111111

292

Günter Nimtz, Recombination in narrow -gap semiconductors

luminescence of a diode as the gap energy approaches the optical phonon energies or the plasmon energy.

In order to search for these recombination channels Cammack et al. [40]used the temperature dependence of the bandgap of PbSnTe to adjust the gap energy to the energy of different phonons and coupled plasmons and phonons. Experimental results of the photoconductivity versus temperature are

shown in fig. 25. The experiment was carried out with Pb0 63Sn6.37Te where the bandgap decreases with decreasing temperature. At temperatures below 100 K where the gap energy becomes resonant with the different elementary excitations, responsivity and lifetime drop dramatically. The experimental data seem to

confirm the predicted importance of the single-excitation recombination channels, in spite of the fact that the results do not show any structure which may be related to a specific recombination channel. In searching for single-elementary excitation recombination transitions Dornhaus and Nimtz [41—43] varied the bandgap by varying an applied magnetic field to establish resonance condition (fig. 26). Starting with semimetallic compositions of n-type Hg1 xCdxTe, with x in the range between 0.14 and

0.165, the magnetically induced bandgap can easily be adjusted to the relevant energies of elementary excitations by applying moderate magnetic fields and in addition the resonance can easily be switched on and off. In this study photoconductivity and time resolved lifetime measurements were carried out at 4.2 K in magnetic fields up to 8 T. Typical results of the photoconductivity are shown in fig. 27. The photoionization was performed with two lasers of different wavelengths. A GaAs laser (A = 0.9 ~i.m)and a CO2 laser (A = 11 p~m)were used in order to rule out any resonance absorption effect which might cause a photoconductivity modulation depending on magnetic field. According to fig. 27a the observed

structures in the photoconductivity signal were not affected by the ionization energy. Thus one can conclude that the observed strong decrease of photoconductivity between 1 and 1.5 T is due to an enhancement of the recombination rate at a distinct bandgap. The structures seen at fields below 0.8 T have been identified as Shubnikov—de Haas-type oscillations, since their position is in agreement with the corresponding structures observed in the magneto-resistance data [3]. The marked drop of the photoconductivity found between 1 and 1ST can be related to a single-LO-phonon transition.

H_

r~ hg~Cd155Te

~7K B(T) Fig. 2( Band structure of HgiimsCdii ssTe. Left side shows the energy-momentum relation without magnetic field Right side shows the magnetic field dependence of the band extrema at the F point [41).

Günter Nimtz, Recombination in narrow-gap semiconductors

1

293

42K x~0.155

Uphoto iarb unitsi

OO

5J01~52D

____ BIT) BIT) Fig. 27. (a) Photoconductivity versus magnetic field (= gap energy). The photoionization was carried out with a GaAs laser (A 0.9 ~m) and a CO2 laser (A = 11 jim), (b) Photoconductivity versus magnetic field (—gap energy) with structures at the one-LO-phonon and at the two-LO-phonon energy resonance [41,42].

Experimental data from three different semimetallic compositions of Hg1 ~Cd~Teare shown together with theoretical values in fig. 28. The numerical plot gives the magnetic field dependent on composition ratio x, at which the LO-phonon energy becomes resonantwith the magneticfield induced bandgap. The HgCdTe mixed crystal represents a two-mode system, having a HgTe-like and a CdTe-like reststrahlen band. Both modes have a polar-optical coupling to the free carriers [2].Since the oscillator strength is proportional to the composition ratio x we expect the HgTe-like phonon to be the most important one in the investigated crystals. Phonon and plasmon energiesand other data ofthe investigated crystals are given in table 2.

~ Eg(B0) —60

—20

0

20

1meV)

0.14 0.15 0.16 0.17 0.1% x Fig. 28. Resonance positions for single LO-phonon recombination, two-phonon transitions and plasmon transitions versus bandgap (at B — 0) and composition of the mixed crystal Hg1 0Cd0Te. Experimental results of three compositions (x = 0.165; 0.155; 0.140) are presented [42].

2t4

Günter Nimtz, Recombination in narrow-gap semiconductors Table 2

Sample characteristics (T

Hgi 1Cd~Te

42K) 2. 41.

Energy gap E~(meV)

42]. Pu1

and m1 F represent the conduction band edge mass and the effective mass at the Fermi energy respectively

oh1

/oi,,

m1 in hw~°(meV)

o

I). lbS ((.155 ((.14)1

-4-0.5 19(1 47.0

4x I)) IS x 10 5.9 x II)

lx 10 10 59 x (0

3.3 x

7.! 17.) 17 1

Phonon bo ~1(meV)

19 1 19.1 19 I

n (cm

~)

2x 2 ~101 (H ~ 101

Plasnsons hw~(meV)

7.2 95

As shown in fig. 27b the photoconductivity versus magnetic field curve has a second structure near 3 T. According to theoretical data the magnetic field position of this structure can be related to a two-phonon transition as presented in fig. 28. Comparing the lifetime deduced from time-resolved decay measurements at the single-phonon transition and at the two-phonon transition it was found that the lifetime related to the two-phonon transition exceeds that of the single-phonon transition at least by two orders of magnitude. This result seems reasonable bearing in mind that a two-phonon resonance transition corresponds to a higher order perturbation interaction than the one-phonon transition. According to the experimental observations [40—43]and in agreement with the predictions [38,39] the

single LO-phonon transitions dominate the recombination behaviour, if energy-resonance between bandgap and single-phonons takes place. These recombination channels cause a severe limitation of the

detectivity in photoconductivity detectors at wavelengths above 50 ~m. On the other hand the channels may be used to construct very fast detectors or modulators and also to generate high densities of LO-phonons.

4.6. Plasmon recombination Another important frequency in a semiconductor is the plasma frequency of conduction electrons. The frequency of this collective oscillation is given by the formula 2/em * (28) w n0q where n 0 is the free carrier density, m* the mass of the carriers in question and e the dielectric constant. Recombining electrons and holes may emit the quantum of this plasma field either spontaneously, or in a stimulated emission process. The emission rate is expected to become very high if the bandgap is adjusted to the plasmon energy in a way similar to single-LO-phonon transitions as has been discussed in the last section. Such phonon recombination channels were predicted and investigated in various theoretical papers [39,44—47].In searching for such transitions Dornhaus and Nimtz [41—43] have carried out experiments with semimetallic n-type Hg1~.86Cd6114Te. The bandgap was magnetically tuned to the resonance condition between plasmon and bandgap energy in the same way as described for the LO-phonon transitions (section 4.5). There was observed a lifetime dependence on bandgap energy qualitatively in agreement with numerical data presented in fig. 29. The values were calculated by adopting and slightly modifying the theoretical results of Fussing et al. [46] and of Elci [45].It was assumed that the magnetic field generates Landau-levels and hence induces a bandgap, other magnetic field effects were neglected. The overlap integral between conduction and valence band states was taken —

Günter Nimtz. Recombination in narrow-gap semiconductors

Tls(

295

/ Plusmon Lifetiii—m

/

iO~-

/ Findu Plusmon I (—lps)

10.10)

~WP(EF)Z~SrneV

/

u~210

~z~e(

1011 0 1 2 3 BIT) 4 Fig. 29. Calculated dependence of the plasmon dominated excess carrier lifetime versus magnetic field for infinite and finite plasmon lifetimes [421.

B—0 B*0 6-0 B~0 Fig. 30. Sketch to illustrate the modification; (a) of the band structure and (b) of the density of states in a magnetic field [11,50].

as constant with the usual value 0.1, and the plasmon occupation number was approximated by 1. The lifetimes presented in fig. 29 have been evaluated under these assumptions for a degenerate extrinsic n-type Hg0.86Cd0.14Te-sample with a carrier concentration of 6.9 x 10~electrons/m3. In the paper of Tussing et al. [46] the plasmon lifetime was taken as infinite and therefore the recombination rate goes to zero for hai 1, E~.However, as Elci [45] pointed out, the plasmon decay in semiconductors like Hg1 ~Cd~Te is very rapid, accordingly plasmon recombination transitions can actually occur for bandgaps larger than hw~as a result of the short lifetime of the plasmons and the consequently broadening of their energy. Adopting the results of Elci [45]one may crudely estimate the effect of this broadening on the electron lifetime as indicated in fig. 29. Comparing these numerical lifetime values with recent results of Gerhardts et al. [31] on Auger recombination, which has been shown to be the dominant process in n-type HgCdTe crystals, plasmon recombination is seen to be extremely effective in the energy range hw~~ Eg. According to the experimental results the plasmon recombination rate is one to two orders of magnitude larger than the Auger recombination rate. Above a critical recombination rate, the emission is expected to become a stimulated oneexperiments [39,47,48]. 3s). In the ElciDornhaus [48] estimated the critical recombination raterecombination to be about 10~° by and Nimtz [41,42] spontaneous rateselectrons/(m up to 1026 electrons/(m3s) were obtained. With some efforts the critical intensity for stimulated plasmon emission should be reached in such an experiment. In any case, both recombination channels, the single LO-phonon and the single plasmon, can be used to obtain extremely high intensities of these quasi-particles.

4.7. Cyclotron-resonance enhanced Auger transitions A new magnetic quantum effect involved in the recombination of free carriers in narrow-gap

Gunter !Vimtz. Recombination in narrow gap semiconductors

29n

semiconductors was predicted by Takeshima [49] in 1973. Takeshima pointed out that an Auger transition should become highly probable at distinct magnetic fields. The effect is illustrated by the sketch of fig. 30. As has been discussed in section 4.4 Auger transitions are handicapped by the simultaneous conservation of both energy and momentum. According to Beattie and Landsberg [28] this resulted in an effective bandgap up to 50% higher than the intrinsic bandgap and correspondingly to an exponentially lower Auger recombination rate (eq. (23)). However, in a magnetic field, as shown in fig. 30a vertical transitions become allowed if the bandgap equals the energy separation between two levels of the Landau spectrum. The Landau levels are represented by the first term of the free carrier energy in magnetic fields E~ (~+n)qBIm*±~p~8gB +~h2k~/rn*.

(29)

Here n —0, 1,2,.. , m* is the effective mass, ~ is the Bohr magneton, g is the effective g-factor and k~is the free carrier wave-vector component parallel to the applied magnetic field B. Besides the modification of the energy states the density of states is also significantly redistributed in a magnetic field. For the density of states the dependence on magnetic field is given by .

Z(E) dE

~

—

{E

(n + ~)hw~±2gILBB}

112dE

(30)

where w. = qB/m * is the cyclotron resonance frequency and the sum is to be taken up to a maximum value, nmax which makes E (n + ~)hw~.The density of states is also plotted schematically in fig. 30b neglecting the spin splitting of the Landau levels. Thus the magnetic quantization of the eigenvalues allows vertical Auger transitions at distinct magnetic fields. These transitions are characterized by the smallest amount of transition energy since they take place at k = 0. In addition such transitions are favoured by the maxima of the density of states function eq. (30). The maxima in the Auger transition rate corresponding to lifetime minima are expected near magnetic fields where the equation holds nhw 0 = Eg(B

—

0)+ ~h(w~0+ w~~) ~.p.8B(g0+ g~)

(31)

where w~.and w~.are the cyclotron resonance frequencies of conduction and valence band respectively. Spin-flip processes are neglected, accordingly only spin-up levels are considered in eq. (31). Adopting Beattie and Landsberg’s model for magnetic fields, Takeshima [49] carried out some numerical calculations for InSb. As was pointed out in sections 4.2 and 4.4 in this semiconductor the recombination of carriers is governed by a Shockley—Read mechanism at low temperatures. However,

magnetic quantum effects usually become observable only at very low temperatures. Searching for these magnetic quantum oscillations Dornhaus et al. [50, 11] carried out magnetic field dependent lifetime experiments with various narrow-gap compositions of the Hg1 Cd~ Te alloy. As shown in fig. 31 for three different Hg1 ~Cd~Tecrystals, there were observed pronounced lifetime oscillations in magnetic fields. Taking into consideration the strong non-parabolicity of the conduction band the expected magnetic field positions of the lifetime minima were calculated [50, 11]. They are indicated by arrows in fig. 31. The agreement between the observed minima positions and the numerical ones is very good. The reader may ask, why these Auger oscillations where not observed in the experiments presented in sections 4.5 and 4.6, where the bandgap was opened by magnetic fields. These experiments, however,

297

GünterNimtz, Recombination in narrow-gap semiconductors

were carried out with semimetallic compositions in which the Landaulevel splitting is always larger than

the induced energy gap. One feature of the observed magnetic quantum oscillations was unexpected: Takeshima’s model predicts a strong lifetime decrease at moderate fields, whereas in the experiments, always an increase by about a factor of 20 was observed. This is the range of fields up to about 1 T as seen from the plots in fig. 30. It was pointed out by Nimtz [11]that this effect opposite to Takeshima’s calculations might be due to the spin polarization taking place in the experiments. Takeshima did not consider the

spin-splitting of the Landau levels. However, in the investigated semiconductors at fields above 1 T only the spin-up state will be occupied. Accordingly at elevated magnetic fields only an interaction between electrons having the same spin takes place and the usually dominant, much stronger interaction between electrons of opposite spins ceases with increasing fields. Gerhardts [51,52] carried out calculations in

which he included non-parabolicity and spin-splitting of the conduction band for various n-type Hg1_~Cd~Te semiconductors. Numerical data are shown in fig. 32a. Both the lifetime increase at low fields caused by the spin polarization and the magnetic field dependent positions of the lifetime minima

are in agreement with the observed data. For comparison in fig. 32b numerical values are shown corresponding to Takeshima’s model without spin-splitting. So these magnetic quantum oscillations of the Auger transition rate show some interesting features of the electron—electron interaction from the

theoretical point of view. In addition it seems to us that this magnetic quantum effect plays an important role also in device applications. The strong increase of lifetime by about a factor of 20 due to the

spin-polarization and the decrease of mobility in the magnetic field can be used to develop a sensitive infrared photoconductivity detector. This detector, however, is expected to be useful only under a reduced background radiation level as may happen e.g. in cosmic problems. Today an excellent 11 ~m n-type HgCdTe infrared photoconductivity detector achieves already 300 K-background radiation limited performance, when operated at 77 K.

0~.-6~0~—5~0~—4~

(ork) uni

0~- Q~5~ ~

30 ~Xr0.216

/

20

U)

xr0.20 10

i/

Hg1

~Cd~Te(T~4:Kl

8

BIT) Fig. 31. Experimental photoconductivity and lifetime versus magnetic field for three different n-type Hg1 uCduTC compositions [11,501.

Fig. 32. Theoretical Auger lifetime data versus magnetic field for Hgo gCdo2Te; (a) with spin-splitting and (b) neglecting the spin-splitting of the Landau levels [51].; represents the lifetime for B — 0.

298

Günter Nimtz, Recornhination in narrow gap semiconductors

0052

0048

>-.

0

-

/

T

7

EGI ~ 0

0056

o

T2

~0052~

°~ 0048

EG~iT2~

-

:0:00

10

20 MAGNETIC

50 FIELD

Fig. 33. Magnetic field dependence of laser emission in Pb1 ~SnuSediodes with x energy gap decreases with increasing magnetic field [53].

60

(kG)

0.10 and x

0.19. In the composition with x

((.19 the effective

There is some evidence that the cyclotron resonance enhanced Auger recombination rate may limit the magnetic tuning of a PbSnSe laser diode at long wavelengths. In Pb1 ~Sn~Semixed crystals with x > 0.15 the effective bandgap shrinks with increasing magnetic field, the Zeeman-splitting being larger than the Landau level splitting. Experimental results of magnetically tuned Pb1 Sn~Se laser diodes were reported by Calawa et al. [53] and some of their data are shown in fig. 33. The energy of the emitted photons is plotted versus magnetic field for two compositions, one having an increasing bandgap and another having a decreasing bandgap in magnetic fields. The term scheme of the transitions is shown in the inserts of fig. 33. Obviously the laser line can easily be tuned, but it was found that the T1-line of Pb0 81Sn0.19Se, which shows a decrease of energy with increasing magnetic field ceased near 8 T at a wavelength of 34 rim. A laser emission at a longer wavelength was not observable. We think that the laser operation died at this wavelength due to a cyclotron resonance enhanced Auger recombination. Near this critical magnetic field the effective bandgap equals the energy difference of two Landau levels resulting in a very high Auger transition rate, which seems to exceed the radiative transition rate. Thus the cyclotron resonance enhanced Auger recombination mechanism can play a limiting role in magnetically tuned laser diodes. 5. Conclusions It was the intention of this article to introduce and to discuss recombination mechanisms which are important for the modern class of narrow-gap semiconductors. It should be mentioned that the

Günter Nimtz, Recombination in narrow-gap semiconductors

299

references discussed and cited are not exhaustive. The present state of art of the research in the field of recombination in semiconductors in general may be obtained from studying the proceedings of the conference of Recombination in Semiconductors which took place recently in Southampton/UK [54]. According to the papers presented at this conference many new recombination effects were discovered and it becomes evident that the study of recombination mechanisms is still a field of current interest. Particularly some new processes were observed, which are associated with narrow-band gaps. So it has become obvious that the Auger effect represents the most efficient intrinsic recombination process

in semiconductors with a small energy gap. In the lead salts often an extrinsic mechanism, namely a Shockley—Read process governs the lifetime of the excess carriers, but this behaviour seems to become less important with improving quality of the crystals. Besides the Auger effect, in narrow-gap

semiconductors single-LO-phonon and single-plasmon recombination channels were observed and a new magnetic quantum effect was discovered, an oscillatory behaviour of the Auger transition rate. All the new recombination mechanisms seem to be established from the experimental point of view, however, there are some theoretical problems left, at least some of the theoretical models have to be improved. For instance the assumption of a quasi-equilibrium becomes highly questionable at low temperatures [31].Here the intraband relaxation time may easily reach a value of the same order of magnitude or may even exceed the excess carrier lifetime. In this case strong deviations from a purely

exponential carrier decay become probable. Another unsolved problem represents the screening in semiconductors having extremely high static dielectric constants. For example the static dielectric constant of the lead salts is about 10~at low temperature. Is there any contribution from the lattice to the screening of the electron—hole interaction? In order to obtain a quantitative understanding of the recombination mechanisms in narrow-gap semiconductors some of the theoretical models have to be refined and on the other hand the experimentalists have to deliver more accurate values of bandparameters for some of the narrow-gap semiconductors. Acknowledgements The author is very grateful to Burghard Schlicht for a critical reading of the manuscript and to Rolf R. Gerhardts for many discussions on recombination in semiconductors. References [1] With kind permission: DFVLR, Deutsche Forschungs- und Versuchsanstalt für Luft- und Raumfahrt E.V. (Oberpfaffenhofen) and SAT, Societe Anonyme de TélCcommunications (Paris). [2] R. Dornhaus and G. Nimtz, The Properties and Applications of the Hgi ~C&Te Alloy System, in: Springer Tracts in Modern Physics, Vol. 78 (1976). [3] R. Dornhaus and G. Nimtz, Solid State Communications 22 (1977) 41. [4] T.C. Harman and I. Melngailis, in: Applied Solid State Science, Vol. 4 (1974). [5] 0. Nimtz and B. Schlieht, Narrow-Gap Leadsalts, in: Springer Tracts in Modern Physics (to be published 1980). [6] IS. Blakemore, Semiconductor Statistics (Pergamon Press, 1962). [7] RH. Bube, Photoconductivity of Solids (John Wiley & Sons, Inc., New York, 1960). [8] T.S. Moss, Optical Properties of Semi-Conductors (Butterworths Scientific Publications, London, 1959). [9] S.M. Ryvkin, Photoelectric Effects in Semiconductors (Consultants Bureau, New York, 1964). [101S.W. Kurnick and R.N. Zitter, J. AppI. Phys. 27 (1956) 278. [11] 0. Nimtz, R. Dornhaus, K-H. Muller and M. Schifferdecker, Lecture Notes, Intern. Conf. The Application of High Magnetic Fields in Semiconductor Physics, Wurzburg 1976, p. 321.

3(X)

[12]G

Günter Nimtz, Recombination in narrow gap semiconductors

Nimtz. G. Bauer. R. Dornhaus and K-H. Muller. Phys. Rev. B 10 (1974) 3302. [13] D.R. Hamann and A.L. McWorther. Phys. Rev 134 A (1964) 250. [14]V.L. Bonch-Bruewich and E.G. Landsberg, Phys. Stat. Sol. 29 (1968) 9. [15] PT. Landsberg, Phys. Stat. Sol 41(1970) 457~ [16] ‘N. Shockley and W.T. Read. Phys. Rev. 87 (1952) 825. [17]C.H. Henry and D.V. Lang, Phys. Rev. B 15(1977) 989. [18]BK. Ridley, J. Phys. C 11(1978)2323 [19]‘N. Schneider, H. Groh and K. Hüber, Z. Physik B 25(1976) 29 [20] R.A. Laff and H.Y. Fan, Phys. Rev. 121 (1961) 53. [21] H.D. Baumgart, G. Nimtz and P. Kokoschinegg, Phys, Stat. So). (a) 12 (1972) 477 [22] B. Schlicht, R. Dornhaus, G. Nimtz, L.D. Haas and T. Jakobus, Solid-State Electronics 21(1978)1481. [23] M.A. Kinch, M.J. Brau and A. Simmons, J Appl. Phys. 44 (1973) 1649. [24]H. Preier, AppI. Phys. 20 (1979) 189 [25]L R. Tomasetta and C.G. Fonstad, IEEE J. Quantum Electronics 11(1975)384. [261J. Hesse, Jap. J. AppI. Physics Suppl 16-1 (1977) 297. [27]J.N. Walpole, A.R. Calawa. T.C. Harman and 5.H. Groves. AppI. Phys. Letters 28 (1976) 552. [281A.R. Beattie and P.T. Landsberg, Proc. Roy. Soc. A 249 (1959) 16. [29]P.T. Landsberg, Abhandlungen der Deutschen Akademie der Wiscenschaften zu Berlin, Nr. 7(196(1) 57. [30]P.E. Petersen, J. AppI. Phys. 41(1970)3465. [31] R.R. Gerhardts, R. Dornhaus and G. Nimtz, Solid-State Electronics 21(1978)1467. [321E. Antoncik and P.T. Landsberg, Proc. Phys. Soc. 82 (1963) 337. [33] A.R. Beattie and G. Smith. Phys. Stat. Sol. 19 (1967) 577. [34] P.R. Emtage, J. AppI. Phys. 47 (1976) 2565. [35] K. Lischka and W. Huber, 1. AppI. Phys. 47 (1976) 2565 [36] P. Berndt, D. Genzow and K.H. Herrmann, Phys. Stat. Sol. (a) 38 (1976) 497. [37]P.T. Landsberg, Proc. R. Soc. Lond. A 331 (1972) 103. [38]JO. Dimmock, I. Melngailis and A.J. Strauss, Phys. Rev. Letters 16 (1966) 1193. [39] P.A. Wolff, Phys. Rev. Letters 24 (1970) 266 [40] D.A. Cammack, A.V. Nurmikko, G.W. Pratt Jr. and J.R. Lowney, J. AppI. Phys. 46 (1975) 3965 [41] R. Dornhaus and G. Nimtz, Solid State Communications 27 (1978) 575 [42] R. Dornhaus and G. Nimtz, Solid-State Electronics 21(1978)1471. [43] R. Dornhaus, Dissertation Mathematisch-Naturwissenschaftliche Fakultlt, UniversitIt zu Köln (1978) [44] V.L. Bonch-Bruevich, Soy. Phys. Solid State 1 (1960) 984. [45] A. Elci, Phys. Rev. B 16 (1977) 5443. [46] P Tussing,w. Rosenthal and A. Haug, Phys. Stat. So). (b) 53(1972)451. [47] N.S. Baryshev, Soy, Phys. Semicond. 9 (1976) 1324. [48] A. Elci. Phys. Rev. B 19 (1979) 4181. [49] M. Takeshima, J. AppI. Phys. 44 (1973) 4717. [50] R. Dornhaus, K-H. Muller, G. Nimtz and M. Schifferdecker. Phys. Rev. Letters 37(1976) 710. [51] R.R. Gerhardts, Solid State Communications 23 (1977) 137. [52] R R. Gerhardts, in: Physics of Narrow Gap Semiconductors. Proc. III Intern. Conf. Warszawa 1977. eds. J. Rauluszkiewic7 M Gorska and F Kaczmarek, p. 103. [53] A.R. Calawa, JO. Dimmock, T.C. Harman and I. Melngailis, Phys. Rev Letters 23(1969) 7 [54] Recombination in Semiconductors, Intern. Conf.. eds. PT. Landsberg and A.F.W. Willoughby, in: Solid State Electronics 21(1978) 1273—1618. [55] L. Huldt, N.G. Nilsson and KG. Svantesson, Appl. Phys. Lett 35(1979) 776 [56] R. Conradt, Festkörperprobleme XII (1972) 449. [57] T.S. Moss, Proc. Phys. Soc. 66 B (1953) 993. [58] J.S. Blakemore, Proc. Intern. Conf. Phys. Semicond. Prague (1960). [59] R.N Zitter, A.J. Strauss and A.E. Attard. Phys. Rev. 115 (1959) 266