P. TUSSING et al. : Recombination by Excitat,ion of Plasmons 45 1

phys. stat. sol. (b) 53, 451 (1972)

Subject classification: 13.5.2; 22.2.1 ; 22.2.3

I I . Institut fur Theoretische Physik der Technischen Universitat Berlin

Recombination in Semiconductors by Excitation of Plasmons

BY P. TUSSING, W. ROSENTHAL, and A. HAUG

In a highly doped semiconductor excitation of plasmons is shown to be an effective mechanism for electron-hole recombination. If the carrier concentration is sufficiently high the corresponding recombination rate overwhelms that for the radiative process by orders of magnitude.

Fur stark dotierte Halbleiter wird gezeigt, da13 die Anregung von Plasmonen ein effek- tiver Mechanismus fur die Band-Band-Rekombination ist. Wenn die Ladungstrager- konzentration genugend hoch ist, uberwiegt die zugehorige Rekombinationsrate die der strahlenden Rekombination um einige GroBenordnungen.

1. Introduction The electron-plasmon interaction has been proposed by several authors [l, 21

for the explanation of non-radiat,ive band-to-band transitions of electrons. We shall investigate this assumption for n-type material in what follows. For this purpose we need the energy of a plasmon 131 :

m,, N , are the effective mass and the concentration of the conduction electrons and E is the dielectric constant. In the case of band-to-band recombination this energy must be at least equal to the energy gap AE between the valence and the conduction band :

h w , l z AE . (1.2) Therefore we get a lower limit of the electron concentration for which the process is possible. Concerning the 111-V semiconductors this limit lies between 5 x l0lS for InSb and 1021 for GaAs. Band-to-band recombination via plasmon excitation is thus only possible in highly doped semiconductors. In addition, it is necessary that the electrons of the conduction band build up their own plasmons, independently of the electrons of the valence band.

We shall calculate the lifetime of the recombination via plasmon excitation in the following sections. Comparing it with the lifetime of corresponding radiative transitions, we show that the plasmon recombination is usually dominating.

2. The Electron-Plasmon Interaction The Hamiltonian for the electrons in a crystal can be written H 1 2 E n ( $ ) ) a,+,*, o an,*, a +

n , p ,

452 P. TUSSING, W. ROSENTNAL, and A. HAUQ

(2.3)

an+,p, u, an,p, state

are the creation and destruction operators of an electron in a BIoch

~ 2 n , p ( r ) = u n , p ( r ) e ip . (2.4)

p is the wave vector, cr the spin quantum number, and n tht, 1 band number con- cerning only the conduction and valence band. This Hamiltonian must be transformed into the plasmon representation. In contrast to the usual trans- formation which refers to a gas of free electrons [3] we have to do with electrons in a crystal and therefore to distinguish between the electrons of the conduction band and those of the valence band. Thus we use a modified transformat,ion [6 I :

U = e tp , ( 2 . 5 )

\

f C I n v i ( P , K ) d, p + K , ~ am,y,u , (2.6) ) n, ?n n i 7 n

where Q,, are the collective coordinates of the plasmons and K , is the critical wave vector length which separates the short range and the long range parts of- the Coulomb potential. Using (2.6) and assuming that

I1nm.I < IInnI for n i 7n we get the new Hamiltonian

n+ni

with the subsidiary condition

( 2 . 7 )

(2.8)

Recombination in Semicon&ctors by Excitation of Plasmons 453

and

The first two terms in (2.7) describe the electrons interacting via the short range part of the Coulomb forces. The next term represents the undisturbed plasmons where the plasmons of the valence band and those of the conduction band are separated corresponding to the band number. Then follows a plasmon-plasmon interaction and a constant term by which the long range part of the electron- electron self interaction is compensated. The last term is the electron-plasmon interaction which we have to deal with. We now introduce creation and destruc- tion operators for the plasmons :

We only consider that part of the electron-plasmon interaction which causes transitions between the conduction and the valence band with the aid of con- duction band plasmons :

P, with

I n the following we will omit the suffix c a t the plasmon operators.

3. Calculation of Lifetime

The probability of a band-to-band transition of an electron with the emission of a plasmon is given by

2n h

w = - l(fI H , li)lz 6 (Ef - Ei). (3.1)

i, f are the initial and final state of the undisturbed system. We get the total recombination probability WR by evaluating the matrix-elements, summing over all transition probabilities and taking into account the occupation prob- abilities :

x [1 - f (Ev(P))I6 ( W P ) - E,(P + K ) + hcu) . (3.2)

In (3.2) we used the abbreviation

Idv, c, c, p , H)I2 = IdC, v, c, p + K , -w2 = Is@, WIZ . (3.3)

/ (En@)) is the probability of finding an electron in a Bloch state with the (pan - tum-numbers n, p , and sK is the number of plasmons. The additional factor 2 results from summing of the spin states.

454 P. Tussrxa, W. ROSENTHAL, and A. HAUG

To obtain the net recombination we subtract from (3 .2) the probability W , for the reverse processes. I n thermal equilibrium we have

w, = W , = wo. (3.4)

I n non-equlibrium we can set

(3.5)

n, p are the electron and hole densities, respectively, no, p , are their values in equilibrium. As a consequence of (3.5) we get the lifetime [4]

Following the calculations of Beattie and Landsberg [5] this result can be derived in a more rigorous way by means of the concept of quasi-Fermi levels

To get Wo we have to introduce the equilibrium expression for f and s, that is the Fermi distribution for the electrons and the Bose distribution for the plas- mons. As the plasmon energies in question are h~ > 0.1 eV and the tempera- tures are T < 400 K we can set

S + l X l . (3.7)

161.

Moreover the Fermi distribution of the holes can be approximated by a Boltz- mann distribution since the Fermi energy lies near or in the conduction band in the case of heavy doping. Changing from sums to integrals and using simple energy bands,

(3.8)

( 3 . 9 )

Recombination in Semiconductors by Excitation of Plasmons 455

A11 integrals of (3.10) can be calculated exactly. The last one has been approxi- mated by the mean value theorem for integrals. The result is

hco - AE k T

-

~ _ _ (2 72 hw)2 hw h o - AE J - 2 -

Eoc --q h~ - AE ’

kT k T 1 (3.11)

The correspondent lifetime (3.6) becomes

[ (Lc) (- AE +Phnr)l (3.12) x exp - +exp ~~

+ ( l + p ) k T ’ kT

1 hw AE =

if - > l + , u

(3.13)

y follows from the limitation of the integration range by the %function.

4. Application (to InSb)

We apply our results to n-dopedInSb with carrier densities n, between lo1* and 1020 em-3. I n this case we can neglect n;l in (3.13) in comparison to p;l whereas p , is given by

Using E = 16.1, p = 0.035 (1 +,u = 1) and IIc,vl = 0.1, the last being the same as with Auger processes [7], we obtain a lifetime for plasmon processes which is plotted in Fig. 1 as a function of electron density.

12, is the limit density which is necessary for such processes (cf. Section 1). In addition there is the lifetime of radiative processes according t o the Roosbroeck-Shockley method. For n > h, the lifetime of plasmon processes is lower by orders of magnitude ; therefore these processes doniinate in this range. This result is also true for other 111-V-compounds. For n < 2, the plasmon processes die out and the radiative processes predominate. Such an effect is ex-

Fig. 1. Lifetime T as a fuiictioii of tho electron density in the con- duction band. The straight line shows the lifetime of the radiation

processes electron density -

456 P. TWSINI et al. : Recombination by Excitation of Plasmons

perimentally well known ; one observes a sharp drop of luminescence efficiency in several 111-V semiconductors if the electron density increases. But the ob- served electron densities are much lower than Gc I S to lo]. This may perhaps be due t o the effects of heavy doping, for instance thc distortion of the band struc- ture, which are not considered in our calculation.

Finally we want to mention the fact that even in cases, where the doping is much lower than necessary to build up plasmons with an energy comparable with the band gap, if there flows a current, there can exist plasmons with such energies [ 11 1.

References [l] V. 1,. BONCH-BRUEVICH and E. G. LANDSBERG, phys. stat. sol. 29, 9 (1968). [2] H. J. QUEISSER, Verh. DPG 1, 179 (1969). [3] D. PINES, Elementary Excitations in Solids, W. A . Benjamin Inc., New York/Amster-

[4] J. 8. BLAKEMORE, Semiconductor Statistics, Perpmon Press, 1962. [5] A . R. BE~TTIE and P. T. LANDSBERG, Proc. Roy. SOC. A249, 16 (1958). [6] P. TUSSING, Diplomarbeit, Technische Universitat Berlin, 1971. 171 E. ANTONE~K and P. T. LANDSBERG, Proc. Phys. SOC. 82, 337 (1962). [8] H. J. QUEISSER and M. B. PANISH, J. Phys. Chem. Solids 28, 1177 (1967). [9] J. C. TSANG, P. J. DEAN, and P. T. LANDSBERG, Phys. Rev. 178, 814 (1968).

darn 1964.

1101 C. J. H w r ~ a , J. appl. Phys. 42, 4408 (1971). ( 1 1 1 D. J. BALLEGEER and C. A. RAUMGARDNER, Solid State Commun. 10, 111 (1972).

(Received June 30, 1972)