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SEMICONDUCTOR PHYSICS | Recombination Processes
SEMICONDUCTOR PHYSICS / Recombination Processes 21
Kelly MJ (1995) Low-Dimensional Semiconductors. Oxford, UK: Clarendon Press. Orton JW and Blood P (1990) The Electrical Characterisation of Semiconductors: Measurement of Minority Carrier Properties. London: Academic Press.
Perkowitz S (1993) Optical Characterisation of Semiconductors. London: Academic Press. Wolfe CM, Holonyak N Jr and Stillman GE (1989) Physical Properties of Semiconductors. Englewood Cliffs, NJ: Prentice Hall.
Recombination Processes P T Landsberg, The University of Southampton, Southampton, UK q 2005, Elsevier Ltd. All Rights Reserved.
Introduction In studies of recombination there occur considerable complications; many are associated with interactions between electrons, electrons and phonons, excitons, bi-excitons, impurity centers, etc. Most of these are not required in the present exposition. Modern work normally assumes that the basics of recombination physics are understood and the present exposition offers an appropriate outline.
These electron –boson interactions result in transitions of single electrons in an energy band scheme. We shall attach a superfix ‘S’ (for single-electron) to the recombination coefficients for such processes. These are then denoted by B S or T S depending whether only bands are involved or whether traps are also involved. They are illustrated in Figure 1, where the solid horizontal lines represent the conduction and valence band edges and the dashed line refers to traps. Note that the two-electron transitions do not have the superfix ‘S’, while n and p refer to the electron and hole concentrations; N0 and N1 are the concentration of unoccupied and occupied trap states, respectively. Electron –Electron Interaction in Traps
Basic Assumptions Electron – Electron Interactions for Electrons in Bands
The recombination problem in semiconductors is greatly complicated by the interaction of the electrons with each other. This allows one to speak only of the quantum states of the semiconductor crystal as a whole. However, as in metals, so also in semiconductors, a simplified picture is successful. In this, the electron interactions, and other interactions, are first neglected, but are later taken into account as a perturbation. Thus, the electrons are first treated as if they can move through each other; the fact that they collide and deflect each other by virtue of their Coulomb interaction is treated as a perturbation. The transitions are still described within the framework of the single-particle states of the unperturbed problem. The Effect of Electron – Boson Interactions
Our first approximation is to neglect most (but not all) electron interactions. Later we take into account the two-electron transitions that arise. This twoparticle recombination process is referred to as Auger recombination and its inverse as the generation of current carriers by impact ionization. In addition, electrons interact with the radiation and lattice fields and emit or absorb photons or phonons.
Electron – electron interactions can at least formally be taken into account in connection with electrons trapped in a center: the spectrum is a function of the number, r ð¼ 1; 2; 3; …; MÞ; of electrons captured. The ‘irremovable’ electrons can be included with the ion core. Given that the center captured r electrons (say), it can still be in a variety of quantum states, and they will be denoted by the symbol ‘; yielding a set of quantum states ð‘; rÞ for a center. The energy of such a state, divided by kT to make it dimensionless, is denoted by hð‘; rÞ; yielding the canonical partition function Zr ¼
X
exp½2hð‘; rÞ
½1
‘
The location of a center in space will here be considered to be of no significance. Centers can capture several electrons. This brings in a need for the chemical potentials (or Fermi levels) for r-electron centers. They are here denoted by g (when divided by kT) and the suffix ‘eq’ denotes an equilibrium value. Thus, the equilibrium probability of finding an r-electron center in state ‘ is given by: Pð‘; rÞeq ¼
exp½rgeq 2 hð‘; rÞ M X ðexp sgeq ÞZS s¼0
½2
22
SEMICONDUCTOR PHYSICS / Recombination Processes
Figure 1 Definition of recombination coefficients. Transition rates per unit volume are stated with each process, and, in brackets, for the reverse process. Thus B1, B2, T1 –T4 refer to Auger processes; B s, T1s, T2s refer to single-electron recombination; Y s, X1s, X2s refer to carrier generation processes; Y1, Y2, X1 –X4 refer to impact ionization processes. Arrows indicate transitions made by electrons.
One can identify the dominant energies from optical or electrical experiments. To understand the properties of these centers, suppose we can arrange for the equilibrium Fermi level to rise from the valence band to the conduction band. At first practically no electrons are captured ðr ¼ 0Þ: As the Fermi level rises the states corresponding to r ¼ 1 begin to appear. The states ‘ of the center which correspond to the ground states in equilibrium are always more highly populated than the states ‘ corresponding to excited states, so that we can often confine attention to them. As the Fermi level rises, the ground states for r ¼ 2 become more important, and there may now be hardly any centers which have captured fewer electrons. If a larger number of electrons cannot be captured, by the time the Fermi level reaches the conduction band, then states of the center with r . 2 can (normally) be neglected as unstable. That this is a satisfactory picture is our second approximation.
Assumptions for Nonequilibrium Statistics
The now much simplified recombination problem is still complex because of the many states available to electrons in bands and centers. A key simplification arises from the fact that it is often possible to talk about a small number of groups of quantum states: let I ð¼ 1; 2; …Þ label quantum states in a group i, let J ð¼ 1; 2; …Þ label quantum states in a group j, etc. Within each group it is supposed that the transitions
are much more rapid than they are between groups. In that sense electrons in each group are in equilibrium among themselves. Their quasi-equilibrium is then characterized by what is called a quasiFermi level. The dimensionless version, obtained by dividing it by kT, is denoted by gi ; gj ; etc., for groups i, j, etc. That this is reasonable is our third assumption. Thus gc, gv are quasi-Fermi levels which refer to the quasi-equilibrium of the conduction and valence band, respectively, and gI to an I-electron center. Groups of states with different quasi-Fermi levels are not in equilibrium with each other. Recombination problems can now be discussed by neglecting transitions within one (quasi-equilibrium) group, because they proceed at exactly the same rate as their reverse transitions. The number of transition types to be considered is thereby greatly reduced. Away from equilibrium we shall adopt eqn [3] instead of [2] in order to allow for distinct quasi-Fermi levels: Pð‘; rÞ ¼
exp½rgr 2 hð‘; rÞ M X ðexp sgS ÞZS
½3
s¼0
Equation [3] is not always correct, but is an approximation arising from the quasi-Fermi level assumption. Among the improved theories are, for example, the ‘cascade’ theories.
SEMICONDUCTOR PHYSICS / Recombination Processes 23
The assumption of a quasi-Fermi level for each state of charge of a center implies that all the excited states of an r-electron center are populated according to eqn [2].
The Main Recombination Rates General Results and The Two-Band Case
Recombination and its converse, generation, consist of a transition of an electron from one state to another. The observed rate is the net rate of recombination and is the algebraic sum of the recombination and generation processes. During these processes both total energy and total momentum must be conserved. This is achieved by creation or absorption of photons, or phonons, excitation of secondary electrons, etc. The transition probability per unit time, SIJ, for a single-electron transition from state I in a band to state J requires that state I should be occupied, with probability pI say, and state J should be vacant with probability qJ say. The general expression for the average rate of the transition from I to J then takes the form pI SIJ qJ : For the reverse process electron state J has to be occupied and state I vacant. Thus the rate of this reverse process is pJ SJI qI : The net recombination rate per unit volume of the process I to J can then be written as: uIJ ¼ ðpI SIJ qJ 2 pJ SJI qI ÞV 21
area of the surface involved, is a current density. If i and j denote the states of the conduction and valence bands of a semiconductor, excitation independence may often be assumed, and one then expects a current density proportional to expðlelf=kTÞ 2 1: This is characteristic of the current through pn junctions, metal semiconductor junctions, etc. In these configurations the Fermi level difference between the ends of the device determines the voltage across it. When radiation is involved, however, excitation dependence of some of the parameters introduced above (e.g. SJI ) tends to spoil this simple story. The Case of Defects
So far we have considered only two bands. When a trap is involved matters are rather different. Because of the interactions among the electrons on a center it is not possible to talk of the same level being occupied or vacant. Consequently, identification of forward and reverse processes in terms of levels becomes impossible. Instead one deals with a center, say an r-electron center, as a whole; we then need the probability that a given center is an r-electron center. For example, the capture of an electron converts an ðr 2 1Þ-electron center into an r-electron center. Thus the p’s and q’s must be replaced by more complicated expressions. It is convenient to denote uij by ucv in the simple two-band case and its structure is given (with a sign change) by a recombination coefficient:
½4
By the principle of detailed balance, this expression vanishes at equilibrium. If one puts XJI ; SIJ pI qJ =SJI pJ qI ; one has uIJ ¼ pJ SJI qI ðXJI 2 1ÞV 21. In equilibrium XJI ! ðXJI Þeq ¼ 1; and the recombination rate is zero. One can say a little more if one assumes that states I and J are in conduction or valence bands, each with a quasi-Fermi level (divided by kT to make it dimensionless). If these are denoted by ge and gh, then pI ¼ ½1 þ expðhI 2 ge Þ21 for a conduction band. For the total recombination rate per unit volume between the bands i and j one finds: "X X # lelf pJ SJI qI uij ; 2 1 V 21 ½5 exp I[i J[j kT where f is essentially the difference between the quasi-Fermi levels: ge 2 gh : The first factor depends on the bands involved and it has been assumed that the transition probability ratio SIJ =SJI is independent of the excitation. A transition rate, when multiplied by the charge of current carriers, is a current, and when divided by the
ucv ¼ BS np½1 2 expðgh 2 ge Þ ; BS np 2 Y S
½6
using Figure 1a. Here, n and p are the electron and hole concentrations. Auger effects in Figures 1b and 1c can also be included, at least formally. Then B S has to be replaced in eqn [6] by BS þ B1 n þ B2 p: Analogous replacements are found for recombination processes involving defects. In the simplest case of one type of localized defect one finds a steady-state recombination rate per unit volume of the form:
u¼
np 2 ðnpÞ0 tn0 ðp þ p1 Þ þ tp0 ðn þ n1 Þ
½7
Here tn0 ; tp0 are parameters with the dimension of time and p1, n1 are parameters with the dimension of concentration. The recombination increases with the defect concentration, which is actually in the denominators of tn0 and tp0 : This is an old and much used result associated with the names of W Shockley and W T Read.
24
SEMICONDUCTOR PHYSICS / Recombination Processes
Radiative Transitions Spontaneous and Stimulated Emission
Quantum theory was initiated by Planck’s law for black-body radiation at temperature T. This gives the spatial energy density as a function of frequency n in this system as f ðn; TÞ ¼
8pn2 hn c3 expðhn=kTÞ 2 1
if one is away from equilibrium, the above rule can be suspended and one can have population inversion. For hn ¼ m there is trouble with [10] because the steady-state photon occupation diverges. This does not correspond to a ‘death ray’, but is the result of imperfect modeling; for example, the leakage of photons from the cavity may have been neglected. A formula of type [10] is also needed in connection with solar cells.
½8 Donor – Acceptor Pair Recombination
For low frequencies one finds the Rayleigh– Jeans law which makes [8] proportional to kT in agreement with the classical equipartition theorem. For high temperatures [8] diverges, as required. The result [8] may also be obtained by writing N ¼ ½A þ Bf ðn; TÞNu
½9
for the emission rate of photons by atoms, where Nu is the number of atoms in the upper of two states which are separated by the energy hn. The first term on the right-hand side is due to the normal decay of an excited state (‘spontaneous emission’). The second term refers to additional emission (‘stimulated emission’) induced by the radiation of frequency hn itself. The first factor in [8] is due to the density of states and the second factor is due to the fact that the energy is considered. If one considers the number of photons of energy hn at temperature T one comes up with 21 hn 2 m 21 Nn ¼ exp ½10 kT ðBose – Einstein distributionÞ Here m is a possible chemical potential of the radiation which is non-zero only in nonequilibrium situations such as in a semiconductor laser. In a pn junction the two bulk materials several diffusion lengths away from the junction are approximately in equilibrium even if a modest current is flowing. On one side one has then just one quasiFermi level, say mI, and on the other side one has just one quasi-Fermi level, say mJ. Then the difference
m ; mI 2 mJ ¼ qw
½11
corresponds to the applied voltage w. This is in agreement with eqn [10], in that w ¼ 0 implies no current and hence thermal equilibrium is possible. Statistical mechanics teaches one the rule that, in equilibrium, occupation probabilities of individual quantum states are always less for states of higher energies. This ensures that the total energy of a quantum system converges, the occupation probability p being a kind of convergence factor. However,
A striking demonstration of radiative donor–acceptor transitions in GaP at low temperature (1.6 K) was revealed by sharp lines (Figure 2) at photon energies hni given by hni ¼ EG 2 EA 2 ED þ e2 =1Ri ½2E
½12
where Ri is the distance between the ith-order nearest neighbors. The discrete nature of the peaks is due to the fact that the impurities involved settle in general on lattice sites so that only definite separations Ri are possible. The energy gap is EG; the value of EA þ ED can be inferred from the experimental lines by extrapolation to Ri ! 1: The energy term E is sometimes neglected. The ZnS phosphors were the first materials in which donor–acceptor radiation was hypothesized. However, it is hard to control its stoichiometry and its impurity content, and it has relatively large carrier mass and hence relatively large impurity activation energies. In such cases spectra such as those shown in Figure 2 are hard or impossible to obtain. This applies in general to many II – VI compounds. Quantum Efficiency
The quantum efficiency is the radiative recombination as a fraction of the total recombination. It can also be regarded as the average number of electrons produced per incident photon. Consider an intrinsic semiconductor, i.e., one in which the electron and hole concentrations are equal. Then with the notation of Figure 1 the radiative band – band recombination rate is Bs n2 : The nonradiative rate is ðB1 þ B2 Þn3 : We shall add another nonradiative rate, An say, proportional to the injected carrier density n , p: The quantum efficiency is then
h¼
Bs n2 An þ ðB1 þ B2 Þn3 þ Bs n2
½13
SEMICONDUCTOR PHYSICS / Recombination Processes 25
Figure 2 Comparison of the positions and intensities of the sharp line spectra at 1.6 K corresponding to both ZnS and CdS acceptor – donor pairs with the predicted pair distribution. The lower scales show the pair separation (R) and the Coulombic energy derived from R. The emission energy scales for the two measured spectra are shown at the top. Reproduced with Gershenzon M, Logan RA, Nelson DF, et al. (1968) Proceedings of the International Conference on Luminescence. Budapest, Hungary: Akademia Kiada.
26
SEMICONDUCTOR PHYSICS / Recombination Processes
Traps are here neglected, and one finds a maximum 21 2 hmax ¼ 1 þ s A1=2 ðB1 þ B2 Þ1=2 B
½14
For a high-quality epitaxial AlGaAs/GaAs double heterostructure of great purity we may take A , 0:5 £ 106 s21 ; B , 10210 cm3 s21 ; B1 þ B2 , 10229 cm6 s21
½15
whence n1 , 2 £ 1017 cm23 ; hmax , 0:96: Detailed Balance
It is clear that the radiative recombination rate from a material should be obtainable in terms of its optical ‘constants’, the absorption coefficient a(l) and the refractive index m(l). Both are functions of the wavelength. This connection can be formalized by comparing the optical absorption process with the emission process and this can be done by using the principle of detailed balance. This equates the rate of disappearance by absorption of photons with the rate of production of photons by radiative recombination. The radiative recombination rate can thus be obtained as an integral over the optical functions. This relationship, pioneered by van Roosbroeck and Shockley in 1954, has been used a great deal because the optical quantities are often known with some accuracy. This result corresponds to balancing the rate Bs np and the rate Y s in Figure 1a. Analogous detailed balance results are obtainable for the other pairs of processes in Figure 1. Thus, the Auger recombination rate can be related to an integral over the impact ionization rate. However, this connection is less useful since impact ionization data are generally less well known than the optical absorption information.
Nonradiative Processes Auger Effects
The Auger effect was discovered in 1925 in gases by Pierre Auger. An atom is ionized in an inner shell. An electron drops into the vacancy from a higher orbit and a second electron takes up the energy which is used to eject it from the atom. In solids the effect is roughly analogous. One of its characteristics is that it is not radiative. Since radiation is what is seen in many experiments, the Auger effect is suspected if
and when there is less radiation than expected in the first place. It is rather harder to investigate than radiative transitions. The effect has proved to be important as it limits the performance of semiconductor lasers, light-emitting diodes and solar cells, and it can be crucial in transistors and similar devices whose performance is governed by lifetimes. When heavy doping is required, as it is in the drive towards microminiaturization, its importance tends to increase, since the Auger recombination rate behaves roughly as n2 p or p2 n (see Figure 1) compared with the radiative rate which behaves more like np. The inverse process is impact ionization, and is important in the photodiode, the impact avalanche transit time (IMPATT) diode, and hot electron devices. Impact ionization can be regarded as an autocatalytic reaction of order one: e ! 2e þ h or h ! 2h þ e
½16
i.e., one extra particle is produced of the type present in the first place. This is a key feature for impactinduced nonequilibrium phase transitions in semiconductors. In the theory one needs wavefunctions for four states of two electrons which are relevant after the many-electron problem has been reduced to a twoelectron problem. In a matrix element calculation the four wavevectors imply a 12-fold integration in kspace to cover all possible states of the two electrons. However, momentum and energy conservation usually reduce this to an eight-fold integration. Such a calculation is hard, for it requires (1) good wavefunctions and (2) accurate integrations. Here we shall merely concentrate on the broad principles. In addition, the many-electron nature of the problem implies that another approximation is often inherent in the treatment apart from the use of perturbation theory. This is clear if one considers that the electron interactions are screened twice: once by an exponential screening factor and a second time by the dielectric constant. So there is some double counting, and the treatment of the effect as uncorrelated electronic transitions mediated by screened Coulomb interactions is another approximation. The collective effects that enter require more sophisticated field theoretic methods which take care of electron – hole correlations, plasmon effects and the effect of free excitons, the so-called excitonic Auger effect. These calculations broadly confirm the results obtained by the simpler methods used for energy gaps large compared with the plasmon energies,
SEMICONDUCTOR PHYSICS / Recombination Processes 27
provided the high-frequency dielectric constant is used and doping is not too heavy. Significant corrections are required for the narrow gap lead compounds, for example, and a considerable computational effort is required. Let us now consider lifetimes for high carrier concentrations, as limited by band –band processes. They are best studied by looking at the emitted radiation. In this connection we recall the striking rise of the threshold current density J in a semiconductor laser as the material is changed from a direct material like GaAs to an indirect material, by mixing it with a compound such as GaP. Figure 3 shows this spectacular rise for GaAs12xPx at 77 K near x ¼ 0:38: It is due to the drop in the radiative transition probabilities, as the substance becomes indirect, thus requiring a higher current density for threshold. So we should start with band –band processes in direct materials as most favorable for the experimentally accessible radiative transitions. The essential point here is that the injected carrier density behaves as:
ninj ¼
This brings in band – band Auger effects. These have also been invoked to explain the undesirable increase in J with temperature. Thus the interest in direct band –band Auger effects in III– V compounds is fuelled by the need for better and smaller optoelectronic devices which work at long wavelengths (1.3 – 1.5 mm). It matches the interest in indirect band –band Auger and impurity Auger effects in silicon due to the importance of heavy doping in VLSI (very large scale integration) and transistor technology. Figure 4 shows a typical Auger process, called CHHS, as the conduction band (C) and the split-off band (S) are involved. Two relevant states are in the heavy hole band (H). State 20 is referred to as the Auger carrier, as it has more kinetic energy than the others.
Jt ð5 £ 103 A cm22 Þð1029 sÞ , , 1017 cm23 qd ð1:6 £ 10219 CÞð1024 cmÞ ½17
where J/q is the particle current density at threshold, t is the lifetime of the carriers, and d is the thickness of the active layer. As active layers are made thinner ninj increases to 1019 cm23 or so, far in excess of the defect concentration in the (undoped) material.
Figure 3 Lowest values of laser threshold current density at 77 K as a function of mole fraction of GaP for Ga(As12xPx) junction lasers. Reproduced from Neill CJ, Stillman GE, Sirkis MD, et al. (1966) Gallium arsenide-phosphide: Crystal, diffusion and laser properties. Solid-State Electronics 9: 735 – 742, with permission from Elsevier.
Figure 4 Conduction band, heavy hole and split-off valence bands of GaAs, all treated as parabolic with "2 k02 =2mh ; EG 2 D: x denotes a quadruplet of states for a most probable transition. The two states in the heavy hole band are not shown separately. The arrows indicate electron transitions. Reproduced from Neill CJ, Stillman GE, Sirkis MD, et al. (1966) Gallium arsenide-phosphide: Crystal, diffusion and laser properties. Solid-State Electronics 9: 735–742, with permission from Elsevier.
28
SEMICONDUCTOR PHYSICS / Recombination Processes
Impact Ionization
The CHHS and CHCC processes and their inverses can be written as eC þ 2hH $ hs 2eC þ hH $ eC
½18
where the suffix refers to the band. Viewed from left to right these are Auger processes. Viewed from right to left they are autocatalytic and impact ionizations. Such processes are important for the impactinduced nonequilibrium phase transition in semiconductors. Also reaction rates with autocatalytic elements imply nonlinear equations in the concentrations and this gives rise to much interesting behavior as regards stability and bifurcation phenomena. Momentum and energy conservation are very restrictive conditions on the four states involved in [18], and it is easily seen that they cannot normally be satisfied if in a direct band semiconductor the recombining electron drops from the band minimum to the valence band maximum. This effect raises all the kinetic energies of possible processes. The Auger electron (on the right-hand sides of [18]) must also have a minimum energy in order that the impact-ionization process can proceed. This leads to an activation energy for the process. For Figure 4, for example, and in the case of nondegeneracy, the energetic hole has a kinetic energy at threshold of Eth ¼
me þ 2mh ðE 2 DÞ ðCHSSÞ me þ 2mh 2 ms G
½19
As the band gap increases we see that Eth goes up and so the Auger rate for simple parabolic bands decreases. Values of Eth for other transitions can be obtained from equation [19]. The total kinetic or threshold energy Eth can be converted to an activation energy by subtracting the basic energy which must under all conditions be part of the Auger particle energy. In the case of eqn [19] one finds for EG . D:
weakly temperature dependent, if EG , D: This can occur, for example, for InAs, GaSh and their solid solutions. The connection between Auger processes and impact ionization has also been formalized. Of all Auger quadruplets of states in a set of nondegenerate bands, the most probable Auger transition involves the quadruplet, which yields the threshold for impact ionization. The three crosses in Figure 4 indicate such a quadruplet (the central cross represents two states). Identification of Auger Effects
How does one know that an Auger effect has occurred? In pure but highly excited materials there is the original solid-state Auger effect which leads to a lifetime broadening of the electronic states at the band edge. For a semiconductor this effect causes fuzziness in the band edge which is a contributory factor to the overall bandgap shrinkage. The mechanism is illustrated in Figure 5, which shows how the lifetime of a vacant electron state near the band edge is shortened by the Auger processes within this band. Under normal conditions this effect is small in a semiconductor. But under degenerate conditions the effect leads by lifetime broadening to a low-energy tail in emission. The blanket term ‘Auger effect’ for all Coulombically excited two-particle transitions has been used here. In semiconductor device work the ‘Auger effect’ has become associated with transitions in which one electron bridges an energy gap. But there is no need to limit the concept in this way. Its competition with radiative effects means that it is often detrimental and a full discussion of means of suppressing it has recently been given by Pidgeon et al. More spectacular and more convincing is the detection of the energetic Auger electron or hole. For this purpose one may look for the weak luminescence emitted when this carrier recombines radiatively. This has been done for the band –band process in Si and for the band – impurity process
Ea ¼ Eth 2 ðEG 2 DÞ ¼
ms ðE 2 DÞ me þ 2mh 2 ms G
½20
This can result in a strong temperature dependence in accordance with an Arrhenius factor expð2Eth =kTÞ: However, this is again lost, and the direct band – band Auger effect will certainly be important and only
Figure 5 The filling of a state k by the Auger effect. The arrows indicate electron transitions.
SEMICONDUCTOR PHYSICS / Spin Transport and Relaxation in Semiconductors; Spintronics 29
in GaAs. It leads to so-called 2EG-emission. Its ratelimiting step is the rate of the Auger process per unit volume, B1 n2 p; which populates the highenergy level. The radiative process then proceeds at a rate B0 1 n2 p2 per unit volume with B0 1 , 10254 – 10251 cm9 s21 : More usual is the identification of the band – band Auger recombination mechanism from the minority carrier lifetime t, which behaves as 1 1 ¼ B1 n2 or ¼ B2 P 2 tp tn
also be overcome if there is an indirect minimum near a Brillouin zone edge at about half the direct energy gap.
Acknowledgments The author wishes to acknowledge support from NATO under their grant PST.CLG 975758.
½21
where B1 and B2 are Auger coefficients when the Auger particle is an electron or hole, respectively. A more complete expression allows also for trapping processes. The departure from parabolic bands plays an important part for the energetic Auger particle (in state 20 ). Theory shows that when this effect is taken into account, it tends to lower the band – band Auger coefficient. In fact in GaAs the CHCC process is ruled out altogether at 0 K for nonparabolic bands, suggesting that it should be an ideal material for radiative transitions. However, the possibility of phonon participation brings the effect back again, though it is still comparatively weak. The difficulty of conserving electron energy and momentum, which gives rise to the activation energies (eqn [20]) is greatly alleviated if phonons can take up some of the momentum in a direct gap semiconductor. We have seen that the activation energy barrier against the band– band Auger effect can be overcome by a suitable disposition ðEG ¼ DÞ of the three direct bands and by phonon participation. It can
See also Semiconductor Physics: Band Structure and Optical Properties; Excitons; Impurities and Defects; Infrared Lattice Properties; Outline of Basic Properties; Quantum Wells and GaAs-Based Structures.
Further Reading Fraser DA (1986) The Physics of Semiconductor Devices, 4th edn. Oxford, UK: Clarendon Press. Hangleiter A (1985) Experimental proof of impurity Auger recombination in silicon. Physical Review Letter 55: 2976 – 2978. Harrison D, Abram RA and Brand S (1999) Characteristics of impact ionization rates in direct and indirect gap semiconductors. Journal of Applied Physics 85: 8186. Landsberg PT (1991) (paperback edn., 2003) Recombination in Semiconductors. Cambridge, UK: Cambridge University Press. Nimtz G (1980) Recombination in narrow gap semiconductors. Physics Reports 63: 265. Pidgeon CR, Ciesla CM and Murdin BN (1997) Suppression of non-radiative processes in semiconductor midinfrared emitters and detectors. Progress in Quantum Electronics 21: 361 – 419.
Spin Transport and Relaxation in Semiconductors; Spintronics M E Flatte´, University of lowa, lowa City, IA, USA D D Awschalom, University of California, Santa Barbara, CA, USA q 2005, Elsevier Ltd. All Rights Reserved.
Control of the generation and transport of coherent electron spin in semiconductors suggests new ways to probe the fundamentals of quantum decoherence in solids and to explore device applications that rely on coherence, including quantum computation.
The focus here will be on the physical phenomena that govern spin transport and relaxation, with a particular emphasis on those phenomena that are amenable to direct manipulation. Manipulation of material properties in general has been taken to its most advanced level in the design of semiconductor electronic devices, in which operations are performed by perturbing the motion of electrons through their electric charge; hence, ‘spintronics’ is a term introduced to describe both the manipulation of electrons by their spin and control of their spin properties, such as g-factors.
Kelly MJ (1995) Low-Dimensional Semiconductors. Oxford, UK: Clarendon Press. Orton JW and Blood P (1990) The Electrical Characterisation of Semiconductors: Measurement of Minority Carrier Properties. London: Academic Press.
Perkowitz S (1993) Optical Characterisation of Semiconductors. London: Academic Press. Wolfe CM, Holonyak N Jr and Stillman GE (1989) Physical Properties of Semiconductors. Englewood Cliffs, NJ: Prentice Hall.
Recombination Processes P T Landsberg, The University of Southampton, Southampton, UK q 2005, Elsevier Ltd. All Rights Reserved.
Introduction In studies of recombination there occur considerable complications; many are associated with interactions between electrons, electrons and phonons, excitons, bi-excitons, impurity centers, etc. Most of these are not required in the present exposition. Modern work normally assumes that the basics of recombination physics are understood and the present exposition offers an appropriate outline.
These electron –boson interactions result in transitions of single electrons in an energy band scheme. We shall attach a superfix ‘S’ (for single-electron) to the recombination coefficients for such processes. These are then denoted by B S or T S depending whether only bands are involved or whether traps are also involved. They are illustrated in Figure 1, where the solid horizontal lines represent the conduction and valence band edges and the dashed line refers to traps. Note that the two-electron transitions do not have the superfix ‘S’, while n and p refer to the electron and hole concentrations; N0 and N1 are the concentration of unoccupied and occupied trap states, respectively. Electron –Electron Interaction in Traps
Basic Assumptions Electron – Electron Interactions for Electrons in Bands
The recombination problem in semiconductors is greatly complicated by the interaction of the electrons with each other. This allows one to speak only of the quantum states of the semiconductor crystal as a whole. However, as in metals, so also in semiconductors, a simplified picture is successful. In this, the electron interactions, and other interactions, are first neglected, but are later taken into account as a perturbation. Thus, the electrons are first treated as if they can move through each other; the fact that they collide and deflect each other by virtue of their Coulomb interaction is treated as a perturbation. The transitions are still described within the framework of the single-particle states of the unperturbed problem. The Effect of Electron – Boson Interactions
Our first approximation is to neglect most (but not all) electron interactions. Later we take into account the two-electron transitions that arise. This twoparticle recombination process is referred to as Auger recombination and its inverse as the generation of current carriers by impact ionization. In addition, electrons interact with the radiation and lattice fields and emit or absorb photons or phonons.
Electron – electron interactions can at least formally be taken into account in connection with electrons trapped in a center: the spectrum is a function of the number, r ð¼ 1; 2; 3; …; MÞ; of electrons captured. The ‘irremovable’ electrons can be included with the ion core. Given that the center captured r electrons (say), it can still be in a variety of quantum states, and they will be denoted by the symbol ‘; yielding a set of quantum states ð‘; rÞ for a center. The energy of such a state, divided by kT to make it dimensionless, is denoted by hð‘; rÞ; yielding the canonical partition function Zr ¼
X
exp½2hð‘; rÞ
½1
‘
The location of a center in space will here be considered to be of no significance. Centers can capture several electrons. This brings in a need for the chemical potentials (or Fermi levels) for r-electron centers. They are here denoted by g (when divided by kT) and the suffix ‘eq’ denotes an equilibrium value. Thus, the equilibrium probability of finding an r-electron center in state ‘ is given by: Pð‘; rÞeq ¼
exp½rgeq 2 hð‘; rÞ M X ðexp sgeq ÞZS s¼0
½2
22
SEMICONDUCTOR PHYSICS / Recombination Processes
Figure 1 Definition of recombination coefficients. Transition rates per unit volume are stated with each process, and, in brackets, for the reverse process. Thus B1, B2, T1 –T4 refer to Auger processes; B s, T1s, T2s refer to single-electron recombination; Y s, X1s, X2s refer to carrier generation processes; Y1, Y2, X1 –X4 refer to impact ionization processes. Arrows indicate transitions made by electrons.
One can identify the dominant energies from optical or electrical experiments. To understand the properties of these centers, suppose we can arrange for the equilibrium Fermi level to rise from the valence band to the conduction band. At first practically no electrons are captured ðr ¼ 0Þ: As the Fermi level rises the states corresponding to r ¼ 1 begin to appear. The states ‘ of the center which correspond to the ground states in equilibrium are always more highly populated than the states ‘ corresponding to excited states, so that we can often confine attention to them. As the Fermi level rises, the ground states for r ¼ 2 become more important, and there may now be hardly any centers which have captured fewer electrons. If a larger number of electrons cannot be captured, by the time the Fermi level reaches the conduction band, then states of the center with r . 2 can (normally) be neglected as unstable. That this is a satisfactory picture is our second approximation.
Assumptions for Nonequilibrium Statistics
The now much simplified recombination problem is still complex because of the many states available to electrons in bands and centers. A key simplification arises from the fact that it is often possible to talk about a small number of groups of quantum states: let I ð¼ 1; 2; …Þ label quantum states in a group i, let J ð¼ 1; 2; …Þ label quantum states in a group j, etc. Within each group it is supposed that the transitions
are much more rapid than they are between groups. In that sense electrons in each group are in equilibrium among themselves. Their quasi-equilibrium is then characterized by what is called a quasiFermi level. The dimensionless version, obtained by dividing it by kT, is denoted by gi ; gj ; etc., for groups i, j, etc. That this is reasonable is our third assumption. Thus gc, gv are quasi-Fermi levels which refer to the quasi-equilibrium of the conduction and valence band, respectively, and gI to an I-electron center. Groups of states with different quasi-Fermi levels are not in equilibrium with each other. Recombination problems can now be discussed by neglecting transitions within one (quasi-equilibrium) group, because they proceed at exactly the same rate as their reverse transitions. The number of transition types to be considered is thereby greatly reduced. Away from equilibrium we shall adopt eqn [3] instead of [2] in order to allow for distinct quasi-Fermi levels: Pð‘; rÞ ¼
exp½rgr 2 hð‘; rÞ M X ðexp sgS ÞZS
½3
s¼0
Equation [3] is not always correct, but is an approximation arising from the quasi-Fermi level assumption. Among the improved theories are, for example, the ‘cascade’ theories.
SEMICONDUCTOR PHYSICS / Recombination Processes 23
The assumption of a quasi-Fermi level for each state of charge of a center implies that all the excited states of an r-electron center are populated according to eqn [2].
The Main Recombination Rates General Results and The Two-Band Case
Recombination and its converse, generation, consist of a transition of an electron from one state to another. The observed rate is the net rate of recombination and is the algebraic sum of the recombination and generation processes. During these processes both total energy and total momentum must be conserved. This is achieved by creation or absorption of photons, or phonons, excitation of secondary electrons, etc. The transition probability per unit time, SIJ, for a single-electron transition from state I in a band to state J requires that state I should be occupied, with probability pI say, and state J should be vacant with probability qJ say. The general expression for the average rate of the transition from I to J then takes the form pI SIJ qJ : For the reverse process electron state J has to be occupied and state I vacant. Thus the rate of this reverse process is pJ SJI qI : The net recombination rate per unit volume of the process I to J can then be written as: uIJ ¼ ðpI SIJ qJ 2 pJ SJI qI ÞV 21
area of the surface involved, is a current density. If i and j denote the states of the conduction and valence bands of a semiconductor, excitation independence may often be assumed, and one then expects a current density proportional to expðlelf=kTÞ 2 1: This is characteristic of the current through pn junctions, metal semiconductor junctions, etc. In these configurations the Fermi level difference between the ends of the device determines the voltage across it. When radiation is involved, however, excitation dependence of some of the parameters introduced above (e.g. SJI ) tends to spoil this simple story. The Case of Defects
So far we have considered only two bands. When a trap is involved matters are rather different. Because of the interactions among the electrons on a center it is not possible to talk of the same level being occupied or vacant. Consequently, identification of forward and reverse processes in terms of levels becomes impossible. Instead one deals with a center, say an r-electron center, as a whole; we then need the probability that a given center is an r-electron center. For example, the capture of an electron converts an ðr 2 1Þ-electron center into an r-electron center. Thus the p’s and q’s must be replaced by more complicated expressions. It is convenient to denote uij by ucv in the simple two-band case and its structure is given (with a sign change) by a recombination coefficient:
½4
By the principle of detailed balance, this expression vanishes at equilibrium. If one puts XJI ; SIJ pI qJ =SJI pJ qI ; one has uIJ ¼ pJ SJI qI ðXJI 2 1ÞV 21. In equilibrium XJI ! ðXJI Þeq ¼ 1; and the recombination rate is zero. One can say a little more if one assumes that states I and J are in conduction or valence bands, each with a quasi-Fermi level (divided by kT to make it dimensionless). If these are denoted by ge and gh, then pI ¼ ½1 þ expðhI 2 ge Þ21 for a conduction band. For the total recombination rate per unit volume between the bands i and j one finds: "X X # lelf pJ SJI qI uij ; 2 1 V 21 ½5 exp I[i J[j kT where f is essentially the difference between the quasi-Fermi levels: ge 2 gh : The first factor depends on the bands involved and it has been assumed that the transition probability ratio SIJ =SJI is independent of the excitation. A transition rate, when multiplied by the charge of current carriers, is a current, and when divided by the
ucv ¼ BS np½1 2 expðgh 2 ge Þ ; BS np 2 Y S
½6
using Figure 1a. Here, n and p are the electron and hole concentrations. Auger effects in Figures 1b and 1c can also be included, at least formally. Then B S has to be replaced in eqn [6] by BS þ B1 n þ B2 p: Analogous replacements are found for recombination processes involving defects. In the simplest case of one type of localized defect one finds a steady-state recombination rate per unit volume of the form:
u¼
np 2 ðnpÞ0 tn0 ðp þ p1 Þ þ tp0 ðn þ n1 Þ
½7
Here tn0 ; tp0 are parameters with the dimension of time and p1, n1 are parameters with the dimension of concentration. The recombination increases with the defect concentration, which is actually in the denominators of tn0 and tp0 : This is an old and much used result associated with the names of W Shockley and W T Read.
24
SEMICONDUCTOR PHYSICS / Recombination Processes
Radiative Transitions Spontaneous and Stimulated Emission
Quantum theory was initiated by Planck’s law for black-body radiation at temperature T. This gives the spatial energy density as a function of frequency n in this system as f ðn; TÞ ¼
8pn2 hn c3 expðhn=kTÞ 2 1
if one is away from equilibrium, the above rule can be suspended and one can have population inversion. For hn ¼ m there is trouble with [10] because the steady-state photon occupation diverges. This does not correspond to a ‘death ray’, but is the result of imperfect modeling; for example, the leakage of photons from the cavity may have been neglected. A formula of type [10] is also needed in connection with solar cells.
½8 Donor – Acceptor Pair Recombination
For low frequencies one finds the Rayleigh– Jeans law which makes [8] proportional to kT in agreement with the classical equipartition theorem. For high temperatures [8] diverges, as required. The result [8] may also be obtained by writing N ¼ ½A þ Bf ðn; TÞNu
½9
for the emission rate of photons by atoms, where Nu is the number of atoms in the upper of two states which are separated by the energy hn. The first term on the right-hand side is due to the normal decay of an excited state (‘spontaneous emission’). The second term refers to additional emission (‘stimulated emission’) induced by the radiation of frequency hn itself. The first factor in [8] is due to the density of states and the second factor is due to the fact that the energy is considered. If one considers the number of photons of energy hn at temperature T one comes up with 21 hn 2 m 21 Nn ¼ exp ½10 kT ðBose – Einstein distributionÞ Here m is a possible chemical potential of the radiation which is non-zero only in nonequilibrium situations such as in a semiconductor laser. In a pn junction the two bulk materials several diffusion lengths away from the junction are approximately in equilibrium even if a modest current is flowing. On one side one has then just one quasiFermi level, say mI, and on the other side one has just one quasi-Fermi level, say mJ. Then the difference
m ; mI 2 mJ ¼ qw
½11
corresponds to the applied voltage w. This is in agreement with eqn [10], in that w ¼ 0 implies no current and hence thermal equilibrium is possible. Statistical mechanics teaches one the rule that, in equilibrium, occupation probabilities of individual quantum states are always less for states of higher energies. This ensures that the total energy of a quantum system converges, the occupation probability p being a kind of convergence factor. However,
A striking demonstration of radiative donor–acceptor transitions in GaP at low temperature (1.6 K) was revealed by sharp lines (Figure 2) at photon energies hni given by hni ¼ EG 2 EA 2 ED þ e2 =1Ri ½2E
½12
where Ri is the distance between the ith-order nearest neighbors. The discrete nature of the peaks is due to the fact that the impurities involved settle in general on lattice sites so that only definite separations Ri are possible. The energy gap is EG; the value of EA þ ED can be inferred from the experimental lines by extrapolation to Ri ! 1: The energy term E is sometimes neglected. The ZnS phosphors were the first materials in which donor–acceptor radiation was hypothesized. However, it is hard to control its stoichiometry and its impurity content, and it has relatively large carrier mass and hence relatively large impurity activation energies. In such cases spectra such as those shown in Figure 2 are hard or impossible to obtain. This applies in general to many II – VI compounds. Quantum Efficiency
The quantum efficiency is the radiative recombination as a fraction of the total recombination. It can also be regarded as the average number of electrons produced per incident photon. Consider an intrinsic semiconductor, i.e., one in which the electron and hole concentrations are equal. Then with the notation of Figure 1 the radiative band – band recombination rate is Bs n2 : The nonradiative rate is ðB1 þ B2 Þn3 : We shall add another nonradiative rate, An say, proportional to the injected carrier density n , p: The quantum efficiency is then
h¼
Bs n2 An þ ðB1 þ B2 Þn3 þ Bs n2
½13
SEMICONDUCTOR PHYSICS / Recombination Processes 25
Figure 2 Comparison of the positions and intensities of the sharp line spectra at 1.6 K corresponding to both ZnS and CdS acceptor – donor pairs with the predicted pair distribution. The lower scales show the pair separation (R) and the Coulombic energy derived from R. The emission energy scales for the two measured spectra are shown at the top. Reproduced with Gershenzon M, Logan RA, Nelson DF, et al. (1968) Proceedings of the International Conference on Luminescence. Budapest, Hungary: Akademia Kiada.
26
SEMICONDUCTOR PHYSICS / Recombination Processes
Traps are here neglected, and one finds a maximum 21 2 hmax ¼ 1 þ s A1=2 ðB1 þ B2 Þ1=2 B
½14
For a high-quality epitaxial AlGaAs/GaAs double heterostructure of great purity we may take A , 0:5 £ 106 s21 ; B , 10210 cm3 s21 ; B1 þ B2 , 10229 cm6 s21
½15
whence n1 , 2 £ 1017 cm23 ; hmax , 0:96: Detailed Balance
It is clear that the radiative recombination rate from a material should be obtainable in terms of its optical ‘constants’, the absorption coefficient a(l) and the refractive index m(l). Both are functions of the wavelength. This connection can be formalized by comparing the optical absorption process with the emission process and this can be done by using the principle of detailed balance. This equates the rate of disappearance by absorption of photons with the rate of production of photons by radiative recombination. The radiative recombination rate can thus be obtained as an integral over the optical functions. This relationship, pioneered by van Roosbroeck and Shockley in 1954, has been used a great deal because the optical quantities are often known with some accuracy. This result corresponds to balancing the rate Bs np and the rate Y s in Figure 1a. Analogous detailed balance results are obtainable for the other pairs of processes in Figure 1. Thus, the Auger recombination rate can be related to an integral over the impact ionization rate. However, this connection is less useful since impact ionization data are generally less well known than the optical absorption information.
Nonradiative Processes Auger Effects
The Auger effect was discovered in 1925 in gases by Pierre Auger. An atom is ionized in an inner shell. An electron drops into the vacancy from a higher orbit and a second electron takes up the energy which is used to eject it from the atom. In solids the effect is roughly analogous. One of its characteristics is that it is not radiative. Since radiation is what is seen in many experiments, the Auger effect is suspected if
and when there is less radiation than expected in the first place. It is rather harder to investigate than radiative transitions. The effect has proved to be important as it limits the performance of semiconductor lasers, light-emitting diodes and solar cells, and it can be crucial in transistors and similar devices whose performance is governed by lifetimes. When heavy doping is required, as it is in the drive towards microminiaturization, its importance tends to increase, since the Auger recombination rate behaves roughly as n2 p or p2 n (see Figure 1) compared with the radiative rate which behaves more like np. The inverse process is impact ionization, and is important in the photodiode, the impact avalanche transit time (IMPATT) diode, and hot electron devices. Impact ionization can be regarded as an autocatalytic reaction of order one: e ! 2e þ h or h ! 2h þ e
½16
i.e., one extra particle is produced of the type present in the first place. This is a key feature for impactinduced nonequilibrium phase transitions in semiconductors. In the theory one needs wavefunctions for four states of two electrons which are relevant after the many-electron problem has been reduced to a twoelectron problem. In a matrix element calculation the four wavevectors imply a 12-fold integration in kspace to cover all possible states of the two electrons. However, momentum and energy conservation usually reduce this to an eight-fold integration. Such a calculation is hard, for it requires (1) good wavefunctions and (2) accurate integrations. Here we shall merely concentrate on the broad principles. In addition, the many-electron nature of the problem implies that another approximation is often inherent in the treatment apart from the use of perturbation theory. This is clear if one considers that the electron interactions are screened twice: once by an exponential screening factor and a second time by the dielectric constant. So there is some double counting, and the treatment of the effect as uncorrelated electronic transitions mediated by screened Coulomb interactions is another approximation. The collective effects that enter require more sophisticated field theoretic methods which take care of electron – hole correlations, plasmon effects and the effect of free excitons, the so-called excitonic Auger effect. These calculations broadly confirm the results obtained by the simpler methods used for energy gaps large compared with the plasmon energies,
SEMICONDUCTOR PHYSICS / Recombination Processes 27
provided the high-frequency dielectric constant is used and doping is not too heavy. Significant corrections are required for the narrow gap lead compounds, for example, and a considerable computational effort is required. Let us now consider lifetimes for high carrier concentrations, as limited by band –band processes. They are best studied by looking at the emitted radiation. In this connection we recall the striking rise of the threshold current density J in a semiconductor laser as the material is changed from a direct material like GaAs to an indirect material, by mixing it with a compound such as GaP. Figure 3 shows this spectacular rise for GaAs12xPx at 77 K near x ¼ 0:38: It is due to the drop in the radiative transition probabilities, as the substance becomes indirect, thus requiring a higher current density for threshold. So we should start with band –band processes in direct materials as most favorable for the experimentally accessible radiative transitions. The essential point here is that the injected carrier density behaves as:
ninj ¼
This brings in band – band Auger effects. These have also been invoked to explain the undesirable increase in J with temperature. Thus the interest in direct band –band Auger effects in III– V compounds is fuelled by the need for better and smaller optoelectronic devices which work at long wavelengths (1.3 – 1.5 mm). It matches the interest in indirect band –band Auger and impurity Auger effects in silicon due to the importance of heavy doping in VLSI (very large scale integration) and transistor technology. Figure 4 shows a typical Auger process, called CHHS, as the conduction band (C) and the split-off band (S) are involved. Two relevant states are in the heavy hole band (H). State 20 is referred to as the Auger carrier, as it has more kinetic energy than the others.
Jt ð5 £ 103 A cm22 Þð1029 sÞ , , 1017 cm23 qd ð1:6 £ 10219 CÞð1024 cmÞ ½17
where J/q is the particle current density at threshold, t is the lifetime of the carriers, and d is the thickness of the active layer. As active layers are made thinner ninj increases to 1019 cm23 or so, far in excess of the defect concentration in the (undoped) material.
Figure 3 Lowest values of laser threshold current density at 77 K as a function of mole fraction of GaP for Ga(As12xPx) junction lasers. Reproduced from Neill CJ, Stillman GE, Sirkis MD, et al. (1966) Gallium arsenide-phosphide: Crystal, diffusion and laser properties. Solid-State Electronics 9: 735 – 742, with permission from Elsevier.
Figure 4 Conduction band, heavy hole and split-off valence bands of GaAs, all treated as parabolic with "2 k02 =2mh ; EG 2 D: x denotes a quadruplet of states for a most probable transition. The two states in the heavy hole band are not shown separately. The arrows indicate electron transitions. Reproduced from Neill CJ, Stillman GE, Sirkis MD, et al. (1966) Gallium arsenide-phosphide: Crystal, diffusion and laser properties. Solid-State Electronics 9: 735–742, with permission from Elsevier.
28
SEMICONDUCTOR PHYSICS / Recombination Processes
Impact Ionization
The CHHS and CHCC processes and their inverses can be written as eC þ 2hH $ hs 2eC þ hH $ eC
½18
where the suffix refers to the band. Viewed from left to right these are Auger processes. Viewed from right to left they are autocatalytic and impact ionizations. Such processes are important for the impactinduced nonequilibrium phase transition in semiconductors. Also reaction rates with autocatalytic elements imply nonlinear equations in the concentrations and this gives rise to much interesting behavior as regards stability and bifurcation phenomena. Momentum and energy conservation are very restrictive conditions on the four states involved in [18], and it is easily seen that they cannot normally be satisfied if in a direct band semiconductor the recombining electron drops from the band minimum to the valence band maximum. This effect raises all the kinetic energies of possible processes. The Auger electron (on the right-hand sides of [18]) must also have a minimum energy in order that the impact-ionization process can proceed. This leads to an activation energy for the process. For Figure 4, for example, and in the case of nondegeneracy, the energetic hole has a kinetic energy at threshold of Eth ¼
me þ 2mh ðE 2 DÞ ðCHSSÞ me þ 2mh 2 ms G
½19
As the band gap increases we see that Eth goes up and so the Auger rate for simple parabolic bands decreases. Values of Eth for other transitions can be obtained from equation [19]. The total kinetic or threshold energy Eth can be converted to an activation energy by subtracting the basic energy which must under all conditions be part of the Auger particle energy. In the case of eqn [19] one finds for EG . D:
weakly temperature dependent, if EG , D: This can occur, for example, for InAs, GaSh and their solid solutions. The connection between Auger processes and impact ionization has also been formalized. Of all Auger quadruplets of states in a set of nondegenerate bands, the most probable Auger transition involves the quadruplet, which yields the threshold for impact ionization. The three crosses in Figure 4 indicate such a quadruplet (the central cross represents two states). Identification of Auger Effects
How does one know that an Auger effect has occurred? In pure but highly excited materials there is the original solid-state Auger effect which leads to a lifetime broadening of the electronic states at the band edge. For a semiconductor this effect causes fuzziness in the band edge which is a contributory factor to the overall bandgap shrinkage. The mechanism is illustrated in Figure 5, which shows how the lifetime of a vacant electron state near the band edge is shortened by the Auger processes within this band. Under normal conditions this effect is small in a semiconductor. But under degenerate conditions the effect leads by lifetime broadening to a low-energy tail in emission. The blanket term ‘Auger effect’ for all Coulombically excited two-particle transitions has been used here. In semiconductor device work the ‘Auger effect’ has become associated with transitions in which one electron bridges an energy gap. But there is no need to limit the concept in this way. Its competition with radiative effects means that it is often detrimental and a full discussion of means of suppressing it has recently been given by Pidgeon et al. More spectacular and more convincing is the detection of the energetic Auger electron or hole. For this purpose one may look for the weak luminescence emitted when this carrier recombines radiatively. This has been done for the band –band process in Si and for the band – impurity process
Ea ¼ Eth 2 ðEG 2 DÞ ¼
ms ðE 2 DÞ me þ 2mh 2 ms G
½20
This can result in a strong temperature dependence in accordance with an Arrhenius factor expð2Eth =kTÞ: However, this is again lost, and the direct band – band Auger effect will certainly be important and only
Figure 5 The filling of a state k by the Auger effect. The arrows indicate electron transitions.
SEMICONDUCTOR PHYSICS / Spin Transport and Relaxation in Semiconductors; Spintronics 29
in GaAs. It leads to so-called 2EG-emission. Its ratelimiting step is the rate of the Auger process per unit volume, B1 n2 p; which populates the highenergy level. The radiative process then proceeds at a rate B0 1 n2 p2 per unit volume with B0 1 , 10254 – 10251 cm9 s21 : More usual is the identification of the band – band Auger recombination mechanism from the minority carrier lifetime t, which behaves as 1 1 ¼ B1 n2 or ¼ B2 P 2 tp tn
also be overcome if there is an indirect minimum near a Brillouin zone edge at about half the direct energy gap.
Acknowledgments The author wishes to acknowledge support from NATO under their grant PST.CLG 975758.
½21
where B1 and B2 are Auger coefficients when the Auger particle is an electron or hole, respectively. A more complete expression allows also for trapping processes. The departure from parabolic bands plays an important part for the energetic Auger particle (in state 20 ). Theory shows that when this effect is taken into account, it tends to lower the band – band Auger coefficient. In fact in GaAs the CHCC process is ruled out altogether at 0 K for nonparabolic bands, suggesting that it should be an ideal material for radiative transitions. However, the possibility of phonon participation brings the effect back again, though it is still comparatively weak. The difficulty of conserving electron energy and momentum, which gives rise to the activation energies (eqn [20]) is greatly alleviated if phonons can take up some of the momentum in a direct gap semiconductor. We have seen that the activation energy barrier against the band– band Auger effect can be overcome by a suitable disposition ðEG ¼ DÞ of the three direct bands and by phonon participation. It can
See also Semiconductor Physics: Band Structure and Optical Properties; Excitons; Impurities and Defects; Infrared Lattice Properties; Outline of Basic Properties; Quantum Wells and GaAs-Based Structures.
Further Reading Fraser DA (1986) The Physics of Semiconductor Devices, 4th edn. Oxford, UK: Clarendon Press. Hangleiter A (1985) Experimental proof of impurity Auger recombination in silicon. Physical Review Letter 55: 2976 – 2978. Harrison D, Abram RA and Brand S (1999) Characteristics of impact ionization rates in direct and indirect gap semiconductors. Journal of Applied Physics 85: 8186. Landsberg PT (1991) (paperback edn., 2003) Recombination in Semiconductors. Cambridge, UK: Cambridge University Press. Nimtz G (1980) Recombination in narrow gap semiconductors. Physics Reports 63: 265. Pidgeon CR, Ciesla CM and Murdin BN (1997) Suppression of non-radiative processes in semiconductor midinfrared emitters and detectors. Progress in Quantum Electronics 21: 361 – 419.
Spin Transport and Relaxation in Semiconductors; Spintronics M E Flatte´, University of lowa, lowa City, IA, USA D D Awschalom, University of California, Santa Barbara, CA, USA q 2005, Elsevier Ltd. All Rights Reserved.
Control of the generation and transport of coherent electron spin in semiconductors suggests new ways to probe the fundamentals of quantum decoherence in solids and to explore device applications that rely on coherence, including quantum computation.
The focus here will be on the physical phenomena that govern spin transport and relaxation, with a particular emphasis on those phenomena that are amenable to direct manipulation. Manipulation of material properties in general has been taken to its most advanced level in the design of semiconductor electronic devices, in which operations are performed by perturbing the motion of electrons through their electric charge; hence, ‘spintronics’ is a term introduced to describe both the manipulation of electrons by their spin and control of their spin properties, such as g-factors.