- Email: [email protected]
Recombination at dislocations
RECOMBINATION T.
AT DISLOCATIONS RGIELSKI
Institute of Physics, Polish Academy of Sciences, Warsaw, Poland Abstract-A dislocation in semiconductors behaves as a recombination flaw having a large number of charge states. The effective cross section of the dislocation is therefore a variable parameter depending by the electrostatic interaction on the occupation factor. This property manifests itself through peculiarities of photoconductivity of dislocated crystals e.g. lcgarithmic type decay of excess current carriers at low temperature. The elementary centres of recombination are most probably dangling bonds as indicated by spin-dependent effects. The magnitude of the spindependent cross section can be explained only if one takes into account the exchange interaction between neighbouring dangling electrons on a dislocation.
1.
IBTROMICTION
The role of dislocations in recombination processes in semiconductors is of considerable importance at least at low and moderate injection levels. It was often underestimated in the past which is the reason, as we believe, for serious errors committed by the people who dealt in ’60s with the recombination processes occuring via different impurities introduced into Ge and Si. As a result we have inherited a lot of useless data conceming’the capture cross sections of almost all elements of the periodic table. Moreover, as far as it is known to this writer, nobody who has ever studied seriously the recombination processes has succeeded in providing a complete explanation of experimental results on the basis of recombination statistics for isolated centres. Usually, the best way out of the difficulties was to assume an interference due to other kinds of recombination- or trap-centres (of unknown origin at best). Most of the difficulties could, however, be easily eliminated taking into account the individual character of dislocations whose action usually overlaps the action of other recombination centres. Unfortunately the knowledge of dislocation properties is not sufficiently familiar to people who deal with excess current carrier effects in semiconductors. For instance, one is usually surprized when one realizes that the dislocation limited lifetime of majority carriers in silicon at 77 K is about one year (independently of a dislocation density) and that the greatest part of carrier decay is not exponential vs time but logarithmic. These features mentioned above are direct consequences of the particular recombination statistics governed by an electrostatic interaction between recombining current carriers and charged dislocations. It is most important that the charge of a dislocation which controls an influx of current carriers into the dislocation is not a constant parameter but depends on the excess carrier density. The subject of this article is confined to the nonradiative recombination via dislocations in Ge and Si. The situation in III-V compounds is probably similar to that in elemental semiconductors. However, the amount of information concerning these materials is too small to draw any definite conclusions. 1403
In the first part of this paper the models of electron states for 60”-dislocation are briefly discussed. The second part contains the principles and consequences of the phenomenological “barrier model” of recombination via dislocations as well as their experimental verification. In the third part, the results of spin dependent recombination are presented. The article is closed with some remarks on possible mechanisms of energy dissipation by recombining charges.
The dominant dislocations in diamond- and sphaleritestructure semiconductors are: the 6tY-dislocation with Burgers vector (a/2)(1 10) and with glide plane {I 1I}, and the screw dislocation. The 6@-dislocation has dangling bonds with unpaired electrons at the edge of an extra lattice halfplane, which are believed to be mainly responsible for electrical activity of the dislocation. It usually appears as dissociated into two Shockley partials according to the reaction
with a stacking fault between them. The split components have the character of the pure edge- and the 30”-dislocation; both having dangling bonds. As found by transmission electron microscopy the dissociated dislocation is composed of alternate split segments and constricted parts [ I]. Since the dangling bonds are spaced along dislocations at interatomic distances, it was formerly believed that they form a one-dimensional half-filled energy band[2]. However, as the dangling bonds originate from atomic sp3 orbitals and they dangle perpendicularly or almost perpendicularly to the dislocation axis-the overlap of neighbouring wave functions is slight. Thus the individual dangling electrons are rather strongly localized along the dislocation line and then the conventional Hartree-Fock model, developed for a wide energy band, is not applicable. For a narrow energy band the Hubbard model[3] is much more realistic. The Hubbard hamiltonian takes into account the repulsion energy of two
T. RCIELSKI
1404 Table I. Majority lifetimes measured at the temperature shown in the third column, and corresponding capture cross sections per unit length of dislocation *Ill., [sl
Ge-n Ge-p Si-n Si-p
TGl,,,
Kl
L
i [cm 1
4x lo-8 154 lack of exact data 195 2 x lo-’ 80 25 I95 I4 x lo-’
?
at the same place. The model involves splitting of the energy band into two subbands with a variable number of states. It explains in a natural way a transition from the energy band scheme to the discrete energy levels when the distances between the dangling bonds increase infinitely. In the last case each dangling bond may be treated as a D-centre with three possible charge states: D- when the second electron is accepted into the dangling bond, D“ when one unsaturated electron is present, and D’ when the dangling electron is taken away from the bond. Such a system is described by two energy levels, which correspond to the Hubbard subbands. Experimental confirmation of the Hubbard model is obtained from microwave conductivity[4] and spindependent photoconductivity[S]. The equilibrium charge state of the whole dislocation (i.e. the net charge per one dangling bond) is not a very sensitive function of temperature or doping. The main factor which controls the population of dislocation states is an increase in the electrostatic energy due to the interacting charges.
potential difference @ can be defined by the work which has to be done against electrostatic forces in order to increase by one more charge the number of elementary charges on the line (in isothermal process). In the case of thermal equlibrium this work has to be exactly balanced by the reduction of free energy of the system due to the localization of that electron charge at the dislocation states. If we picture the situation on the band scheme, there will appear a potential barrier for electrons and a potential well for holes at the dislocation (Fig. I).
eleftrons
3. RECOMUMATION
PROCESS
The carrier recombination through dislocations is discussed here on a basis of the barrier model[6-91. This model has been formulated mainly as a result of experimental studies of photoconductivity and related phenomena in plastically deformed Ge and Si[lO-161. The intention of this author is to discuss that subject as generally as possible without referring to any particular structure of dislocation electron states. Let us consider a system consisting of undisturbed crystal-matrix-and a dislocation with edge component. There are certain electron quantum states associated with a dislocation which are localized at least in two dimensions and have energy levels inside the forbidden energy gap. For a neutral dislocation a number of those states are occupied by electrons, the others are empty. Such a neutral dislocation in order to reach equillibrium with the matrix has, in general, to exchange electrons with it. Depending on the Fermi energy in the matrix, the dislocation either accepts electrons to become negatively charged or loses them to become positively charged. nType material and negatively charged dislocations will be discussed here, but the results can be directly adopted for p-type as well. The dislocation line is surrounded by a space charge of opposite sign consisting of ionized donors and possibly holes. Thus, there appears a difference in the macroscopic electrostatic potential betwe-r ++e charged line and the bulk of material. This
Fig. 1. Potential
barrier
at dislocation.
@ is a function of linear density of charge n. Within a limited range of variation, Q, may be assumed to be proportional to 7, where @ and n are not necessarily equilibrium values. Thus
This relation is a basis for all further considerations which will be valid as far as the potential of dislocation may be linearized with respect to its charge. We denote by m. the mean volume density of the electrons which have been accepted by dislocations to equilibrate the system (and which constitute the total charge of ail dislocations in the crystal in thermal equilibrium). If N is the density of dislocations [cm-*], and q is the electron charge, then we have an evident relation TON
=
qmo.
(1)
The actual potential at the dislocation line can be expressed through the equilibrium value Q. and the relative variation of dislocation charge Am/m0 as follows
(g!!?!>
@=@0
m0
(2)
where Am = m. - m is the mean concentration of positive charge trapped at dislocations. Am includes the excess charge localized at the inherent quantum states of dislocations as well as that of excess holes contributing to the space charge. It is assumed that there is always an equilibrium among electrons in the conduction band, including the regions of dislocation core. Then the local concentration of free electrons at dislocation core n*, can be expressed by the electron concentration in the bulk, n, as II* = n exp (-
q@/kT).
(3)
1405
Recombinationat dislocations It is assumed here that the electron gas is nondegenerate and there is no overlap of space charge regions of dislocations.
After transforming the eqns (4) and (6) to a more convenient form and limiting their validity to the case when An % no one obtains R = C,Nn, exp (- qQo/kT)[exp (q%Am/kTmo)-
3.1 Capture rates We are looking for the net rate of electron capture at dislocations per unit volume and time, L The number of electrons which may be captured is proportional to the concentration n*. The number of empty dislocation states available for electrons slightly depends on external conditions. Thus, for simplicity, it may be assumed to be a constant in comparison with the strongly variable exponential factor (3). Then R = C,N[n exp (-qQ/kT)-
noexp (-q%/kT)l. (4)
The fust term in (4) corresponds to the capture of conduction electrons at dislocation, while the second one describes the reverse process-reemission of localized electrons into the conduction band. The last term is chosen in such a way that the resulting capture rate in thermal equilibrium is zero, according to the principle of detailed balancing. The coefficient C. denotes the mean probability of an elementary act of capture and may be expressed by the cross section Z. for this process per unit length of dislocation
where u. is the mean thermal velocity of electrons which can penetrate into the dislocation core. It is supposed that the excess holes trapped in the potential well are localized in the sense that they do not participate in the d.c. conductivity. They contribute only to the electric charge of the dislocation. The excess holes which are generated in the bulk sink into potential wells and subsequently recombine with electrons. So there is certain flux of holes between the bulk and the well which in the thermal equilibrium equals zero. We suppose that the influx of holes into the well is a “bottle neck” for the process of hole recombination. (This assumption is not necessary but it simplifies the further calculation.) The magnitude and direction of the flux is determined by the deviation of the dislocation potential from the equihbrium value. Thus the net influx of holes into the well may be expressed approximately as follows Rh = C,,NAp - p. exp (q@lkT)
(6)
neglecting the term which would lead to a quadratic dependence of Ap with generation rate. Thus the eqn (6) is valid strictly for small deviation from equilibrium. The recombination coefficient Ch may be formally expressed by the cross section for that process per unit length of dislocation cl# = IhUh where UI,is the mean thermal velocity of holes.
(7)
II (8)
R,, = C,,NAp - C,,NpoIexp (q%Am/kTmo)-
11. (9)
3.2 Steady state The problem of recombination via dislocations has been reduced to two differential equations describing the rates at which the current carriers leave their original bands. Now let us consider the case in which electronhole pairs are homogenously generated in the bulk at the rate G. At the steady state we have, R = Rh = G. Then eqn (8) determines the value of Am and combination of eqns (8) and (9) results in Ap. Moreover, from the condition of crystal neutrality one has An =Ap+Am.
(10)
The equation system (8)-(10) determines the excess concentrations of current carriers in the bands and excess charges at dislocations. Therefore the statistical problem of recombination via dislocations has been approximately solved and now nothing remains to be done except finding out what physical conclusions can be drawn from this extremely simple theory. We introduce the following notation: 6CF is the difference between the energy of the conduction band edge and Fermi energy at dislocation core, and l,SV= & - EC,+where & is the forbidden energy gap. The exponential forms in (8) and (9) may be then written as no exp (-qQo/kT)
= NC exp (-&kT)
(11)
p. exp (-qQo/kT)
= N, exp (-r&kT)
(12)
where N,, N, are the effective densities of states in the conduction and valence bands. We are interested in the variation in excess concentration depending upon the temperature. Let us start discussing excess holes. By using (12) the eqns (8) and (9) may be rewritten in the following form Ap
=
G nNvexp (-w/kT)
m
<
0
+&
h
(13)
There are several factors in (13) which vary with temperature but the exponential factor varies the most. Thus a slight variation of N,, N, and probably C, and Ch may be neglected at the tirst inspection compared with that of the exponent. At sufficiently high temperature, but still within the extrinsic range of conductivity, the first term in (13) dominates the second one. As the temperature is lowered the first term decreases indicating the activation energy c~v and below a certain temperature becomes negligible compared with the second one So, the expected dependence of Ap follows the lower dotted line in the Fig. 2. In the approximation used here Ap is proportional to the generation rate within the
T. FIGIELSKI
1406
with electron concentration lOI crnm3 and dislocation densities ranging from lo3 to lO’cm_’ one obtains by (16) T, varying between 220 and 300K for Ge, and between 330 and 490 K for Si. These values are in a good agreement with experiment (Fig. 3). Since T, does not depend on the generation rate it means that above this critical temperature the dislocation charge cannot be changed by illumination. This important property may have interesting consequences in some phenomena different from photoconductivity, for instance in the photoplastic effect (17).
extrinsic conductivity
T(‘K) x)0
Recomb.
IM
120
03
91
nonllneor mnge
linear ronge T,-’
200
Trapping range
range
I
303
4
I
Ti’
x-1 I/T
Fig. 2. Dependences of excess charge concentrations on reciprocal temperature expected on the barrier model.
whole range of temperature. Then linear recombination occurs and the relation Ap = GTI, is valid for the steady state by which the lifetime of holes can be determined. This T,, is inversely proportional to the dislocation density as really observed in experiments. Now let us consider the charges trapped at the dislocations. We have from (8) and (I 1) &$ <
exp &cd&T) . (14) c I
The run of the curve described by (14) is shown in the Fig. 2 by the upper dotted line. When the relative deviation of the dislocation charge from equilibrium is small Am/m0 < &T/q@,, then (14) is reduced to Am=!$&$exp(&kT). c
(15)
Fig. 3. Temperature dependence of the photoconductivity in a plastically bent Ge at different rates of generation of electronhole pairs. (a) G = 10” cme3 s-‘, (b) G = 5 X IO’s cm-’ s-’ and (c) G = 10”cm-‘s-‘[12].
Consequently we may introduce another characteristic temperature Tz, below which the Am (or An) deviates from exponential increase to saturation. This T2 is not sharply defined, but a convention may be used according to which
l
and this describes the linear part of the curve in Fig. 2. The concentration of excess electrons equal to Ap + Am is shown in Fig. 2 by the full line. Two different temperature ranges are clearly seen m this figure: the first, where Ap = An, corresponds to the bipolar photoconductivity (PC), and the second one, where A.n4 Ap, corresponds to the quasimonopolar PC. Within the last range an essential trapping of excess holes at dislocations occurs. The temperature of transition from the 8rst to the second region is given by the expression (16) This temperature increases with the dislocation density and is almost proportional to &. For typical materials
(17) T2 depends on the generation rate per one dislocation G/N. With increasing dislocation density Tz shifts towards lower values but this shift can be compensated by simultaneous increase in the light intensity. The temperature T2 divides the trapping range of PC in two parts with quite different properties. Within the range TI > T > Tz An is proportional to the generation rate. Thus a linear recombination occurs and consequently the concept of electron lifetime can be introduced in the usual way. By lowering the temperature below Tt one enters the most interesting nonlinear region of photoconductivity. The relative deviation of dislocation charge from equilibrium is significant and the barrier height is consider-
Recombination ably
reduced. In that case nonlinear properties of dislocation recombination flaws are clearly seen. The excess electron concentration is here described by
It follows that the steady state photoconductivity grows with the dislocation density, although more slowly than linearly. It should be noticed here that photoconductivity increases with illumination proportionally to the logarithm of the generation rate (Fig. 4). This relation is valid of course within the limited range of light intensities. It is important that the dislocation density N enters into the proportionality coefficient in the relation An-InG.
a
e tb 4o a
at dislocations
1407
where (21)
kT rlo
1
(22)
The to is a parameter determined by the initial condition in the sample and to< T< for T < Tz. The T< is the characteristic time of the decay process and has the same analytical form as the electron lifetime in the linear range where the decay is an exponential process. r< does not depend on dislocation density but is a very sensitive function of temperature. According to (20) the main part of the decay is linear with the logarithm of time. Plotting An vs In t one obtains a straight line, which extrapolated to the t-axis intersects it at the point t = 7r - to which practically equals 7. for a sufficiently low temperature (Fig. 5). After a time comparable with T., An decreases to such a small value at which the criterion of linear recombination becomes fuliiled. Thus the further decay occurs exponentially with the lifetime T_
20
.o
I (orb. units)
Fig. 4. Steady state photoconductivity in plastically deformed Si vs illumination intensity after[l5]. (I) reference sample, (2) heated but unbent sample, (3). (4) and (5) sample bent to the radii of curvature 12, 6 and 3 cm respectively. 5
IO
50
)O
t (5)
3.3 Kinetics of recombination in nonlinear region The nonlinear character of the recombination process occuring through dislocations is most clearly seen in the kinetics of the decay of photoconductance. Let us start from a steady state for an illuminated sample and assume that at the moment t = 0 ihumination is switched off. The excess charges decay in time and we look for the timedependence of An. The rate of electron capture is given by the eqn (8) which in the considered range is reduced to the following form
Fig. 5. Decay of photoconductivity after illumination of the sample by a rectangular light pulse for n-type silicon, after1151.
(1) 150fkm sample bent to the curvatureradius r, = 3 cm, (2) and (4) IOOOfkmunbent sample at two different illumination levels, (3) 35 km,
unbent sample.
The decay described by (20) has another interesting property. By introducing a new time variable t’ = to+ t into (20) it becomes transformed into a form independent of the generation rate (23)
$
= WN,
exp (-ecdkT)
exp (q%Am/kTq,N). (19)
It is solved by An=Am=-7
kT2N,nfo+ 4 @o
7.
(20)
This equation describes an universal decay (Fig. 6) which includes any particular illumination level. In this picture the decay starts at different values t’= to corresponding to the different initial steady states. The rise of PC, not, discussed here, is quite well described by the formula n(t) = An(a)[ 1- exp (-t/to)].
1408
Fig. 6. Universal decay curve a-b for majority current carriers. I, > 1, > I,different illumination levels.
By lowering temperature r. rises to extremely large values. An example of photodecay at 77K in plastically bent Si is shown in Fig. 7 in two different time scales. The To extrapolated from this picture equals 3. 10’s. Thus a long time period after illumination of the crystal the excess carrier density remains at some, practically unchanging, level. This property is usually called photomemory. As the temperature is lowered below that at which r. reaches the value of about 103s the conditions required for a proper experiment become more and more rigorous. The crystal exhibits then a very high photosensitivity and even the lowest background radiation can considerably change the experimental zero level of PC[8]. The actual values of T for majority carriers in Ge and Si, measured in the range where it can be done with great accuracy, as well as the estimated cross sections are shown in the table.
3.4 Supplementary remarks The temperature dependence of PC in samples with different dislocation density under definite illumination is schematically shown in Fig. 8. The family of curves for different illuminations levels is sketched in Fig. 9. It can be seen that the illumination level is just as important a parameter in PC as the temperature, and it can even change the shape of the curve observed. If one wants to get some definite information from a single measurement, one must apply as low illumination as possible. The mistake often committed in the past was doing an experiment at some arbitrary but relatively high injection level, i.e. under conditions which wouid correspond to
N3
NZ
1 N,
I=corw N, ‘NH
Fig. 7. Time response of photoconductivity in plastically deformed Si after illumination of the sample by a rectangular light pulse. Internal picture-linear time scale, external picturelogarithmic time scale [ IS].
I/T
Fig. 8. Theoretical dependence of photoconductivity vs reciprocal temperature for different dislocation densities.
Recombination
at dislocations
1409
10’2-10’4crnw3,which means about a half of the energy gap. Glaentzer and Jordan[l3] found out from the kinetics of decay of PC the value 0.52 eV in n-Si and the values varying in the range 0.38-0.5 eV in p-Si. However, they have not paid due attention to the nonlinearity in the decay. On the contrary Golacki[lS] has found almost full symmetry of the kinetic process with respect to the inversion of type, which is in accordance with the position of the Fermi level at the dislocation in the middle of the forbidden gap.
N=amsi.
Fii.
9. Theoretical dependence of photoconductivity on the reciprocal temperatures for different illumination levels.
the curves above the curve (a) in Fig. 3. To this category of measurements belong, for instance, those by Wertheim and Pearson [ 18). The energy factors in the exponents of (13) and (14) enter through the local concentrations of majority carriers at the dislocation and, therefore, are defined by the Fermi level position at the dislocation core lCF or em. What are their experimental values? One does not have to pay much attention to the activation energy which appears within the recombination range of PC, as the extension of this range is usually very small and, moreover, the model utilized turns out to be of little use. Just the opposite situation occurs in the linear trapping range. The extension of this range can be almost arbitrarily enlarged by using extremely low light intensities, and the exponential increase in PC can be observed within several orders of magnitude. In n-type germanium the mean value of the activation energy obtained from a great number of experiments is -0.42eV. A large majority of results do not exceed the mean value by +O.O2eV. No definite shift due to doping within the range of concentrations 10’3-10’5cm-3 has been observed. The activation energy in p-type Ge can not be determined as accurately as in n-type, mainly due to much less photosensitivity of those crystals, however, the values reported lie between 0.24 and 0.30 eV, and are close to the value & - EN resulting from the theory. The activation energies in Si, in the steady state PC, are independent of the type of conductivity and amount to 0.6 2 0.03 eV for the crystal within the doping range ssEvol.?I.No. ,,,,
I-,
The spin-dependent recombination process 151is hoped to be an effective tool to examine more thoroughly the mechanism of carrier recombination at dislocations as well as the nature of electron states of the dislocation. This phenomenon is revealed in spin-dependent photoconductivity (SDP). The philosophy of this effect is the following. The two-electron stable state of a dangling bond is most probably the singlet state in which the spins of both the electrons are antiparallel. Its energy corresponds to the upper level (subband) in the forbidden energy gap while the lower level corresponds to the one-electron state. A possible triplet state-with parallel spin orientation of both the electrons-is probably degenerated with the conduction band states, and therefore does not form a stable configuration. If the spin of the recombining electron is preserved in the recombination transition, then the only intermediate state in the process of recombination is the singlet state. (The situation for the recombining hole is similar although the holes are mostly captured at the lower energy level). In that case the cross section for an electron capture P. depends on the spin polarisation of both the electron systms: that of recombining electrons p, and the one of dangling electrons P, in the following way [ 191
z, =Zo(l -
pP)
(24)
where Z0 is the cross section for zero polarisation. It can be seen from this formula that the capture cross section is reduced due to the spin polarisation. If the product of the spin polarisation degrees of both the electron systems is changed by 6(pP), then a relative change of the cross section is given by
sz._--
I,-
&PP)
I-pP
(25)
which is practically equal to -S(pP) under usual experimental conditions. The SDP is observed in plastically deformed silicon under the conditions of electron spin resonance. When the sample is placed in the magnetic field, then some polarisation appears which can be destroyed by input of microwave power of the Larmor frequency corresponding to one of the two electron systems. In the basic experiment the sample placed in the microwave resonator is steadily illuminated by a light generating electronhole pairs in the bulk. The microwave power is modulated with audio frequency and a change in photoconductivity is detected at the same frequency. When
1410
T.
~GELSKI
sweeping the magnetic field over the resonance region a resonant decrease in photoconductivity is observed. Two other modifications of this basic method may be used in order to record the derivative of the SDP-signal which makes it possible to reveal a certain structure of the spectrum (201. The SW-signal is observed on silicon of both nand p-type in a wide range of temperatures (M-340 K), Fig. 10. The signal maximum corresponds to a g-factor equal to 2.004, which is close to that of the central line in the EPR spectrum due to the dislocations in Si but far from the g-factor of free carriers. The half-width of the resonant line (of about 6 Oe) is independent of temperature over the range of measurement. The room temperature ratio of the SDP signal amplitude, 60, to photoconductivity, Au is of the order of so/Au = lo-’ and independent of the intensity of illumination.
I
I
I
I
I
6
6
10
12
c 4
Fig. 11. Dependece of relative photoconductivity AU/Uand reiative resonant change of photoconductivity &/a in silicon on temperature at the same illumination level. Au/o was measured using light chopped at the same frequency as that of the microwave modulation[21].
one gets the maximum value of the ratio sojho upon saturation by microwave power
photoconductivity Fig. 10. Resonant decrease of silicon sample photoconductivity observed at room temperature when sweeping the d.c. magnetic field and irradiating the sample with microwaves of X-band. The microwave power was square-wave modulated 220Hz. and the conductivity change at this frequency was recorded(211.
The value Wo called the relative resonant change of photoconductivity, where u is the equilibrium conductivity, and the relative photoconductivity Aoiu of the same sample are shown in Fig. 11 as a function of reciprocal temperature. Both the signals are measured at the same ihumination level. At high temperatures, when the photoconductivity in dislocated Si increases exponentially vs the reciprocal temperature, the courses of both the curves are similar. At low temperatures, when the photoconductivity variation saturates, the discrepancy between the courses of both the curves becomes more pronounced. The variations of solo and AU/C with the illumination level and with the light wavelength show analogous dependences known typically for disiocation-controlled recombination in silicon[21]. It must be. pointed out, to demonstrate the advantages of the SDP technique, that it can be successfully applied even in the case when the number of dislocation paramagnetic centres is too small to be detected by the conventional EPR method. Contrary to the EPR, the SDP signal is independent of the number of spins as long as the recombination process through the paramagnetic centres dominates. Let us estimate the maximum value of the SDP signal on the ground of the simplest model described by the formula (25). Passing from the capture cross sections to the
E=pP where Au is the steady-state photoconductivity and Sa is its change due to the microwave power. For noninteracting recombination centres with spin l/2 this yields a value of = IO4 at room temperature in the magnetic field of 30000e. The value obtained in the experiment was one order of magnitude greater despite the nonsaturating microwave power used. 4.1 Exchange interaction
To explain the unexpected large magnitude and temperature independence of the half-width of the SDP signal we shall analyze the exchange interaction between paramagnetic recombination centres spaced along the dislocation line. In a perfect silicon crystal each atom forms four covalent bonds of sp’ hybridized atomic orbitals. Up to now no reasonable approximation has been proposed for the eigenfunctions on unpaired electrons in the dislocation core. Thus, we assume that the states of unpaired electrons at a 60” dislocation are described by undisturbed sp’ orbitals dangling perpendicularly to the dislocation axis and spaced along the dislocation line at interatomic distances of 3.82 A. The dangling orbital can be built of atomic functions of s- and p-type as follows: 9’,,’ =
$s+p.+py+PJ.
Recombination
The exchange integral J between two such states connected with the orbitals localized at the neighbouring sites on the dislocation line was calculated. The positive value of J = 85 meV was obtained[21). The value of the exchange integral decreases steeply with the increasing distance between the sites. Thus the nearest-neighbour approximation seems to be applicable to describe the exchange interaction along the whole dislocation line. On the other hand, the obtained value of / is large enough to enhance the spin polarization of the system of dislocation centres. The dislocation recombination flaw consists of a one dimensional chain of exchange-interacting Dcentres. The knowledge of the chain length is an essential thing in the calculation of its spin polarization. The defects of dislocation structure as well as electrons or holes captured at a dislocation break the chain of exchange-coupled centres, and interactions between these segments can be neglected. For an analysis of the polarization of undistrubed dislocation segments we adopt the theory by Opechowski and Bryan[22] who considered a finite linear chain of paramagnetic centres with an exchange interaction between the nearest neighbours in an external magnetic field. They calculated the partition function Z of such a chain in an Ising approximation. By differentiating In Z with respect to the magnetic field the spin polarization as a function of the number of the centres in the chain, exchange integral, magnetic field, and temperature can be obtained in an explicit form(21]. According to this theory, in zero magnetic field the spin polarisation of the system is zero at all temperatures T > OK. Such a result excludes an existance of a ferromagnetic phase with a spontaneous magnetization. The value MAa has been calculated according to the above model assuming the value of the exchange integral as the one obtained for unperturbed sp’ orb&Is J = 85 meV. The solid line shown in Fig. 12 represents the experimental dependence of the value &/Ao on reciprocal temperature while the dashed one is the theoretically predicted curve. The best fit has been obtained assuming the number of coupled spins in the single segment of a dislocation to be 20 which is a reasonable value.
Fig. 12. Dependence of the radio &/Ao on temperature. The experimental dependence for the sample of n = 6 x IO” cm-’ is shown by the solid line and :he theoretical dependence by the dashed line[21].
at dislocations
1411 5. ENERGYDESIPA’IION
In the
process of recombination dislocations offer some nonconventional mechanisms of energy losses. Two of them are briefly discussed below. The first mechanism is a kind of Auger effect in which the energy of recombining carrier is expended on excitation of an electron from the lower Hubbard subband to the upper one. In other words: we consider an excitation in which the dangling electron is transferred from one @‘-centre to another separated fl-centre resulting in creation of the D-D’ pair or charge dipole. The energy spent on such a transition is E2- El + $/CR- A(R), where Ez- El is the energy difference between the two- and one-electron states of the centre, the second term describes the Coulomb interaction between the charged pair separated by the distance R, and A(R) characterizes a deviation of the interaction from the coulomb one. The value of A(R) is determined by overlap of the wavefunctions of interacting centres. According to this formula an electrostatic interaction between the first-, the second-, and the successive farther neighbours offers a variety of energy quanta which can be taken away from recombining charge in a one-step process. Taking into account the value of Ez - E, = 0.25 eV obtained by Grazhulis et al. [4] for dislocations in Si one finds these quanta of energy to amount from 0.25 to 0.55 eV. The second possible mechanism is much more sophisticated[22]. It employs a special mode of the valence electrons excitation. Let us consider the librational crystal deformation consisting in a rotation of an lattice atom by a certain angle around some axis. This deformation can be transferred throughout the crystal due to the directional character of covalent bonds linking the neighbouring atoms in the lattice. it has been recognized by KoYakowskiI231 that the librational deformation in diamond structure crystals may be transferred resonantly (i.e. without any change in its amplitude) only through (llO)-directed channels formed by the zig-zag paths of covalent bonds and if the axis vector of deformation is perpendicular to the zig-zag plane. This type of deformation has an important dynamical behaviour. Since the valence electrons only take part in the librational motion, the librational wave is expected to propagate at the frequencies comparable with plasma frequency in the crystal. Thus the librational mode plasmon can carry much more energy than the usual lattice phonons. Let us consider this in a little more detail. One can decompose the librational deformation into two parts: the twisting of an lattice atom at which the tetrahedral angles between the valence electron orbitals remain unchanged, and the bond bending. The force constant for bending /3 can be evaluated from the elastic coefficients of the crystal. It is much more difficult to estimate the force constant for twisting motion a, nevertheless there are reasonable arguments that it is about one order of magnitude smaller than /3. Assuming the masses m of valence electrons at equal-ibrium to be concentrated just in the middle of the bridges linking the neighbouring lattice ions one can find
T.
FIGIELSKI
k
Fig. 13. Dispersion curve of the librational mode crystal ex-
citation. the solution of the equation of motion for librational modes in the form of a travelling wave. It obeys the following dispersion formula (Fig. 13)
W= where k is the wave vector and a is the interatomic
distance projected on the (110) axis. The frequency w,,= (a/m)“’ corresponding to the pure twisting oscillations is the most uncertain parameter in this model, however the iho is expected to be a fraction of an eV. The librational mode plasmon can carry out the whole energy of the recombining carrier. Moreover, the librational plasmon energy may be spent on a gradual generation along (1lO)direction of a linear defect called the screw bidislocation[23] having the net Burgers vector equal to zero. This defect in turn can be easily transformed into a dislocation loop or dipole. The mechanism outlined above may give an important contribution to the process of degradation of LED’s. In this mechanism the dislocation offers an effective coupling between the current carriers and librational plasmons. The carriers recombining through dislocations transfer their energy to the librational modes via dangling b_nd oscillations. 6,ElEUMUSANDsuMMApY
In most of the experiments which, have been done hitherto one dealt with macroscopic pieces of crystal. Therefore the information obtained was that averaged over a great number of dislocations with diierent geometries. Nevertheless the experimental data obtained from photoconductivity in various laboratories and on the samples prepared in a different way are excellently
coherent. This is quite opposite to the Hall and conductivity measurements which display an enormous dispersion of results. One can understand it because the recombination occurs selectively; it selects the charged dislocations, while the current carrier concentration is affected by dislocation accompanied defects as well as by the dislocations. The recent development of the EBIC technique provides now facilities for studying properties of individual dislocations. This is perhaps the direct way to obtaining a correspondence between the core structure of dislocations and their electronic properties. Recombination induced dislocation multiplication or generation is a new problem of a great interest both to the solid state physicists and semiconductor technologists. There is still a lack of a microscopic theory of recombination through dislocations. A progress in this field may be achieved only if one’s understanding of the electronic states of dislocations will much improve. It is the present author’s opinion that the Hubbard model is the best approach to doing this. W‘FERENCES 1. D. J. H. Cockayne, I. L. F. Ray and M. J. Whelan, 1. Microsc. 98. 170(1973). 2. 2. W. Schroter and R. Labusch, Phys. Status Solidi 86. 539 (1%9). 3. J. Hubbard, Proc. R. Sot.
London 276,238 (1%3). 4. V. A. Grazhulis, V. V. Kveder and V. Yu. Mukhina, Phys. Status Solidi (a) 44, 107 (1977). 5.
T. Wosirislriand T. figielski, Phys. Status Solidi (b) 71, K73
(1975). 6. S. R. Morrison, Phys. Reu. 104,619 (1956). 7. T. Figielski, Phys. Status Solidi 6. 529 (19&Q 8. T. Figieiski,Phys. Status Solidi 9,555 (1%5). 9.
T. Figielski, Pmt. IV. ht. Summer School Defects. p. 251.
Zakoiane (1973).Polish Scientific, Warsaw (1974). 10. L. 1. Kolesnik and Yu. A. Kontsevoi. Fiz. Tu. Tela 6. 164 (1964. II. T. Figielski and A. D. Belyaev, Fiz. TV.Tela 6, 2146 (1964). 12. M. Jastrzqbska and T. Figielski. Phys. Status Solidi 14. 381 (1966). 13. R. H. Glaenzer and A. G. Jordan, Solid-St. Electron. 12, 247 t 1%9). 14. M. Jastrzebska and T. Figielski, Phys. Status Solidi 32, 791 ( 1%9). IS. Z. Golacki, Thesis, Institute of Physics Polish Academy of Sciences, Warsaw (1971). 16. H. R. Weber. Phys. Status Solidi (a) 25, 445 (1974). Ii. Yu. Osiovan and T. B. Savchenko. JETP Left. 7. 100(1%8). 18. G. K. Gertheim and G. L. Pearson, Phys. Reo. 107. 694 (1957). 19. D. J. Lepine. Phys. Rev. 96,436 (1972). 20. T. Wosiriski, T. Figielski and A. Makosa, Phys. Status sofidi (a) 37, KS7 (1976). 21. T. Wosidski and T. Figielski. Phys. Status Solidi (6) 83, 93 (1977). 2?. T. Figielski, Phys. Status Soolidi In press. 23. B. Kolakowski. in the press in Phys. Status Solidi.
AT DISLOCATIONS RGIELSKI
Institute of Physics, Polish Academy of Sciences, Warsaw, Poland Abstract-A dislocation in semiconductors behaves as a recombination flaw having a large number of charge states. The effective cross section of the dislocation is therefore a variable parameter depending by the electrostatic interaction on the occupation factor. This property manifests itself through peculiarities of photoconductivity of dislocated crystals e.g. lcgarithmic type decay of excess current carriers at low temperature. The elementary centres of recombination are most probably dangling bonds as indicated by spin-dependent effects. The magnitude of the spindependent cross section can be explained only if one takes into account the exchange interaction between neighbouring dangling electrons on a dislocation.
1.
IBTROMICTION
The role of dislocations in recombination processes in semiconductors is of considerable importance at least at low and moderate injection levels. It was often underestimated in the past which is the reason, as we believe, for serious errors committed by the people who dealt in ’60s with the recombination processes occuring via different impurities introduced into Ge and Si. As a result we have inherited a lot of useless data conceming’the capture cross sections of almost all elements of the periodic table. Moreover, as far as it is known to this writer, nobody who has ever studied seriously the recombination processes has succeeded in providing a complete explanation of experimental results on the basis of recombination statistics for isolated centres. Usually, the best way out of the difficulties was to assume an interference due to other kinds of recombination- or trap-centres (of unknown origin at best). Most of the difficulties could, however, be easily eliminated taking into account the individual character of dislocations whose action usually overlaps the action of other recombination centres. Unfortunately the knowledge of dislocation properties is not sufficiently familiar to people who deal with excess current carrier effects in semiconductors. For instance, one is usually surprized when one realizes that the dislocation limited lifetime of majority carriers in silicon at 77 K is about one year (independently of a dislocation density) and that the greatest part of carrier decay is not exponential vs time but logarithmic. These features mentioned above are direct consequences of the particular recombination statistics governed by an electrostatic interaction between recombining current carriers and charged dislocations. It is most important that the charge of a dislocation which controls an influx of current carriers into the dislocation is not a constant parameter but depends on the excess carrier density. The subject of this article is confined to the nonradiative recombination via dislocations in Ge and Si. The situation in III-V compounds is probably similar to that in elemental semiconductors. However, the amount of information concerning these materials is too small to draw any definite conclusions. 1403
In the first part of this paper the models of electron states for 60”-dislocation are briefly discussed. The second part contains the principles and consequences of the phenomenological “barrier model” of recombination via dislocations as well as their experimental verification. In the third part, the results of spin dependent recombination are presented. The article is closed with some remarks on possible mechanisms of energy dissipation by recombining charges.
The dominant dislocations in diamond- and sphaleritestructure semiconductors are: the 6tY-dislocation with Burgers vector (a/2)(1 10) and with glide plane {I 1I}, and the screw dislocation. The 6@-dislocation has dangling bonds with unpaired electrons at the edge of an extra lattice halfplane, which are believed to be mainly responsible for electrical activity of the dislocation. It usually appears as dissociated into two Shockley partials according to the reaction
with a stacking fault between them. The split components have the character of the pure edge- and the 30”-dislocation; both having dangling bonds. As found by transmission electron microscopy the dissociated dislocation is composed of alternate split segments and constricted parts [ I]. Since the dangling bonds are spaced along dislocations at interatomic distances, it was formerly believed that they form a one-dimensional half-filled energy band[2]. However, as the dangling bonds originate from atomic sp3 orbitals and they dangle perpendicularly or almost perpendicularly to the dislocation axis-the overlap of neighbouring wave functions is slight. Thus the individual dangling electrons are rather strongly localized along the dislocation line and then the conventional Hartree-Fock model, developed for a wide energy band, is not applicable. For a narrow energy band the Hubbard model[3] is much more realistic. The Hubbard hamiltonian takes into account the repulsion energy of two
T. RCIELSKI
1404 Table I. Majority lifetimes measured at the temperature shown in the third column, and corresponding capture cross sections per unit length of dislocation *Ill., [sl
Ge-n Ge-p Si-n Si-p
TGl,,,
Kl
L
i [cm 1
4x lo-8 154 lack of exact data 195 2 x lo-’ 80 25 I95 I4 x lo-’
?
at the same place. The model involves splitting of the energy band into two subbands with a variable number of states. It explains in a natural way a transition from the energy band scheme to the discrete energy levels when the distances between the dangling bonds increase infinitely. In the last case each dangling bond may be treated as a D-centre with three possible charge states: D- when the second electron is accepted into the dangling bond, D“ when one unsaturated electron is present, and D’ when the dangling electron is taken away from the bond. Such a system is described by two energy levels, which correspond to the Hubbard subbands. Experimental confirmation of the Hubbard model is obtained from microwave conductivity[4] and spindependent photoconductivity[S]. The equilibrium charge state of the whole dislocation (i.e. the net charge per one dangling bond) is not a very sensitive function of temperature or doping. The main factor which controls the population of dislocation states is an increase in the electrostatic energy due to the interacting charges.
potential difference @ can be defined by the work which has to be done against electrostatic forces in order to increase by one more charge the number of elementary charges on the line (in isothermal process). In the case of thermal equlibrium this work has to be exactly balanced by the reduction of free energy of the system due to the localization of that electron charge at the dislocation states. If we picture the situation on the band scheme, there will appear a potential barrier for electrons and a potential well for holes at the dislocation (Fig. I).
eleftrons
3. RECOMUMATION
PROCESS
The carrier recombination through dislocations is discussed here on a basis of the barrier model[6-91. This model has been formulated mainly as a result of experimental studies of photoconductivity and related phenomena in plastically deformed Ge and Si[lO-161. The intention of this author is to discuss that subject as generally as possible without referring to any particular structure of dislocation electron states. Let us consider a system consisting of undisturbed crystal-matrix-and a dislocation with edge component. There are certain electron quantum states associated with a dislocation which are localized at least in two dimensions and have energy levels inside the forbidden energy gap. For a neutral dislocation a number of those states are occupied by electrons, the others are empty. Such a neutral dislocation in order to reach equillibrium with the matrix has, in general, to exchange electrons with it. Depending on the Fermi energy in the matrix, the dislocation either accepts electrons to become negatively charged or loses them to become positively charged. nType material and negatively charged dislocations will be discussed here, but the results can be directly adopted for p-type as well. The dislocation line is surrounded by a space charge of opposite sign consisting of ionized donors and possibly holes. Thus, there appears a difference in the macroscopic electrostatic potential betwe-r ++e charged line and the bulk of material. This
Fig. 1. Potential
barrier
at dislocation.
@ is a function of linear density of charge n. Within a limited range of variation, Q, may be assumed to be proportional to 7, where @ and n are not necessarily equilibrium values. Thus
This relation is a basis for all further considerations which will be valid as far as the potential of dislocation may be linearized with respect to its charge. We denote by m. the mean volume density of the electrons which have been accepted by dislocations to equilibrate the system (and which constitute the total charge of ail dislocations in the crystal in thermal equilibrium). If N is the density of dislocations [cm-*], and q is the electron charge, then we have an evident relation TON
=
qmo.
(1)
The actual potential at the dislocation line can be expressed through the equilibrium value Q. and the relative variation of dislocation charge Am/m0 as follows
(g!!?!>
@=@0
m0
(2)
where Am = m. - m is the mean concentration of positive charge trapped at dislocations. Am includes the excess charge localized at the inherent quantum states of dislocations as well as that of excess holes contributing to the space charge. It is assumed that there is always an equilibrium among electrons in the conduction band, including the regions of dislocation core. Then the local concentration of free electrons at dislocation core n*, can be expressed by the electron concentration in the bulk, n, as II* = n exp (-
q@/kT).
(3)
1405
Recombinationat dislocations It is assumed here that the electron gas is nondegenerate and there is no overlap of space charge regions of dislocations.
After transforming the eqns (4) and (6) to a more convenient form and limiting their validity to the case when An % no one obtains R = C,Nn, exp (- qQo/kT)[exp (q%Am/kTmo)-
3.1 Capture rates We are looking for the net rate of electron capture at dislocations per unit volume and time, L The number of electrons which may be captured is proportional to the concentration n*. The number of empty dislocation states available for electrons slightly depends on external conditions. Thus, for simplicity, it may be assumed to be a constant in comparison with the strongly variable exponential factor (3). Then R = C,N[n exp (-qQ/kT)-
noexp (-q%/kT)l. (4)
The fust term in (4) corresponds to the capture of conduction electrons at dislocation, while the second one describes the reverse process-reemission of localized electrons into the conduction band. The last term is chosen in such a way that the resulting capture rate in thermal equilibrium is zero, according to the principle of detailed balancing. The coefficient C. denotes the mean probability of an elementary act of capture and may be expressed by the cross section Z. for this process per unit length of dislocation
where u. is the mean thermal velocity of electrons which can penetrate into the dislocation core. It is supposed that the excess holes trapped in the potential well are localized in the sense that they do not participate in the d.c. conductivity. They contribute only to the electric charge of the dislocation. The excess holes which are generated in the bulk sink into potential wells and subsequently recombine with electrons. So there is certain flux of holes between the bulk and the well which in the thermal equilibrium equals zero. We suppose that the influx of holes into the well is a “bottle neck” for the process of hole recombination. (This assumption is not necessary but it simplifies the further calculation.) The magnitude and direction of the flux is determined by the deviation of the dislocation potential from the equihbrium value. Thus the net influx of holes into the well may be expressed approximately as follows Rh = C,,NAp - p. exp (q@lkT)
(6)
neglecting the term which would lead to a quadratic dependence of Ap with generation rate. Thus the eqn (6) is valid strictly for small deviation from equilibrium. The recombination coefficient Ch may be formally expressed by the cross section for that process per unit length of dislocation cl# = IhUh where UI,is the mean thermal velocity of holes.
(7)
II (8)
R,, = C,,NAp - C,,NpoIexp (q%Am/kTmo)-
11. (9)
3.2 Steady state The problem of recombination via dislocations has been reduced to two differential equations describing the rates at which the current carriers leave their original bands. Now let us consider the case in which electronhole pairs are homogenously generated in the bulk at the rate G. At the steady state we have, R = Rh = G. Then eqn (8) determines the value of Am and combination of eqns (8) and (9) results in Ap. Moreover, from the condition of crystal neutrality one has An =Ap+Am.
(10)
The equation system (8)-(10) determines the excess concentrations of current carriers in the bands and excess charges at dislocations. Therefore the statistical problem of recombination via dislocations has been approximately solved and now nothing remains to be done except finding out what physical conclusions can be drawn from this extremely simple theory. We introduce the following notation: 6CF is the difference between the energy of the conduction band edge and Fermi energy at dislocation core, and l,SV= & - EC,+where & is the forbidden energy gap. The exponential forms in (8) and (9) may be then written as no exp (-qQo/kT)
= NC exp (-&kT)
(11)
p. exp (-qQo/kT)
= N, exp (-r&kT)
(12)
where N,, N, are the effective densities of states in the conduction and valence bands. We are interested in the variation in excess concentration depending upon the temperature. Let us start discussing excess holes. By using (12) the eqns (8) and (9) may be rewritten in the following form Ap
=
G nNvexp (-w/kT)
m
<
0
+&
h
(13)
There are several factors in (13) which vary with temperature but the exponential factor varies the most. Thus a slight variation of N,, N, and probably C, and Ch may be neglected at the tirst inspection compared with that of the exponent. At sufficiently high temperature, but still within the extrinsic range of conductivity, the first term in (13) dominates the second one. As the temperature is lowered the first term decreases indicating the activation energy c~v and below a certain temperature becomes negligible compared with the second one So, the expected dependence of Ap follows the lower dotted line in the Fig. 2. In the approximation used here Ap is proportional to the generation rate within the
T. FIGIELSKI
1406
with electron concentration lOI crnm3 and dislocation densities ranging from lo3 to lO’cm_’ one obtains by (16) T, varying between 220 and 300K for Ge, and between 330 and 490 K for Si. These values are in a good agreement with experiment (Fig. 3). Since T, does not depend on the generation rate it means that above this critical temperature the dislocation charge cannot be changed by illumination. This important property may have interesting consequences in some phenomena different from photoconductivity, for instance in the photoplastic effect (17).
extrinsic conductivity
T(‘K) x)0
Recomb.
IM
120
03
91
nonllneor mnge
linear ronge T,-’
200
Trapping range
range
I
303
4
I
Ti’
x-1 I/T
Fig. 2. Dependences of excess charge concentrations on reciprocal temperature expected on the barrier model.
whole range of temperature. Then linear recombination occurs and the relation Ap = GTI, is valid for the steady state by which the lifetime of holes can be determined. This T,, is inversely proportional to the dislocation density as really observed in experiments. Now let us consider the charges trapped at the dislocations. We have from (8) and (I 1) &$ <
exp &cd&T) . (14) c I
The run of the curve described by (14) is shown in the Fig. 2 by the upper dotted line. When the relative deviation of the dislocation charge from equilibrium is small Am/m0 < &T/q@,, then (14) is reduced to Am=!$&$exp(&kT). c
(15)
Fig. 3. Temperature dependence of the photoconductivity in a plastically bent Ge at different rates of generation of electronhole pairs. (a) G = 10” cme3 s-‘, (b) G = 5 X IO’s cm-’ s-’ and (c) G = 10”cm-‘s-‘[12].
Consequently we may introduce another characteristic temperature Tz, below which the Am (or An) deviates from exponential increase to saturation. This T2 is not sharply defined, but a convention may be used according to which
l
and this describes the linear part of the curve in Fig. 2. The concentration of excess electrons equal to Ap + Am is shown in Fig. 2 by the full line. Two different temperature ranges are clearly seen m this figure: the first, where Ap = An, corresponds to the bipolar photoconductivity (PC), and the second one, where A.n4 Ap, corresponds to the quasimonopolar PC. Within the last range an essential trapping of excess holes at dislocations occurs. The temperature of transition from the 8rst to the second region is given by the expression (16) This temperature increases with the dislocation density and is almost proportional to &. For typical materials
(17) T2 depends on the generation rate per one dislocation G/N. With increasing dislocation density Tz shifts towards lower values but this shift can be compensated by simultaneous increase in the light intensity. The temperature T2 divides the trapping range of PC in two parts with quite different properties. Within the range TI > T > Tz An is proportional to the generation rate. Thus a linear recombination occurs and consequently the concept of electron lifetime can be introduced in the usual way. By lowering the temperature below Tt one enters the most interesting nonlinear region of photoconductivity. The relative deviation of dislocation charge from equilibrium is significant and the barrier height is consider-
Recombination ably
reduced. In that case nonlinear properties of dislocation recombination flaws are clearly seen. The excess electron concentration is here described by
It follows that the steady state photoconductivity grows with the dislocation density, although more slowly than linearly. It should be noticed here that photoconductivity increases with illumination proportionally to the logarithm of the generation rate (Fig. 4). This relation is valid of course within the limited range of light intensities. It is important that the dislocation density N enters into the proportionality coefficient in the relation An-InG.
a
e tb 4o a
at dislocations
1407
where (21)
kT rlo
1
(22)
The to is a parameter determined by the initial condition in the sample and to< T< for T < Tz. The T< is the characteristic time of the decay process and has the same analytical form as the electron lifetime in the linear range where the decay is an exponential process. r< does not depend on dislocation density but is a very sensitive function of temperature. According to (20) the main part of the decay is linear with the logarithm of time. Plotting An vs In t one obtains a straight line, which extrapolated to the t-axis intersects it at the point t = 7r - to which practically equals 7. for a sufficiently low temperature (Fig. 5). After a time comparable with T., An decreases to such a small value at which the criterion of linear recombination becomes fuliiled. Thus the further decay occurs exponentially with the lifetime T_
20
.o
I (orb. units)
Fig. 4. Steady state photoconductivity in plastically deformed Si vs illumination intensity after[l5]. (I) reference sample, (2) heated but unbent sample, (3). (4) and (5) sample bent to the radii of curvature 12, 6 and 3 cm respectively. 5
IO
50
)O
t (5)
3.3 Kinetics of recombination in nonlinear region The nonlinear character of the recombination process occuring through dislocations is most clearly seen in the kinetics of the decay of photoconductance. Let us start from a steady state for an illuminated sample and assume that at the moment t = 0 ihumination is switched off. The excess charges decay in time and we look for the timedependence of An. The rate of electron capture is given by the eqn (8) which in the considered range is reduced to the following form
Fig. 5. Decay of photoconductivity after illumination of the sample by a rectangular light pulse for n-type silicon, after1151.
(1) 150fkm sample bent to the curvatureradius r, = 3 cm, (2) and (4) IOOOfkmunbent sample at two different illumination levels, (3) 35 km,
unbent sample.
The decay described by (20) has another interesting property. By introducing a new time variable t’ = to+ t into (20) it becomes transformed into a form independent of the generation rate (23)
$
= WN,
exp (-ecdkT)
exp (q%Am/kTq,N). (19)
It is solved by An=Am=-7
kT2N,nfo+ 4 @o
7.
(20)
This equation describes an universal decay (Fig. 6) which includes any particular illumination level. In this picture the decay starts at different values t’= to corresponding to the different initial steady states. The rise of PC, not, discussed here, is quite well described by the formula n(t) = An(a)[ 1- exp (-t/to)].
1408
Fig. 6. Universal decay curve a-b for majority current carriers. I, > 1, > I,different illumination levels.
By lowering temperature r. rises to extremely large values. An example of photodecay at 77K in plastically bent Si is shown in Fig. 7 in two different time scales. The To extrapolated from this picture equals 3. 10’s. Thus a long time period after illumination of the crystal the excess carrier density remains at some, practically unchanging, level. This property is usually called photomemory. As the temperature is lowered below that at which r. reaches the value of about 103s the conditions required for a proper experiment become more and more rigorous. The crystal exhibits then a very high photosensitivity and even the lowest background radiation can considerably change the experimental zero level of PC[8]. The actual values of T for majority carriers in Ge and Si, measured in the range where it can be done with great accuracy, as well as the estimated cross sections are shown in the table.
3.4 Supplementary remarks The temperature dependence of PC in samples with different dislocation density under definite illumination is schematically shown in Fig. 8. The family of curves for different illuminations levels is sketched in Fig. 9. It can be seen that the illumination level is just as important a parameter in PC as the temperature, and it can even change the shape of the curve observed. If one wants to get some definite information from a single measurement, one must apply as low illumination as possible. The mistake often committed in the past was doing an experiment at some arbitrary but relatively high injection level, i.e. under conditions which wouid correspond to
N3
NZ
1 N,
I=corw N, ‘NH
Fig. 7. Time response of photoconductivity in plastically deformed Si after illumination of the sample by a rectangular light pulse. Internal picture-linear time scale, external picturelogarithmic time scale [ IS].
I/T
Fig. 8. Theoretical dependence of photoconductivity vs reciprocal temperature for different dislocation densities.
Recombination
at dislocations
1409
10’2-10’4crnw3,which means about a half of the energy gap. Glaentzer and Jordan[l3] found out from the kinetics of decay of PC the value 0.52 eV in n-Si and the values varying in the range 0.38-0.5 eV in p-Si. However, they have not paid due attention to the nonlinearity in the decay. On the contrary Golacki[lS] has found almost full symmetry of the kinetic process with respect to the inversion of type, which is in accordance with the position of the Fermi level at the dislocation in the middle of the forbidden gap.
N=amsi.
Fii.
9. Theoretical dependence of photoconductivity on the reciprocal temperatures for different illumination levels.
the curves above the curve (a) in Fig. 3. To this category of measurements belong, for instance, those by Wertheim and Pearson [ 18). The energy factors in the exponents of (13) and (14) enter through the local concentrations of majority carriers at the dislocation and, therefore, are defined by the Fermi level position at the dislocation core lCF or em. What are their experimental values? One does not have to pay much attention to the activation energy which appears within the recombination range of PC, as the extension of this range is usually very small and, moreover, the model utilized turns out to be of little use. Just the opposite situation occurs in the linear trapping range. The extension of this range can be almost arbitrarily enlarged by using extremely low light intensities, and the exponential increase in PC can be observed within several orders of magnitude. In n-type germanium the mean value of the activation energy obtained from a great number of experiments is -0.42eV. A large majority of results do not exceed the mean value by +O.O2eV. No definite shift due to doping within the range of concentrations 10’3-10’5cm-3 has been observed. The activation energy in p-type Ge can not be determined as accurately as in n-type, mainly due to much less photosensitivity of those crystals, however, the values reported lie between 0.24 and 0.30 eV, and are close to the value & - EN resulting from the theory. The activation energies in Si, in the steady state PC, are independent of the type of conductivity and amount to 0.6 2 0.03 eV for the crystal within the doping range ssEvol.?I.No. ,,,,
I-,
The spin-dependent recombination process 151is hoped to be an effective tool to examine more thoroughly the mechanism of carrier recombination at dislocations as well as the nature of electron states of the dislocation. This phenomenon is revealed in spin-dependent photoconductivity (SDP). The philosophy of this effect is the following. The two-electron stable state of a dangling bond is most probably the singlet state in which the spins of both the electrons are antiparallel. Its energy corresponds to the upper level (subband) in the forbidden energy gap while the lower level corresponds to the one-electron state. A possible triplet state-with parallel spin orientation of both the electrons-is probably degenerated with the conduction band states, and therefore does not form a stable configuration. If the spin of the recombining electron is preserved in the recombination transition, then the only intermediate state in the process of recombination is the singlet state. (The situation for the recombining hole is similar although the holes are mostly captured at the lower energy level). In that case the cross section for an electron capture P. depends on the spin polarisation of both the electron systms: that of recombining electrons p, and the one of dangling electrons P, in the following way [ 191
z, =Zo(l -
pP)
(24)
where Z0 is the cross section for zero polarisation. It can be seen from this formula that the capture cross section is reduced due to the spin polarisation. If the product of the spin polarisation degrees of both the electron systems is changed by 6(pP), then a relative change of the cross section is given by
sz._--
I,-
&PP)
I-pP
(25)
which is practically equal to -S(pP) under usual experimental conditions. The SDP is observed in plastically deformed silicon under the conditions of electron spin resonance. When the sample is placed in the magnetic field, then some polarisation appears which can be destroyed by input of microwave power of the Larmor frequency corresponding to one of the two electron systems. In the basic experiment the sample placed in the microwave resonator is steadily illuminated by a light generating electronhole pairs in the bulk. The microwave power is modulated with audio frequency and a change in photoconductivity is detected at the same frequency. When
1410
T.
~GELSKI
sweeping the magnetic field over the resonance region a resonant decrease in photoconductivity is observed. Two other modifications of this basic method may be used in order to record the derivative of the SDP-signal which makes it possible to reveal a certain structure of the spectrum (201. The SW-signal is observed on silicon of both nand p-type in a wide range of temperatures (M-340 K), Fig. 10. The signal maximum corresponds to a g-factor equal to 2.004, which is close to that of the central line in the EPR spectrum due to the dislocations in Si but far from the g-factor of free carriers. The half-width of the resonant line (of about 6 Oe) is independent of temperature over the range of measurement. The room temperature ratio of the SDP signal amplitude, 60, to photoconductivity, Au is of the order of so/Au = lo-’ and independent of the intensity of illumination.
I
I
I
I
I
6
6
10
12
c 4
Fig. 11. Dependece of relative photoconductivity AU/Uand reiative resonant change of photoconductivity &/a in silicon on temperature at the same illumination level. Au/o was measured using light chopped at the same frequency as that of the microwave modulation[21].
one gets the maximum value of the ratio sojho upon saturation by microwave power
photoconductivity Fig. 10. Resonant decrease of silicon sample photoconductivity observed at room temperature when sweeping the d.c. magnetic field and irradiating the sample with microwaves of X-band. The microwave power was square-wave modulated 220Hz. and the conductivity change at this frequency was recorded(211.
The value Wo called the relative resonant change of photoconductivity, where u is the equilibrium conductivity, and the relative photoconductivity Aoiu of the same sample are shown in Fig. 11 as a function of reciprocal temperature. Both the signals are measured at the same ihumination level. At high temperatures, when the photoconductivity in dislocated Si increases exponentially vs the reciprocal temperature, the courses of both the curves are similar. At low temperatures, when the photoconductivity variation saturates, the discrepancy between the courses of both the curves becomes more pronounced. The variations of solo and AU/C with the illumination level and with the light wavelength show analogous dependences known typically for disiocation-controlled recombination in silicon[21]. It must be. pointed out, to demonstrate the advantages of the SDP technique, that it can be successfully applied even in the case when the number of dislocation paramagnetic centres is too small to be detected by the conventional EPR method. Contrary to the EPR, the SDP signal is independent of the number of spins as long as the recombination process through the paramagnetic centres dominates. Let us estimate the maximum value of the SDP signal on the ground of the simplest model described by the formula (25). Passing from the capture cross sections to the
E=pP where Au is the steady-state photoconductivity and Sa is its change due to the microwave power. For noninteracting recombination centres with spin l/2 this yields a value of = IO4 at room temperature in the magnetic field of 30000e. The value obtained in the experiment was one order of magnitude greater despite the nonsaturating microwave power used. 4.1 Exchange interaction
To explain the unexpected large magnitude and temperature independence of the half-width of the SDP signal we shall analyze the exchange interaction between paramagnetic recombination centres spaced along the dislocation line. In a perfect silicon crystal each atom forms four covalent bonds of sp’ hybridized atomic orbitals. Up to now no reasonable approximation has been proposed for the eigenfunctions on unpaired electrons in the dislocation core. Thus, we assume that the states of unpaired electrons at a 60” dislocation are described by undisturbed sp’ orbitals dangling perpendicularly to the dislocation axis and spaced along the dislocation line at interatomic distances of 3.82 A. The dangling orbital can be built of atomic functions of s- and p-type as follows: 9’,,’ =
$s+p.+py+PJ.
Recombination
The exchange integral J between two such states connected with the orbitals localized at the neighbouring sites on the dislocation line was calculated. The positive value of J = 85 meV was obtained[21). The value of the exchange integral decreases steeply with the increasing distance between the sites. Thus the nearest-neighbour approximation seems to be applicable to describe the exchange interaction along the whole dislocation line. On the other hand, the obtained value of / is large enough to enhance the spin polarization of the system of dislocation centres. The dislocation recombination flaw consists of a one dimensional chain of exchange-interacting Dcentres. The knowledge of the chain length is an essential thing in the calculation of its spin polarization. The defects of dislocation structure as well as electrons or holes captured at a dislocation break the chain of exchange-coupled centres, and interactions between these segments can be neglected. For an analysis of the polarization of undistrubed dislocation segments we adopt the theory by Opechowski and Bryan[22] who considered a finite linear chain of paramagnetic centres with an exchange interaction between the nearest neighbours in an external magnetic field. They calculated the partition function Z of such a chain in an Ising approximation. By differentiating In Z with respect to the magnetic field the spin polarization as a function of the number of the centres in the chain, exchange integral, magnetic field, and temperature can be obtained in an explicit form(21]. According to this theory, in zero magnetic field the spin polarisation of the system is zero at all temperatures T > OK. Such a result excludes an existance of a ferromagnetic phase with a spontaneous magnetization. The value MAa has been calculated according to the above model assuming the value of the exchange integral as the one obtained for unperturbed sp’ orb&Is J = 85 meV. The solid line shown in Fig. 12 represents the experimental dependence of the value &/Ao on reciprocal temperature while the dashed one is the theoretically predicted curve. The best fit has been obtained assuming the number of coupled spins in the single segment of a dislocation to be 20 which is a reasonable value.
Fig. 12. Dependence of the radio &/Ao on temperature. The experimental dependence for the sample of n = 6 x IO” cm-’ is shown by the solid line and :he theoretical dependence by the dashed line[21].
at dislocations
1411 5. ENERGYDESIPA’IION
In the
process of recombination dislocations offer some nonconventional mechanisms of energy losses. Two of them are briefly discussed below. The first mechanism is a kind of Auger effect in which the energy of recombining carrier is expended on excitation of an electron from the lower Hubbard subband to the upper one. In other words: we consider an excitation in which the dangling electron is transferred from one @‘-centre to another separated fl-centre resulting in creation of the D-D’ pair or charge dipole. The energy spent on such a transition is E2- El + $/CR- A(R), where Ez- El is the energy difference between the two- and one-electron states of the centre, the second term describes the Coulomb interaction between the charged pair separated by the distance R, and A(R) characterizes a deviation of the interaction from the coulomb one. The value of A(R) is determined by overlap of the wavefunctions of interacting centres. According to this formula an electrostatic interaction between the first-, the second-, and the successive farther neighbours offers a variety of energy quanta which can be taken away from recombining charge in a one-step process. Taking into account the value of Ez - E, = 0.25 eV obtained by Grazhulis et al. [4] for dislocations in Si one finds these quanta of energy to amount from 0.25 to 0.55 eV. The second possible mechanism is much more sophisticated[22]. It employs a special mode of the valence electrons excitation. Let us consider the librational crystal deformation consisting in a rotation of an lattice atom by a certain angle around some axis. This deformation can be transferred throughout the crystal due to the directional character of covalent bonds linking the neighbouring atoms in the lattice. it has been recognized by KoYakowskiI231 that the librational deformation in diamond structure crystals may be transferred resonantly (i.e. without any change in its amplitude) only through (llO)-directed channels formed by the zig-zag paths of covalent bonds and if the axis vector of deformation is perpendicular to the zig-zag plane. This type of deformation has an important dynamical behaviour. Since the valence electrons only take part in the librational motion, the librational wave is expected to propagate at the frequencies comparable with plasma frequency in the crystal. Thus the librational mode plasmon can carry much more energy than the usual lattice phonons. Let us consider this in a little more detail. One can decompose the librational deformation into two parts: the twisting of an lattice atom at which the tetrahedral angles between the valence electron orbitals remain unchanged, and the bond bending. The force constant for bending /3 can be evaluated from the elastic coefficients of the crystal. It is much more difficult to estimate the force constant for twisting motion a, nevertheless there are reasonable arguments that it is about one order of magnitude smaller than /3. Assuming the masses m of valence electrons at equal-ibrium to be concentrated just in the middle of the bridges linking the neighbouring lattice ions one can find
T.
FIGIELSKI
k
Fig. 13. Dispersion curve of the librational mode crystal ex-
citation. the solution of the equation of motion for librational modes in the form of a travelling wave. It obeys the following dispersion formula (Fig. 13)
W= where k is the wave vector and a is the interatomic
distance projected on the (110) axis. The frequency w,,= (a/m)“’ corresponding to the pure twisting oscillations is the most uncertain parameter in this model, however the iho is expected to be a fraction of an eV. The librational mode plasmon can carry out the whole energy of the recombining carrier. Moreover, the librational plasmon energy may be spent on a gradual generation along (1lO)direction of a linear defect called the screw bidislocation[23] having the net Burgers vector equal to zero. This defect in turn can be easily transformed into a dislocation loop or dipole. The mechanism outlined above may give an important contribution to the process of degradation of LED’s. In this mechanism the dislocation offers an effective coupling between the current carriers and librational plasmons. The carriers recombining through dislocations transfer their energy to the librational modes via dangling b_nd oscillations. 6,ElEUMUSANDsuMMApY
In most of the experiments which, have been done hitherto one dealt with macroscopic pieces of crystal. Therefore the information obtained was that averaged over a great number of dislocations with diierent geometries. Nevertheless the experimental data obtained from photoconductivity in various laboratories and on the samples prepared in a different way are excellently
coherent. This is quite opposite to the Hall and conductivity measurements which display an enormous dispersion of results. One can understand it because the recombination occurs selectively; it selects the charged dislocations, while the current carrier concentration is affected by dislocation accompanied defects as well as by the dislocations. The recent development of the EBIC technique provides now facilities for studying properties of individual dislocations. This is perhaps the direct way to obtaining a correspondence between the core structure of dislocations and their electronic properties. Recombination induced dislocation multiplication or generation is a new problem of a great interest both to the solid state physicists and semiconductor technologists. There is still a lack of a microscopic theory of recombination through dislocations. A progress in this field may be achieved only if one’s understanding of the electronic states of dislocations will much improve. It is the present author’s opinion that the Hubbard model is the best approach to doing this. W‘FERENCES 1. D. J. H. Cockayne, I. L. F. Ray and M. J. Whelan, 1. Microsc. 98. 170(1973). 2. 2. W. Schroter and R. Labusch, Phys. Status Solidi 86. 539 (1%9). 3. J. Hubbard, Proc. R. Sot.
London 276,238 (1%3). 4. V. A. Grazhulis, V. V. Kveder and V. Yu. Mukhina, Phys. Status Solidi (a) 44, 107 (1977). 5.
T. Wosirislriand T. figielski, Phys. Status Solidi (b) 71, K73
(1975). 6. S. R. Morrison, Phys. Reu. 104,619 (1956). 7. T. Figielski, Phys. Status Solidi 6. 529 (19&Q 8. T. Figieiski,Phys. Status Solidi 9,555 (1%5). 9.
T. Figielski, Pmt. IV. ht. Summer School Defects. p. 251.
Zakoiane (1973).Polish Scientific, Warsaw (1974). 10. L. 1. Kolesnik and Yu. A. Kontsevoi. Fiz. Tu. Tela 6. 164 (1964. II. T. Figielski and A. D. Belyaev, Fiz. TV.Tela 6, 2146 (1964). 12. M. Jastrzqbska and T. Figielski. Phys. Status Solidi 14. 381 (1966). 13. R. H. Glaenzer and A. G. Jordan, Solid-St. Electron. 12, 247 t 1%9). 14. M. Jastrzebska and T. Figielski, Phys. Status Solidi 32, 791 ( 1%9). IS. Z. Golacki, Thesis, Institute of Physics Polish Academy of Sciences, Warsaw (1971). 16. H. R. Weber. Phys. Status Solidi (a) 25, 445 (1974). Ii. Yu. Osiovan and T. B. Savchenko. JETP Left. 7. 100(1%8). 18. G. K. Gertheim and G. L. Pearson, Phys. Reo. 107. 694 (1957). 19. D. J. Lepine. Phys. Rev. 96,436 (1972). 20. T. Wosiriski, T. Figielski and A. Makosa, Phys. Status sofidi (a) 37, KS7 (1976). 21. T. Wosidski and T. Figielski. Phys. Status Solidi (6) 83, 93 (1977). 2?. T. Figielski, Phys. Status Soolidi In press. 23. B. Kolakowski. in the press in Phys. Status Solidi.