Spontaneous emission by ballistic electrons in semiconducting heterostructures

Solid State Communications, Vol. 71, No. 12, pp. 1131-1135, 1989. Printed in Great Britain.

0038-1098/89 $3.00 + .00 Pergamon Press plc

S P O N T A N E O U S EMISSION BY BALLISTIC E L E C T R O N S IN S E M I C O N D U C T I N G HETEROSTRUCTURES M. Botton*

Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel and A. Ron Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel

(Received 10 February 1989 by S. Alexander) We investigate the spontaneous emission by electrons traversing ballistically a superlattice in semiconducting heterostructures. It is found that for an injected current of 100 #A into a 5 × 20/~m 2 device there are about 5 x 108/s infra-red photons emitted, mostly perpendicular to the electrons direction of motion. We also estimate the gain of the structure, and find that stimulated emission occurs when the threshold current density is of order 104 A/cm 2. IN R E C E N T YEARS the advent of thin film epitaxial crystal growth techniques led to resurgence of interest in ballistic current transport in semiconductors. A typical experimental set-up consists of a transistor-like heterostructure in which a very thin base (less than 100nm) is sandwiched between the heavily doped emitter and collector layer (e.g., the T H E T A [1] and PDBT [2] devices). Biasing the base-emitter heterojunction, electrons are injected through a thin barrier into the base region and tranverse it ballistically, i.e., with very few scattering (either elastic or inelastic) almost like free electrons in the drift region of a vacuum tube. The energy spectrum of these ballistic electrons is estimated with a second barrier located in the base-collector heterojunction, which under an appropriate external biasing serves as an energy spectrometer (see recent review by Heiblum and Fischetti

[31). In the present paper we report on our investigation of the spontaneous emission by ballistic electrons in semiconducting heterostructures. The injection of these electrons into the transport region is generally done either by thermionic injection [2] or tunneling injector [I]. We however do not examine here these problems but rather consider the ballistic electrons after they were injected into the transport region and moreover, assume that their energy distribution is known. The transport region is made of alternat-

* This work is based partially on a D.Sc. thesis at the Technion-Haifa.

ing layers of intrinsic semiconducting alloys (e.g., GaxAll_xAs/GayAl~yAs with different AI concentration, i.e., x ~ y) so that a periodic superlattice is formed at will. The idea of a superlattic which was introduced by Esaki and Tsu [4] led to an extensive study of the emission by electrons in heterostructure devices (see the review by Luryi [5]). Mainly two types of radiation mechanisms in these "multiple quantum well" devices were investigated during the years. One is based on transitions of electrons between the excited and ground sub-levels formed in each separate quantumwell [6, 71. The other is based on "photon assisted tunneling" of electrons in a tiled potential from the ground state in one well to an excited state in its neighbour followed by a non-radiative relaxation of the electron to the ground state [8, 9]. In both cases the wells are weakly coupled to each other, hence the emitted power is linear with the number of the wells. Moreover, the electrons need not be ballistic in these emission processes. In the present work we assume that the coupling among the wells is significant so that a real periodic superlattice is formed. A case like this arises when the wells are sufficiently close to each other (i.e., the distance between two adjacent wells is of the same order as their width), and more important, when the mean free path of the injected ballistic electrons is longer than several periods of the superlattice (approximately 10). When these conditions are met, the ballistic electrons "feel", while traversing the transport region, an effective potential which is periodic in space along

1131

1132

S P O N T A N E O U S E M I S S I O N BY B A L L I S T I C E L E C T R O N S

0,i

X 0.3

Vol. 71, No. 12

potent~Ziel~j

iEv~lo[ ~eLIEffective -

j

0, =o

o=

lad

tJA

} 02

_~°~ LI.I

la.l

g

g

"~

o~

]xlO /

/

/" O0

........

~

+

oo I

3

2

dNldt (10 9 photonslsec) (a)

-0

i

(b)

-o,

Fig. 1. The mini-band structure of the periodic potential, V(z), which is depicted in the inset. The electrons are assumed to be injected with energy at the second mini-band and therefore can emit by transitions into the lower mini-band (V = 0.1 eV, l = 6 rim, ~ = 0.5). (b) The photon emission rate as a function of the electrons energy as given by equation (9) (J = 100 A/cm 2, S = 10 6 cm 2, L = 10- 5 cm, v,. = 5 x 107 cm/s and V(z) is the same as at (a). The emission from the superlattice (solid line) is two orders of magnitude stronger than the emission from one potential well (dashed line), and is shifted to higher energies. The fact that the superlattice is finite should make the curve rounded.

their direction of flight. A periodic potential is known to enable interaction between free electrons and electromagnetic radiation as happens for example in the free electron laser [10]. In our case, the periodic electrostatic potential causes the splitting of the conduction band in the host material into mini-bands', i.e., allowed and forbidden energy zones. If the ballistic electrons are injected with energy in an excited miniband they can spontaneously emit photons while relaxing to a lower mini-band. Accordingly, we first calculate the energy states of the ballistic electrons in the peridic superlattice. Then we obtain the spontaneous emission rate and compute the number of emitted photons when the injected current is given. Finally, we estimate the electromagnetic gain of the system, and indicated the feasibility of lasing operation. Consider a ballistic electron in the conduction band of the "host" semiconductor. In the effective mass approximation we assume that the electron moves in a "Kronig-Penney'Mike periodic potential V(z) of periodicity l which is given by the superlattice structure along the z-axis (see the inset of Fig. l(a)) while independent of the x and y directions. The energy eigenstates of the electrons, is) (s stands for the "'longitudinal" energy E and the two transversal components of the wave vector k, and k,) can be separated

into an "envelope" function and a Bloch function of the atomic lattice. Since the transitions here are between two different mini-bands within the conduction band, we neglect the changes in the Bloch function of the host material and retain only the envelope function, hut, i.e., tPL~(X, y, Z)

I

,Vr~ exp {ik,.x + ik,y}t~(z),

(1)

where S is the area, and ¢(z) obeys the Schroedinger equation with V(z). Since V(z) is periodic, we use Bloch's theorem and write 0(z) = ul, k(z) exp {ikz}, where u(z) is periodic in l, and k is the wave number in the first Brillouin zone (Jkl < ~/l). We neglect here the effects of the boundaries (emitter and collector), and assume that the superlattice is long enough (at least 10 unit cells) to justify our model. In the first unit cell we write the wave function as:

¢(z)

=

4,(z)

j'A exp {ipz} + B e x p {-ipz}

0<:<~1

C exp {iqz} + D exp { - i q z }

~l<:<1 (2)

with p = (2mE/he) 1'2, and q = [2m(V + E)/~-'] I ~. Furthermore, in the first cell UE.k(Z) = 4~(Z) exp

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S P O N T A N E O U S EMISSION BY BALLISTIC E L E C T R O N S

{ - i k z } (0 < z < l), since u(z + nl) = u(z) for n integer, we have expressed the Bloch wave function O(z) of equation (1) in terms of the four amplitudes A, B, C, D of equation (2). We now use the continuity conditions on W~ and dUd,/dz at z = 0 and z = ~1 to determine the ratios A/C, B/C, D/C. This leaves one constant, C, to be determined by the normalization of qJ~. At this stage we normalize ud, to account for one electron in the superlattice volume V - SL (L = Nl is the length of the superlattice). The connection between k and E is also found and is given by the following dispersion equation: cos (kl)

= cos (p~l) cos [q(l -- 3)l] p2 + q2 sin (p~l) sin [q(l -- ~1]. (3) 2pq

For propagating electron states, we consider only the real solutions for k, and thus equation (3) describes the formation of the energy min#bands (see Fig. l(a) for a superlattice with l = 6nm, ~ = 0.5 and V = 0.1 eV. We now turn to calculate, using Fermi's golden rule, the rate of spontaneous emission due to transitions from an excited mini-band to a lower energy state of the ballistic electron. We shall denote by 12) (energy e2) the energy state of the injected electron, moving in the positive z-direction, e.g., ~/J2(z)= u~(z) exp {ik2z}, k2 > 0; and denote by I1) (energy eL) the final state of this electron after it emit spontaneously a photon of energy h(o = ~: -~ e~, wave vector/~ and polarization vector a. The Hamiltonian of interaction, tt~, between electrons and photons, is taken to be the linear term p - d (p is the particle momentum, and A is the vector potential of the field). The rate of spontaneous emission of a/L a-photon is therefore: ~r U) [7R 12

x

f d 3 r ug*(r) exp { - i l l " rla" VV2r)

x a(e2 - < - ho~).

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the form

N2 cos20 nl

x

i

/I erO~VR

t ~ 2 f dz q)*(z) G ~/)2(z) 0

1

X 15(E 2 - - E I - - h u ) ) ~

t~(k 2 -

k 1 ),

(5)

where k2 (k~) is the wave-vector of the electron in the initial (final) state, 0 is the angle between a and the z axis, ~b(z) is determined by equation (2) and the integration is carried only over the first unit cell. We note that the final state of the electron is determined by its initial state due to the conservation of momentum in equation (5), i.e., k2 = k,. Since the electron is "free" in the tranverse directions, momentum conservation immediately leads to energy conservation in these directions. Consequently, the transverse state of the electron remains unchanged and only the longitudinal energy is affected during its passage in the periodic potential. The emission rate of photons in the/~ direction, per solid angle dD is then

c 3N2[AI212c0s20 a l l

dW,p =

(6) with the vertical transition matrix A,2 =

j dz qS*(z) ~ q52(z).

(7)

0

Here k~ of the final electron state is equal to the initial value of k2 (i.e., k~ = k2) and the energy of the emitted photons is determined by this "vertical transition" rule, according to the equation h(o = E2(k) - E~ (k). Notice that the radiation is directed mostly perpendicular to the motion of the ballistic electrons (i.e., the x-y plane) as expressed by the cos 20 factor in equation (6). The total rate of spontaneous emission is obtained by integrating over the solid angles, i.e.,

(4)

Here e is the electronic charge, m the effective mass (m ~ 0.067 m e in GaAs where m,, is the electron mass), h the Planck's constant, gr is the dielectric constant of the host semiconductor, VR is the normalization volume of the radiation field, and co = cfl/x/-~ is the photon's circular frequency (c is the speed of light in vacuum). Substituting equation (1) into equation (4), exploiting the periodicity of the u's in the Bloch functions, and assuming that the radiation wave-lengths are much larger than the electron's wave-lengths (dipole approximation) we can cast equation (4) into

W,, =

W,,,~\m

/

N2I/A,2I 2.

(8) Here the subscript H stands for the hydrogen atom, hcoH ~ 13.6eV, aH ~ 0.53A and Wu = ~3COH "~ 10m/s, with :( the fine structure constant. We observe from equation (8) that the transition probability for one electron in the superlattice, is reduced, with respect to WH, by the length scale squared and by the frequency ratio. However, this is somewhat compensated by the effects of the mass ratio, and the dielectric

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S P O N T A N E O U S E M I S S I O N BY B A L L I S T I C E L E C T R O N S

constant. When the injected current density J is given, we can estimate the total number of photons emitted per unit time by dNph _ dt

J S L W~r. e V~

(9)

Here (JSL/eV~) is the average number of electrons in the active region, ~ is the average velocity of the injected electrons. To obtain an order of magnitude estimate for the number of photons, we consider the following parameters: m and er are taken for GaAs, l = 6nm, L = 60nm and NlA~ 2 is of order 1. If we further take the characteristic parameters of the hotelectron GaAs devices [1-3], V~ = 5 x 107cm/s, h ~ = 0.1 eV, S = 5 x 2 0 # m 2 = 10 6 cm 2, and J = 100A/cm 2 we find that the rate of infrared photons emission by one electron [equation (8)] is _~5 x 106/s and by the current [equation (9)] is 5 x 108/s. The frequency characteristics of the spontaneous emission is depicted in Fig. l(b). Equation (9) is plotted as a function of the energy of the injected electrons for a square potential with 0.1 eV depth, 3 mn width (i.e., l = 6 nm, ~ = 0.5) and S, L, V~, and J are the same as above. For comparison, we have also calculated the transition probability for spontaneous emission due to one potential well of the same size, namely when the injected ballistic electrons are initially free, and there is only one final bound state in the well. The main difference, besides a reduction of two orders of magnitude in intensity, is the frequency spectrum of the emitted radiation. The fact that the radiation is more enhanced in the superlattice is understood in terms of a larger effective interaction length, and as a result of resonance behaviour in the periodic structure due to the coupling among the wells so that the emission is proportional to N 2. The frequency difference is essentially a result of the "band structure" of the superlattice. In the potential well transitions may occur from all "free" states above the well (i.e., E~ > 0) into the bound state inside it. In the superlattice however, transitions may occur only between different allowed bands, therefore the initial energy of the electrons must be in an excited mini-band. As seen from Fig. l(b), the peak of the emission in the superlattice is shifted to higher energies of the electrons. This, in turn, should help to increase the rate of radiation in the ballistic-electron devices, to which we referred above, since the energy of the injected electrons is centered around 0.3 eV above the bottom of the conduction band of the host. Our calculations indicate that the ballistic electrons, in the present days mesostructure devices, are able to produce a sizable amount of spontaneous

Vol. 71, No. 12

radiation due to electronic transitions between the mini-bands in superlattices. These emitted infrared photons may be used, in conjunction of other diagnostic methods (see [3] for more details) to investigate the properties of the hot-electron in those carefully grown semiconductor devices. In particular, the spectrum of the emitted radiation, can be correlated with the distribution of the injected electrons, and much can be learned about the ballistic electrons in mesosystems, it should be emphasized that only if the electrons are truly ballistic so that their mean free path is longer than several periods of the superlattice, the proposed radiation mechanisms will be effective. Before concluding this report on spontaneous emission by ballistic electrons, we wish to offer few remarks concerning the feasibility of turning these systems into lasers operating in the IR. In order to estimate the gain of a radiation field which propagates in perpendicular to the motion of the hot electrons in the superlattice, we borrow from the theory of semiconductor lasers (see, e.g., Yariv [11]). The energetically injected electrons with their energy distribution function, f(E~) (normalized to one), are in the inversion population situation. If we take their "life time". T, (or dephasing time for that matter) to be either as the result of the nonradiative transitions (e.g., LO phonons inelastic scattering) or due to the finite time they spend in the active region, we can introduce a "homogeneous line shape" function, g(o9 - COl2) of spectral width A~o ~ l/T, and h~o~2 = E2 - E~. The grain can now be written in terms of ~p(E2) of equation (8) as z-

~;p(E2)f(E2)g(~o - (ol2), (10)

where )o is the wave-length of the radiation, 3 is a form factor of order L/)o (active region/wave-length), and the summation is carried over the initial wavevectors. To obtain amplification one should make larger than the absorption loss per unit length of the host semiconductor, which we take to be of the order 1 cm- t, in the case o f h ~ = 0.1 eV. We further assume T ~ 10 L~s(hA~o ~ 40 meV), and use AE ~ 60meV for the width of the "electron beam" (see Ref. [3]), and find a threshold current density, Ji~,, of the order of 104(A/cm 2). This figure for J,h is considered to be attainable for the ballistic electron devices. In conclusion, we find that a considerable number of infrared photons can be emitted by ballistic electrons that traverse a superlattice. The radiation is shown to be mainly in perpendicular directions to the motion of the electrons, whereas the wave-length is determined by the initial energy of the electrons and

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SPONTANEOUS EMISSION BY BALLISTIC ELECTRONS

the mini-band structure of the superlattice. Since the radiation can be emitted only if the electrons are ballistic, the spectrum can be used, in conjunction with other methods, to analyze the properties of the ballistic electrons. Finally, we indicate that electromagnetic gain is feasible in this type of devices. Acknowledgements - One of use (AR) wishes to acknowledge Dr P.M. Platzman for introducing him to the ballistic electrons devices, Dr M. Heiblum for his patience offering many advices, and to Dr E. Cohen and Dr Arza Ron for many stimulating discussions. This work was supported by the fund for Encouragement of Research at the Technion and by the Israeli Academy of Sciences and Humanities.

4. 5.

6. 7. 8.

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Transport in Hot Electron Transistors, to appear in Physics of Quantum Electron Devices (edited by F. Capasso) in Topics in Current Physics, Springer, Berlin (1987). L. Esaki & R. Tsu, IBM Jour. Res. Develop. 14, 61 (1970). S. Luryi, Hot Electron Injection and Resonant Tunneling Heterojunction Devices, in Heterojunction Band Discontinuity (Edited by F. Capasso & G. Margaritondo) North-Holland, Amsterdam (1987). L.C. West & S.J. Eglash, Appl. Phys. Lett. 46, 1156 (1985). B.F. Levine, R.J, Malik, J. Walker, K.K. Choi, C.G. Bathea, D.A. Kleinman & J.M. Vandenberg, Appl. Phys. Lett. 50, 273 (1987). R.F. Kazarinov & R.A. Suris, Soy. Phys. Semicond. 6, 120 (1972). F. Capssso, K. Mohamad & A Y . Cho, Appl. Phys. Lett. 48, 478 (1986). J.M.J. Madey, J. Appl. Phys. 42, 1906 (1971). A. Yariv, Optical Electronics. Holt-Saunders, New York (1985).