Spontaneous photonemission spectrum of tunneling electrons in a doublebarrierstructureGuang Bai, Kai Shum, and R. R. Alfano Citation: Journal of Applied Physics 70, 1025 (1991); doi: 10.1063/1.349684 View online: http://dx.doi.org/10.1063/1.349684 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/70/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Tunneling assisted thermionic emission in doublebarrier quantum well structures J. Appl. Phys. 77, 2537 (1995); 10.1063/1.358783 Electron tunneling lifetime of a quasibound state in a doublebarrier resonant tunneling structure J. Appl. Phys. 76, 606 (1994); 10.1063/1.357054 Photonassisted tunneling in doublebarrier superconducting tunnel junctions Appl. Phys. Lett. 64, 921 (1994); 10.1063/1.110996 Resonant tunneling of electrons in Si/Ge strainedlayer doublebarrier tunneling structures Appl. Phys. Lett. 61, 1405 (1992); 10.1063/1.107552 Transport in doublebarrier resonant tunneling structures J. Appl. Phys. 61, 2693 (1987); 10.1063/1.337909

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COMMUNlCATlQNS

Spontaneous photon-emission spectrum of tunneling electrons in a double-barrier structure

Guang Bai, Kai Shum, and R. R. Alfano Institute for Ultrafast Spectroscopy and Lasers, Electrical Engineering Department and Physics Department, The City College of the City University of New York, New York, New York 10031

(Received 4 January 1991; accepted for publication 14 April 1991)

The spontaneous photon-emission spectrum of an electron tunneling through a semiconductor double-barrier structure is calculated using the Green’s-function approach. The peak photon-emission power of lo- ’ W per well can be achieved for an injected current of 50 mA. Our results are consistent with the measured value of lo- 9 W per well by Helm et al. [Phys. Rev. Lett. 63, 74 (1989)] when the nonradiative process and other factors are taken into account.

Infrared (IR) photodetectors and lasers are of most importance for many photonic applications in the optical interconnection, communication, and computation areas. Intersubband transitions in a superlattice or in a multiple- quantum-well structure can be used for a variety of IR detectors and lasers. There are many advantages to using intersubband transitions because the transition energy can be tuned by changing the structure parameters such as the well and barrier width and the barrier height. The theoret- ical study in this field was initiated over 20 years ago, soon after Esaki and Tsu’s original idea’ of designing a semicon- ductor superlattice. Kazarinov and Suris2 addressed the possibility of generation and amplification of infrared light in a superlattice using sequential electron tunneling be- tween adjacent wells. Recently, experimental evidence for optical intersubband absorption3 and sequential electron tunneling’ in a quantum-well structure were published. These observations have stimulated further studies in this important area. Various novel semiconductor quantum- well or superlattice structures were proposed to develop an IR laser. Yuh and Wang’ suggested using a band-aligned superlattice structure in which the population inversion can be achieved by a current injection. A multiple-quan- tum-well structure consisting of alternating narrow and wide wells was studied by Liu.6 Spontaneous emission by electrons transversing ballistically in a superlattice was in- vestigated by Botton and Ron.’ Most recently, Helm et a(.* made a significant contribution toward the realization of an intersubband IR laser where the first time IR-light emis- sions from intersubband transitions in a semiconductor multiple-quantum-well structure were observed. The IR- emission power was detected on an order of 10 - lo W per well9 However, none of the above-mentioned theoretical results can be compared with this important measured value of power.

In this communication, we present a theoretical calcu- lation of the spontaneous photon-emission spectrum of a tunneling electron across a double-barrier semiconductor structure using a Green’s-function approach. Our calcula-

tion method differs from the previous methods of calculat- ing the emission spectrum.5-7 First, Fermi’s Golden Rule was used in the previous calculations where the electron wave functions of initial and final states must be properly normalized. Furthermore, it is usually very involved be- cause present structure is an open system (from - CO to + CO ). The normalization mathematically isolates the re-

gion where electrons make optical transitions from an ex- ternal current injection source. In our calculation, elec- trons are allowed to propagate through the external current injection source. Second, the Greens-function ap- proach permits electrons to make optical transitions be- tween any continuous and quasibound states; therefore, a complete emission spectrum can be obtained.

The double-barrier structure used in our calculation is shown in Fig. 1. This structure can be viewed as a basic

- I + II-L III - 7--

Vb _-

I-

--___-___-

4aleb----dcb- FIG. 1. GaAs-GaAlAs double-barrier structure. The parameters used in our calculation are a = 3O.b;, b = 110 A, V, = 250 meV, V, = 100 meV, mods = O.O67m,, and mAlOaAa = O.O92m,; where m. is the free-electron mass. The quasibound state is at E;= - 67.3 meV, and the first resonant tunneling state is at E0 = 27.5 meV.

1025 J. Appl. Phys. 70 (2), 15 July 1991 0021-8979/91/021025-03$03.00 @ 1991 American Institute of Physics 1025 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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unit in a multiple-quantum-well structure under electrical bias. We deal with the following physical process of an electron in the structure: An electron is injected from left lead (region I) and enters the double-barrier region (re- gion II) where it interacts with ground photon states, re- sulting in spontaneous IR-photon emissions. The electron then arrives at the right lead (region III) and finally goes back to the current injection source. The electron-photon interaction under dipole approximation is given by

The vector potential operator A has a form of

A= T &[~(k)e~(~‘r-Y*‘)

+at(,qe-‘W--okf)]~, (2)

where 5? is the unit vector of polarization which is parallel to the electron-tunneling direction, Y is the volume of re- gion II, wk is the angular frequency of a photon with wave vector k, and a(k) and CZ? (k) are annihilation and creation operators for k-mode photons, respectively.

From Green’s-function theory, the wave function can be written as

s t Y (z,t) =YO(z,t> + 4 --m

X I

dzl Go+ (z,t,zl,td V&dY(z,t), (3)

where G$ and Y’(z,t) are the retarded Green’s function and the wave function without electron-photon interaction. Because kin Eq. (2) is very small, etir - 1, Y’(z,t)can be written as Y’(z,t) = Y”(z)e-‘EifI...nk...>, where

“‘nk *mm) denotes the photon state. By iteration, the first- and second-order corrections of the wave functions \I,’ and Y2 can be obtained.

With Y’ and Y*, we get a correction to the current. Since we are considering the spontaneous emission at very low temperature ( -0 K), the number of photons in the initial state is zero. From the relations a IO) = O,at 10) = 1 l), to first order of V,,, the correction to the current, Y” + Y ‘, is zero. While the second-order correction is given by

A=Y’+Y1+(Yo+Y~*fcc) - - ,

in which

(4)

\yl+\yl

dY” * dzl Gz (WI,Ei - h@/c)x 3 (5)

1

where Ei is energy of an incident electron, hmk is energy of an emitting photon, and G$ (z,zl,Ei - hwk) is Fourier transformation of the Green’s function Go+ (z,t,zl,tl). GO+ (z,zl,E) can be written aslo

FIG. 2. Spontaneous photon-emission spectra for the structure in Fig. 1. The material parameters are the same as those listed in Fig. 1. The emit- ted photon power densities are plotted at different incident electron en- ergies. The injection current is 50 mA.

G$ (z,zl,E) = lim c @f(ZI)@k&)

E- Ek + ie . (6) E-O+ Sa

a&(z) is a complete orthogonal set of eigenfunction of elastic Hamiltonian, k = ,I-, and cx denotes the degeneracy at each eigenvalue Ek. If an electron is injected with energy Ei into region I, after emitting a photon of energy hak in region II, the probability of the electron with energy (Ei - hwk) iS

D(hok) =;(Y’ + Y”)k i

2a-h3e2 =VjnZwkd@k)

k

si

dY” * xk7, dzl Gz (z,zl,Ei- hWk)z 9

1

(7)

where ki = Jw, and p(wk) is the photon density at frequency wk. The cross term (Y”+ Y* + c.c.) in Rq. (4) represents the feedback mechanism to the incident channel Ei; it affects the transmission probability at channel Ep But it does not contribute photon emission for E < Ep

From the above analysis, we can attribute D(h@k) to the probability of emitting IR photons with energy hwk per unit energy when electrons travel across the double-barrier structure. If the electric current I through the double-bar- rier structure is given, using D( hwk), we are able to obtain the photon-emission power density P(hw,) i.e., the num- ber of photons emitted per unit time per unit energy):

P(hq) =$hOk)hWk. (8)

1026 J. Appl. Phys., Vol. 70, No. 2, 15 July 1991 Bai, Shum, and Alfano 1026

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,

As a specific structure, we consider the GaAlAs-GaAs- GaAlAs double-barrier structure in Fig. 1 with the follow- ing parameters: a = 30 A, b = 110 A, and c = 30 A; the height of the barrier is Ve = 350 meV and V, = 100 meV. There is one quasibound state at Eg = - 67.3 meV for this structure. The first resonant tunneling channel is at E. = 27.5 meV. With an injection current I = 50 mA, the calculated spontaneous photon-emission power densities P(hw,) at different incident electron energies Ei are plotted in Fig. 2. Sharp emission peaks appear at the photon en- ergy of hmk = EI - Eq, which correspond to the electron transitions from different incident states to the quasibound state. The radiative transition rates associated with the quasibound state as the final state are about four to five orders of magnitude larger than with all other continuous states. For incident electron energies above the first reso- nant tunneling channel (El > E,), there is another peak with much smaller amplitude on curves of Ei > Ep This second peak is associated with the transition from incident channel to first resonant channel at Ep It can be seen from Fig. 2 that the transition from resonant channel E. to the quasibound state Eq has the maximum emission power density, which shows the strong coupling enhancement be- tween E,, and Eq states. For this transition, the width of the emission peak is about 0.05 meV, from which it is inferred that the lifetime of the quasibound state is about 7 ps.

The peak power Wp for IR photons from energy Ep - AE to Ep + AE can be obtained by

wp= s

E,+AE P( E)dE. (9)

Ep-AE

For the above parameters, our result of W, gives a maxi- mum value of about 5 x 10 - ’ W. In order to compare with the experimental value of 10 - 9-10 - lo W per well mea- sured by Helm et a1.,8 the photon energy has to be scaled to the experiment condition (20 meV). This scaling is al- lowed since we have neglected LO-phonon-emission pro- cesses. After scaling, the calculated value for Wp is still two or three orders larger than the experimental value. This difference is not surprising because our model used in the calculation is much simpler than the real experiment. First, we assume that the quasibound state is empty. Occupation of the quasibound state may reduce the transition events. Second, our calculated transition is directly from incident channel to a quasibound state, while in Ref. 8 electrons first relax to a quasibound state below the incident channel;

then, the transition takes place between this quasibound state and another lower quasibound state. Third, only the radiative interaction was considered in our calculation. In a real structure the nonradiative relaxation process cannot be separated from the radiative process. In spite of the simplicity of our model, the calculated value for W, is consistent in order of magnitude with the experiment re- sults when the radiative relaxation time is taken to be on the order of microseconds7’8 and the nonradiative relax- ation time is on the order of nanoseconds.11P’2

Finally, it is worthwhile to compare our results with the other theoretical estimations in Ref. 7. The rate of infrared photon emission for an injected current of 0.1 mA was calculated to be 5X 104/s per well between two mini- bands. When their calculated value is scaled to the exper- imental conditions (1-50 mA, hw = 20 meV) of Ref. 8, a value of about lo-i2 W per well is obtained, which is considerably smaller than the experimental value.8 More- over, if nonradiative processes are included, the discrep- ancy is even larger.

In summary, a Green’s-function approach was used to calculate the spontaneous IR-photon-emission spectrum of tunneling electrons in a double-barrier structure. Our cal- culated results are consistent with the existing experimen- tal data.

We would like to thank Dr. W. Cai and Dr. Il. Wang for their valuable advice. This research is supported by Army Research Office under Grant No. DAAL-89-G- 0110.

‘L. Esaki and R. Tsu, IBM J. Res. Dev. 14, 61 (1970). *R. F. Kazarinov and R. A. Suris, Sov. Phys. Semicond. 5, 707 (1971);

6, 120 (1972). 3L. C. West and S. J. Eglash, Appl. Phys. Lett. 46, 1156 (1985). 4F. Capasso, K. Mohammed, and A. Y. Cho, Appl. Phys. Lett. 48, 478

(1986); IEEE J. Quantum Electron. 22, 1853 (1986). ‘P. Yuh and K. L. Wang, Appl. Phys. Lett. 51, 1404 (1987). 6H. C. Liu, J. Appl. Phys. 63, 2865 (1988). ‘M. Botton and A. Ron, Solid State Commun. 71, 1131 (1989). ‘M. Helm, P. England, E Colas, F. DeRosa, and S. J. Allen, Jr., Phys.

Rev. Lett. 63, 74 (1989). 9S. J. Allen, Jr. (private communication).

“W. Cai, T. P. Zheng, P. Hu, M. Lax, K. Shum, and R. R Alfano, Phys. Rev. Lett. 65, 104 (1990).

‘ID. Y. Oberli, E. R. Wake, M. V. Klein, J. Klem, T. Henderson, and H. Morkog Phys. Rev. Lett. 59, 696 (1987).

‘*Although the photon energy in our calculation is greater than longitu- dinal-optical photon, we have assumed that the nonradiative relaxation rate is dominated by acoustic phonons.

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