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Inter-subband plasmon-emission-based THz Lasers
Physica E 7 (2000) 63–68
www.elsevier.nl/locate/physe
Inter-subband plasmon-emission-based THz Lasers P. Bakshi ∗ , K. Kempa Department of Physics, Boston College, Chestnut Hill, MA 02467, USA
Abstract We have proposed stimulated generation of plasmons (plasma instabilities) as a novel way to generate THz radiation from low-dimensional semiconductor systems. The microcharge oscillations of such plasmons become the source of electromagnetic radiation in the THz range. This plasma-instability-based concept o ers distinct advantages as it relies on a collective phenomenon, which is less susceptible to disruption due to higher temperatures and various scattering e ects. Also, since the plasmons are created internally and form a (coherent) collective mode of the electron gas itself, there is no need for an external feedback mechanism. A self-consistent calculational scheme shows the feasibility of plasma instabilities in appropriately c 2000 Published by Elsevier Science B.V. All rights reserved. designed quantum well structures. Keywords: Terahertz emission; Plasma instabilities; Plasmon lasers; Inter-subband plasmons
1. Introduction There is an increasing interest in and need for compact, coherent and tunable sources of terahertz (THz) radiation, in view of their many possible applications. Neither conventional electronic devices nor lasers have been able to provide sources in this dicult, intermediate frequency range. The conventional electronic devices (transistors, IMPATT diodes, etc.) cannot reach this frequency range due to the so-called impedance limitation [1– 4]. While the quantum cascade structures have been extended to THz frequencies, only spontaneous emission has been obtained so far [5,6]. Stimulated generation of plasmons (plasma instabilities) has been proposed by ∗
Corresponding author. Fax: +1-617-552-8478. E-mail address: [email protected] (P. Bakshi)
us as a novel way to generate THz radiation from low-dimensional semiconductor systems [7]. 1 The microcharge oscillations of such plasmons can become the source of electromagnetic radiation in the THz range. This mechanism is a robust phenomenon, since the microcharge oscillations are intrinsically coherent. Analogs of this phenomenon are well known in gaseous plasmas [8–10], where plasma instabilities have been studied theoretically and observed experimentally in many situations and have led to device applications [11]. Even high temperatures and various scattering e ects cannot easily disrupt these coherent collective oscillations. Since the natural plasma oscillations of the carriers in typical low-dimensional semiconductor systems are in the THz range, we have 1 This paper provides a review of our earlier work and a perspective for future work.
c 2000 Published by Elsevier Science B.V. All rights reserved. 1386-9477/00/$ - see front matter PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 3 1 0 - 0
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systematically investigated the possibility of generating plasma instabilities in several such systems [12–21]. Nonequilibrium plasmas can spontaneously develop growing plasma oscillations (plasma instabilities) in many situations [8–10]. This happens because plasmons constitute a natural energy transfer channel in a plasma, and provide an easy way to relax the excess free energy of a nonequilibrium plasma. A suciently strong population inversion in the carrier distribution function is needed for this to happen. Such a population inversion can often be achieved by driving a constant current through the plasma. However, in solid-state plasmas the drift velocity required to achieve such a population inversion is prohibitively large, of the order of the Fermi velocity [12–17]. In contrast to the uniform systems [12–17] introducing a periodic density modulation [19] in a high mobility quantum wire leads to a dramatic reduction in the threshold drift velocity required to generate a plasma instability, making this system a suitable candidate for experimental veri cation of this phenomenon. Bounded plasmas o er distinct advantages [7] and can be employed as active media to generate strong plasma instabilities by selective extraction and injection of carriers [20,21]. Experimentally, radiative decays of plasmons from two-dimensional electron channels [22] and parabolic quantum wells [23] have been observed, emitting radiation in THz regime at low temperature. Emission from coherent (but decaying) plasmon oscillations even at room temperature, and in the presence of bulk doping, on a picosecond time scale has also been observed [24]. These results suggest that plasma oscillations can survive even at room temperature in doped semiconductors. If a plasma instability is generated along the ideas mentioned above, it would then provide the means to sustain coherent plasma oscillations and the ensuing radiation. This phenomenon can thus be used for the realization of practical semiconductor THz radiation sources. From our investigation of plasma instabilities in various systems, we nd that quantum well structures (QWS) operating under bias in a nonequilibrium steady state, with appropriate carrier injection and extraction [25,26], may be the best candidates for a realization of this idea. The simplest scenario for the generation of plasma instabilities requires [21]
a three-subband system, with the rst and third subband well populated and the second nearly empty (or vice-versa). The essential instability mechanism is the resonant interaction of two plasma modes, due to the up and down depolarization shifted intersubband plasmons, in such a structure [21]. We have developed a fully self-consistent computational scheme for designing such structures. The eigenstates are determined by the Schroedinger– Poisson scheme, the subband populations by rate balance equations, inter-subband transfer rates through a RPA self-energy calculation, and the injection– extraction rates by the transfer matrix for complex energies. The feasibility of plasma instability for the resulting nonequilibrium steady state is examined for each bias, by obtaining the full spectral response in a RPA formalism. The I –V curves, and the domains of instability (as a function of bias) are obtained for any structure. Experimental results on rst generation structures con rm our predictions of the dynamic conductance versus bias [25], and (spontaneous) emission frequencies [26]. Population inversion was insucient to generate a plasma instability in these bulk-doped structures. Preliminary experimental results on the newly designed, second-generation structures, which are remote-doped, show quantitative agreement with our calculation of the current versus bias. Calculations also show that sucient population inversion can be achieved in such structures to obtain plasma instability, and ensuing THz radiation. 2. Calculations 2.1. Non-equilibrium steady state We use the Hartree approximation in which one solves self-consistently the Schroedinger equation of the form ˜2 d 2 (1) − ∗ 2 + Vext + VHartree j (z) = j j (z) 2m d z with the Hartree potential given by 4e2 X d2 VHartree = − nj | j (z)|2 ; 2 dz j
(2)
nj being the electron density in the jth sub-band. This calculation provides the energy levels j and the cor-
P. Bakshi, K. Kempa / Physica E 7 (2000) 63–68
responding wavefunctions Vtot = Vext + VHartree .
j (z),
for the total potential
2.2. Scattering and transmission rates We have developed a full RPA formalism, and an ecient computational scheme for calculating the electron–electron scattering rates in quantum wells, which include both single particle (Auger) and collective (plasmon) processes [27,28]. The intersubband scattering rates, ee (j → j 0 ); can be obtained from the imaginary part of the corresponding self-energy (see Refs. [27,28]). These rates depend on the wave functions and energies of Section 2.1. We similarly obtain electron–phonon rates LO due to the dominant LO phonon emission, through essentially evaluating the imaginary part of the self-energy using the unscreened phonon interaction, and assuming the “bulk” (3D) LO phonon dispersion. The injection and extraction rates j and j are determined by the transfer matrix method [29]. The complex energy poles of the transmission coecient, Ej = Erj − i j =2, determine [30] the total widths j = j + j of the quasi-bound energy levels Erj . This condition is equivalent to M11 (Ej ) = 0 in terms of the transfer matrix M [23]. The ratio ( j = j ) is related to |M21 (Ej )|2 , and along with j determines the values of j and j for any given quantum well structure [29]. We determine the transfer matrix numerically from the shape of the total potential Vtot (z) of Section 2.1 2.3. Balance equations for the subband populations The subband populations are determined by the rate balance equations dni = i (NiL − ni ) − i (ni − NiR ) dt X [ ee (i → j) + LO (i → j)]ni − +
ee
LO
[ (j → i) + (j → i)]nj ;
NiR the populations of the regions to the left and right of the active region. These rate equations are based on integrating over the in-plane momentum transfers, and are adequate at low temperatures. More detailed equations in terms of the distribution functions become necessary at higher temperatures. In the steady state, dni =dt = 0. The and terms provide the net particle ow through each level, and the net current in the steady state is given by X X J =e i (NiL − ni ) = e i (ni − NiR ): (4) i
i
The ee and LO terms represent all the interlevel transitions through the scatterings. The matrix elements entering ee and LO coecients, as well as the rates and depend on the nonequilibrium steady-state energies and wave functions (Section 2.1), which in turn, through the Hartree potential, depend on the subband populations nj obtained from the balance equations. The self-consistent solution is obtained by iteration until the input nj in Eq. (2) and the solutions nj of Eq. (3) agree. 2.4. Response calculations We obtain the electromagnetic response using our RPA formalism for inhomogeneous systems [25,31,32]. The energies and wave functions from the nonequilibrium steady-state calculations (Section 2.1) are used to obtain the density response of the entire system. In general, one can write the induced density as Z p(z; q⊥ ; !) = d z 0 0 (z; z 0 ; q⊥ ; !)VT (z 0 ; q⊥ ; !);(5) where 0 (z; z 0 ; q⊥ ; !) is the single-electron susceptibility given by X X X f − f0 0 (z; z 0 ; q⊥ ; !) = 2 − 0 + ˜! 0 j k⊥
j
X
65
× j (z) (3)
j
where ni is the population (sheet density) of the ith level in the active region, obtained by summing the distribution function over the in-plane momenta. i are the injection and i the extraction rates, NiL and
j
j 0 (z) j (z
0
)
j 0 (z
0
);
(6)
where VT is the total dynamic potential for an external perturbation at frequency !, and includes the dynamical Hartree term. Also, f = fj; k⊥ ; = j + 2 =2m∗ ; 0 = j0 + (k⊥ + q⊥ )2 =2m∗ , where k⊥ is the k⊥ electron wave vector and q⊥ the plasmon wave vector in the perpendicular (x − y) plane.
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The density response becomes in nite at the plasma mode frequency. Our formalism [31,32], with proper analytical continuation [32–34], allows us to study all relevant roots of the response functions in the entire complex frequency plane. Characteristics of the unstable plasma modes are determined, with imaginary part of the frequency = Im ! (when positive) representing the growth rate. Our present calculations show that there is very little dispersion of ! with q⊥ , the in-plane wave vector of the plasmon, and that the growth rate diminishes rapidly with q⊥ . Thus the mode arising from “vertical transitions” (q⊥ = 0) will dominate, providing frequency coherence in spite of the in-plane electron energy continuum. 3. Comments 3.1. Plasma wave growth saturation If, for a given applied bias, the condition for a plasma instability (i.e. ¿ 0) is satis ed, the induced plasma wave density amplitude begins to grow exponentially in time, with time constant 1= . The growing energy in the induced charge oscillation can be viewed to be due to a growing plasmon population. The larger this population becomes, the greater is the rate of creation of additional plasmons; this is why the plasma instability phenomenon can be described as the stimulated generation of plasmons. As the instability develops, the induced plasma wave density grows suciently strong to a ect the population inversion, through the inter-level transitions which generate the plasmons. The reduced population inversion diminishes the instantaneous growth rate ; (Section 2.4), even as the mode amplitude keeps growing. Finally, this continuing wave– particle interaction makes → 0. In this new steady state, the plasma mode has grown to a saturation level, providing a signi cant plasmon population in the system. Determination of the properties of this new steady state will require enlargement of the self-consistency scheme to include the full rami cations of this large amplitude coherent charge oscillation for various collision rates, and for the previous nonequilibrium steady state. If this amplitude is large enough, quasi-energy sidebands appear at multiples
of the plasma wave frequency in this new dynamic nonequilibrium oscillating state. 3.2. Radiation emission characteristics The plasma charge oscillations in the bounded system are dipole active, and directly emit narrow width electromagnetic radiation at the same frequency. The power level of the electromagnetic radiation is related to the dipole strength, and thus to the plasma mode amplitude at saturation, determined through various loss mechanisms and the onset of nonlinear e ects (Section 3.1). Even when the conditions for a plasma instability are not met, the emissive plasma modes of the nonequilibrium plasma will couple to and decay into electromagnetic radiation at the frequencies of the plasma modes. The line widths, however, will be broader, re ecting the uncompensated loss mechanisms. The power levels will also be much lower, since there is no build-up of energy from the driving current. 3.3. Other experimental signatures Even though our goal is the direct observation of plasma instabilities through the ensuing emission of THz radiation (Section 3.2), we can also obtain indirect evidence for an instability, through the identi cation and detection of speci c signatures in the transport and emission characteristics. It is necessary to make these observations in small steps of the applied bias, since the predicted domains of plasma instability cover narrow bias ranges. For some biases, the plasma instability occurs, but being damped out due to larger (intrinsic) dissipative losses in the given structure. If we do not take into account these damping e ects, we would have ¿ 0. However, the e ective growth rate is only e = − , where represents all the damping e ects. So, in a regime where ¿ , plasma wave growth and saturation will occur, with strong radiation (Sections 3.1 and 3.2). On the other hand, if ¡ , the mode is effectively damped with e = – . (1) This e ect will show up as line narrowing for the range of biases where ¿ 0. Other neighboring lines which are due to real-emissive modes ( = 0), will show a larger broadening corresponding to . (2) We nd that the mode
P. Bakshi, K. Kempa / Physica E 7 (2000) 63–68
frequency does not vary much in the range of instability. This is a general feature of the plasma instability phenomenon arising from the merging of two modes in the range of biases where the instability occurs. Thus d!=dVb will show a dip in that range of biases (3). Since the plasmon generation during an instability opens a new channel for transport, we can expect an enhanced current in the range of biases where an instability occurs. This can be seen in the I –V characteristics. 3.4. Concluding remarks As shown in Ref. [26], radiation emission has been observed, in agreement with the theoretical calculation, and shows that a population inversion has been achieved in the previously designed structures. The next generation of structures, with no doping in the active region, should considerably reduce the broadening seen in the emission. The measured current in such structures is much larger, in agreement with the theoretical calculations. Our present self-consistent calculations show instabilities with upto 0.5 meV. Enhancing the growth rate to over 1 meV by varying the structure parameters should be sucient to generate observable plasma instabilities in clean samples. The plasma instability under consideration has shown itself to be both ubiquitous and robust. Thus we expect that our designed structures will lead to a “proof of principle” for the phenomenon of plasma instability, to be detected through an abrupt increase in the current as a function of the bias potential in the measured I –V curve, by line narrowing in emission, by reduction of d!=dVb in the range of biases where instability occurs and nally, for a suciently strong plasma instability, through the ensuing electromagnetic radiation. Acknowledgements This work has been supported by USARO grant no. DAAG55-97-1-0021. References [1] M. Tschernitz, J. Freyer, Electron. Lett. 31 (1995) 582. [2] M.E. Elta et al., IEEE Electron Dev. Lett. EDL-6 (1980) 694.
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[3] H. Eisele, G.C. Haddad, IEEE Microwave Guided Wave Lett. 5 (1995) 385. [4] H. Eisele, G.I. Haddad, Electron. Lett. 30 (1994) 1950. [5] M. Rochat, J. Faist, M. Beck, U. Oesterle, M. Ilegems, Appl. Phys. Lett. 73 (1998) 3724. [6] B. Xu, Q. Hu, M. Melloch, Appl. Phys. Lett. 71 (1997) 440. [7] P. Bakshi, K. Kempa, Superlattices Microstruct. 17 (1995) 363. [8] A.B. Mikhailovskii, in: Theory of Plasma Instabilities, Vol. 1, Consultant Bureau, New York, 1974. [9] N. Krall, A. Trivelpiece, in: Principles of Plasma Physics, McGraw-Hill, New York, 1973. [10] S. Ichimaru, Basic Principles of Plasma Physics – a Statistical Approach, Benjamin=Cummings, Reading, MA, 1973. [11] A. Gover, A. Yariv, in: S.F. Jacobs et al. (Eds.), Novel Sources of Coherent Radiation, Addison-Wesley, London, 1978, p. 197. [12] P. Bakshi, J. Cen, K. Kempa, J. Appl. Phys. 64 (1988) 2243. [13] J. Cen, K. Kempa, P. Bakshi, Phys. Rev. B 38 (1988) 10 051. [14] P. Bakshi, J. Cen, K. Kempa, Solid State Commun. 76 (1990) 835. [15] J. Cen, K. Kempa, P. Bakshi, Solid State Commun. 78 (1991) 433. [16] K. Kempa, P. Bakshi, J. Cen, H. Xie, Phys. Rev. B 43 (1991) 9273. [17] H. Xie, K. Kempa, P. Bakshi, J. Appl. Phys. 72 (1992) 4767. [18] K. Kempa, P. Bakshi, H. Xie, W.L. Schaich, Phys. Rev. B 47 (1993) 4532. [19] K. Kempa, P. Bakshi, H. Xie, Phys. Rev. B 48 (1993) 9158. [20] K. Kempa, P. Bakshi, E. Gornik, Phys. Rev. B 54 (1996) 8231. [21] P. Bakshi, K. Kempa, in: J. Clark, P. Panat (Eds.), Condensed Matter Theories, Vol. 12, Nova Science Publishers, New York, 1997, p. 399. [22] M. Vossebuerger, H.G. Roskos, F. Wolter, C. Waschke, H. Kurz, K. Hirakawa, I. Wilke, K. Yamanaka, J. Opt. Soc. Am. B 13 (1996) 1045. [23] K.D. Maranowski, A.C. Gossard, K. Unterrainer, E. Gornik, Appl. Phys. Lett. 69 (1996) 3522, and further experimental work in Vienna. [24] R. Kersting, K. Unterrainer, G. Strasser, H.F. Kau mann, E. Gornik, Phys. Rev. Lett. 79 (1997) 3038. [25] K. Kempa, P. Bakshi, C.G. Du, G. Feng, A. Scorupsky, G. Strasser, C. Rauch, K. Unterrainer, E. Gornik, J. Appl. Phys. 85 (1999) 3708. [26] P. Bakshi, K. Kempa, A. Scorupsky, C.G. Du, G. Feng, R. Zobl, G. Strasser, C. Rauch, Ch. Pacher, K. Unterrainer, E. Gornik, Appl. Phys. Lett. 75 (1999) 1685. [27] K. Kempa, P. Bakshi, J. Engelbrecht, Y. Zhou, Intersubband electron transitions due to electron–electron interactions in quantum well structures, submitted. [28] K. Kempa, P. Bakshi, in these Proceedings (ITQW ’99), Physica E 7 (2000). [29] C.G. Du, P. Bakshi, K. Kempa, Injection and extraction rates for quantum well structures through the transfer matrix at complex energies, to be published.
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[30] E. Merzbacher, Quantum Mechanics, 2nd Edition, Wiley, New York (Chapter 7) 1970. [31] K. Kempa, D.A. Broido, C. Beckwith, J. Cen, Phys. Rev. B 40 (1989) 8385. [32] K.D. Tsuei, E.W. Plummer, A. Libsch, E. Phelke, K. Kempa, P. Bakshi, Surf. Sci. 247 (1991) 302.
[33] P. Bakshi, K. Kempa, Phys. Rev. B 40 (1989) 3433. [34] K.D. Tsuei, E.W. Plummer, A. Liebsch, K. Kempa, P. Bakshi, Phys. Rev. Lett. 64 (1990) 44.
www.elsevier.nl/locate/physe
Inter-subband plasmon-emission-based THz Lasers P. Bakshi ∗ , K. Kempa Department of Physics, Boston College, Chestnut Hill, MA 02467, USA
Abstract We have proposed stimulated generation of plasmons (plasma instabilities) as a novel way to generate THz radiation from low-dimensional semiconductor systems. The microcharge oscillations of such plasmons become the source of electromagnetic radiation in the THz range. This plasma-instability-based concept o ers distinct advantages as it relies on a collective phenomenon, which is less susceptible to disruption due to higher temperatures and various scattering e ects. Also, since the plasmons are created internally and form a (coherent) collective mode of the electron gas itself, there is no need for an external feedback mechanism. A self-consistent calculational scheme shows the feasibility of plasma instabilities in appropriately c 2000 Published by Elsevier Science B.V. All rights reserved. designed quantum well structures. Keywords: Terahertz emission; Plasma instabilities; Plasmon lasers; Inter-subband plasmons
1. Introduction There is an increasing interest in and need for compact, coherent and tunable sources of terahertz (THz) radiation, in view of their many possible applications. Neither conventional electronic devices nor lasers have been able to provide sources in this dicult, intermediate frequency range. The conventional electronic devices (transistors, IMPATT diodes, etc.) cannot reach this frequency range due to the so-called impedance limitation [1– 4]. While the quantum cascade structures have been extended to THz frequencies, only spontaneous emission has been obtained so far [5,6]. Stimulated generation of plasmons (plasma instabilities) has been proposed by ∗
Corresponding author. Fax: +1-617-552-8478. E-mail address: [email protected] (P. Bakshi)
us as a novel way to generate THz radiation from low-dimensional semiconductor systems [7]. 1 The microcharge oscillations of such plasmons can become the source of electromagnetic radiation in the THz range. This mechanism is a robust phenomenon, since the microcharge oscillations are intrinsically coherent. Analogs of this phenomenon are well known in gaseous plasmas [8–10], where plasma instabilities have been studied theoretically and observed experimentally in many situations and have led to device applications [11]. Even high temperatures and various scattering e ects cannot easily disrupt these coherent collective oscillations. Since the natural plasma oscillations of the carriers in typical low-dimensional semiconductor systems are in the THz range, we have 1 This paper provides a review of our earlier work and a perspective for future work.
c 2000 Published by Elsevier Science B.V. All rights reserved. 1386-9477/00/$ - see front matter PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 3 1 0 - 0
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systematically investigated the possibility of generating plasma instabilities in several such systems [12–21]. Nonequilibrium plasmas can spontaneously develop growing plasma oscillations (plasma instabilities) in many situations [8–10]. This happens because plasmons constitute a natural energy transfer channel in a plasma, and provide an easy way to relax the excess free energy of a nonequilibrium plasma. A suciently strong population inversion in the carrier distribution function is needed for this to happen. Such a population inversion can often be achieved by driving a constant current through the plasma. However, in solid-state plasmas the drift velocity required to achieve such a population inversion is prohibitively large, of the order of the Fermi velocity [12–17]. In contrast to the uniform systems [12–17] introducing a periodic density modulation [19] in a high mobility quantum wire leads to a dramatic reduction in the threshold drift velocity required to generate a plasma instability, making this system a suitable candidate for experimental veri cation of this phenomenon. Bounded plasmas o er distinct advantages [7] and can be employed as active media to generate strong plasma instabilities by selective extraction and injection of carriers [20,21]. Experimentally, radiative decays of plasmons from two-dimensional electron channels [22] and parabolic quantum wells [23] have been observed, emitting radiation in THz regime at low temperature. Emission from coherent (but decaying) plasmon oscillations even at room temperature, and in the presence of bulk doping, on a picosecond time scale has also been observed [24]. These results suggest that plasma oscillations can survive even at room temperature in doped semiconductors. If a plasma instability is generated along the ideas mentioned above, it would then provide the means to sustain coherent plasma oscillations and the ensuing radiation. This phenomenon can thus be used for the realization of practical semiconductor THz radiation sources. From our investigation of plasma instabilities in various systems, we nd that quantum well structures (QWS) operating under bias in a nonequilibrium steady state, with appropriate carrier injection and extraction [25,26], may be the best candidates for a realization of this idea. The simplest scenario for the generation of plasma instabilities requires [21]
a three-subband system, with the rst and third subband well populated and the second nearly empty (or vice-versa). The essential instability mechanism is the resonant interaction of two plasma modes, due to the up and down depolarization shifted intersubband plasmons, in such a structure [21]. We have developed a fully self-consistent computational scheme for designing such structures. The eigenstates are determined by the Schroedinger– Poisson scheme, the subband populations by rate balance equations, inter-subband transfer rates through a RPA self-energy calculation, and the injection– extraction rates by the transfer matrix for complex energies. The feasibility of plasma instability for the resulting nonequilibrium steady state is examined for each bias, by obtaining the full spectral response in a RPA formalism. The I –V curves, and the domains of instability (as a function of bias) are obtained for any structure. Experimental results on rst generation structures con rm our predictions of the dynamic conductance versus bias [25], and (spontaneous) emission frequencies [26]. Population inversion was insucient to generate a plasma instability in these bulk-doped structures. Preliminary experimental results on the newly designed, second-generation structures, which are remote-doped, show quantitative agreement with our calculation of the current versus bias. Calculations also show that sucient population inversion can be achieved in such structures to obtain plasma instability, and ensuing THz radiation. 2. Calculations 2.1. Non-equilibrium steady state We use the Hartree approximation in which one solves self-consistently the Schroedinger equation of the form ˜2 d 2 (1) − ∗ 2 + Vext + VHartree j (z) = j j (z) 2m d z with the Hartree potential given by 4e2 X d2 VHartree = − nj | j (z)|2 ; 2 dz j
(2)
nj being the electron density in the jth sub-band. This calculation provides the energy levels j and the cor-
P. Bakshi, K. Kempa / Physica E 7 (2000) 63–68
responding wavefunctions Vtot = Vext + VHartree .
j (z),
for the total potential
2.2. Scattering and transmission rates We have developed a full RPA formalism, and an ecient computational scheme for calculating the electron–electron scattering rates in quantum wells, which include both single particle (Auger) and collective (plasmon) processes [27,28]. The intersubband scattering rates, ee (j → j 0 ); can be obtained from the imaginary part of the corresponding self-energy (see Refs. [27,28]). These rates depend on the wave functions and energies of Section 2.1. We similarly obtain electron–phonon rates LO due to the dominant LO phonon emission, through essentially evaluating the imaginary part of the self-energy using the unscreened phonon interaction, and assuming the “bulk” (3D) LO phonon dispersion. The injection and extraction rates j and j are determined by the transfer matrix method [29]. The complex energy poles of the transmission coecient, Ej = Erj − i j =2, determine [30] the total widths j = j + j of the quasi-bound energy levels Erj . This condition is equivalent to M11 (Ej ) = 0 in terms of the transfer matrix M [23]. The ratio ( j = j ) is related to |M21 (Ej )|2 , and along with j determines the values of j and j for any given quantum well structure [29]. We determine the transfer matrix numerically from the shape of the total potential Vtot (z) of Section 2.1 2.3. Balance equations for the subband populations The subband populations are determined by the rate balance equations dni = i (NiL − ni ) − i (ni − NiR ) dt X [ ee (i → j) + LO (i → j)]ni − +
ee
LO
[ (j → i) + (j → i)]nj ;
NiR the populations of the regions to the left and right of the active region. These rate equations are based on integrating over the in-plane momentum transfers, and are adequate at low temperatures. More detailed equations in terms of the distribution functions become necessary at higher temperatures. In the steady state, dni =dt = 0. The and terms provide the net particle ow through each level, and the net current in the steady state is given by X X J =e i (NiL − ni ) = e i (ni − NiR ): (4) i
i
The ee and LO terms represent all the interlevel transitions through the scatterings. The matrix elements entering ee and LO coecients, as well as the rates and depend on the nonequilibrium steady-state energies and wave functions (Section 2.1), which in turn, through the Hartree potential, depend on the subband populations nj obtained from the balance equations. The self-consistent solution is obtained by iteration until the input nj in Eq. (2) and the solutions nj of Eq. (3) agree. 2.4. Response calculations We obtain the electromagnetic response using our RPA formalism for inhomogeneous systems [25,31,32]. The energies and wave functions from the nonequilibrium steady-state calculations (Section 2.1) are used to obtain the density response of the entire system. In general, one can write the induced density as Z p(z; q⊥ ; !) = d z 0 0 (z; z 0 ; q⊥ ; !)VT (z 0 ; q⊥ ; !);(5) where 0 (z; z 0 ; q⊥ ; !) is the single-electron susceptibility given by X X X f − f0 0 (z; z 0 ; q⊥ ; !) = 2 − 0 + ˜! 0 j k⊥
j
X
65
× j (z) (3)
j
where ni is the population (sheet density) of the ith level in the active region, obtained by summing the distribution function over the in-plane momenta. i are the injection and i the extraction rates, NiL and
j
j 0 (z) j (z
0
)
j 0 (z
0
);
(6)
where VT is the total dynamic potential for an external perturbation at frequency !, and includes the dynamical Hartree term. Also, f = fj; k⊥ ; = j + 2 =2m∗ ; 0 = j0 + (k⊥ + q⊥ )2 =2m∗ , where k⊥ is the k⊥ electron wave vector and q⊥ the plasmon wave vector in the perpendicular (x − y) plane.
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The density response becomes in nite at the plasma mode frequency. Our formalism [31,32], with proper analytical continuation [32–34], allows us to study all relevant roots of the response functions in the entire complex frequency plane. Characteristics of the unstable plasma modes are determined, with imaginary part of the frequency = Im ! (when positive) representing the growth rate. Our present calculations show that there is very little dispersion of ! with q⊥ , the in-plane wave vector of the plasmon, and that the growth rate diminishes rapidly with q⊥ . Thus the mode arising from “vertical transitions” (q⊥ = 0) will dominate, providing frequency coherence in spite of the in-plane electron energy continuum. 3. Comments 3.1. Plasma wave growth saturation If, for a given applied bias, the condition for a plasma instability (i.e. ¿ 0) is satis ed, the induced plasma wave density amplitude begins to grow exponentially in time, with time constant 1= . The growing energy in the induced charge oscillation can be viewed to be due to a growing plasmon population. The larger this population becomes, the greater is the rate of creation of additional plasmons; this is why the plasma instability phenomenon can be described as the stimulated generation of plasmons. As the instability develops, the induced plasma wave density grows suciently strong to a ect the population inversion, through the inter-level transitions which generate the plasmons. The reduced population inversion diminishes the instantaneous growth rate ; (Section 2.4), even as the mode amplitude keeps growing. Finally, this continuing wave– particle interaction makes → 0. In this new steady state, the plasma mode has grown to a saturation level, providing a signi cant plasmon population in the system. Determination of the properties of this new steady state will require enlargement of the self-consistency scheme to include the full rami cations of this large amplitude coherent charge oscillation for various collision rates, and for the previous nonequilibrium steady state. If this amplitude is large enough, quasi-energy sidebands appear at multiples
of the plasma wave frequency in this new dynamic nonequilibrium oscillating state. 3.2. Radiation emission characteristics The plasma charge oscillations in the bounded system are dipole active, and directly emit narrow width electromagnetic radiation at the same frequency. The power level of the electromagnetic radiation is related to the dipole strength, and thus to the plasma mode amplitude at saturation, determined through various loss mechanisms and the onset of nonlinear e ects (Section 3.1). Even when the conditions for a plasma instability are not met, the emissive plasma modes of the nonequilibrium plasma will couple to and decay into electromagnetic radiation at the frequencies of the plasma modes. The line widths, however, will be broader, re ecting the uncompensated loss mechanisms. The power levels will also be much lower, since there is no build-up of energy from the driving current. 3.3. Other experimental signatures Even though our goal is the direct observation of plasma instabilities through the ensuing emission of THz radiation (Section 3.2), we can also obtain indirect evidence for an instability, through the identi cation and detection of speci c signatures in the transport and emission characteristics. It is necessary to make these observations in small steps of the applied bias, since the predicted domains of plasma instability cover narrow bias ranges. For some biases, the plasma instability occurs, but being damped out due to larger (intrinsic) dissipative losses in the given structure. If we do not take into account these damping e ects, we would have ¿ 0. However, the e ective growth rate is only e = − , where represents all the damping e ects. So, in a regime where ¿ , plasma wave growth and saturation will occur, with strong radiation (Sections 3.1 and 3.2). On the other hand, if ¡ , the mode is effectively damped with e = – . (1) This e ect will show up as line narrowing for the range of biases where ¿ 0. Other neighboring lines which are due to real-emissive modes ( = 0), will show a larger broadening corresponding to . (2) We nd that the mode
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frequency does not vary much in the range of instability. This is a general feature of the plasma instability phenomenon arising from the merging of two modes in the range of biases where the instability occurs. Thus d!=dVb will show a dip in that range of biases (3). Since the plasmon generation during an instability opens a new channel for transport, we can expect an enhanced current in the range of biases where an instability occurs. This can be seen in the I –V characteristics. 3.4. Concluding remarks As shown in Ref. [26], radiation emission has been observed, in agreement with the theoretical calculation, and shows that a population inversion has been achieved in the previously designed structures. The next generation of structures, with no doping in the active region, should considerably reduce the broadening seen in the emission. The measured current in such structures is much larger, in agreement with the theoretical calculations. Our present self-consistent calculations show instabilities with upto 0.5 meV. Enhancing the growth rate to over 1 meV by varying the structure parameters should be sucient to generate observable plasma instabilities in clean samples. The plasma instability under consideration has shown itself to be both ubiquitous and robust. Thus we expect that our designed structures will lead to a “proof of principle” for the phenomenon of plasma instability, to be detected through an abrupt increase in the current as a function of the bias potential in the measured I –V curve, by line narrowing in emission, by reduction of d!=dVb in the range of biases where instability occurs and nally, for a suciently strong plasma instability, through the ensuing electromagnetic radiation. Acknowledgements This work has been supported by USARO grant no. DAAG55-97-1-0021. References [1] M. Tschernitz, J. Freyer, Electron. Lett. 31 (1995) 582. [2] M.E. Elta et al., IEEE Electron Dev. Lett. EDL-6 (1980) 694.
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[3] H. Eisele, G.C. Haddad, IEEE Microwave Guided Wave Lett. 5 (1995) 385. [4] H. Eisele, G.I. Haddad, Electron. Lett. 30 (1994) 1950. [5] M. Rochat, J. Faist, M. Beck, U. Oesterle, M. Ilegems, Appl. Phys. Lett. 73 (1998) 3724. [6] B. Xu, Q. Hu, M. Melloch, Appl. Phys. Lett. 71 (1997) 440. [7] P. Bakshi, K. Kempa, Superlattices Microstruct. 17 (1995) 363. [8] A.B. Mikhailovskii, in: Theory of Plasma Instabilities, Vol. 1, Consultant Bureau, New York, 1974. [9] N. Krall, A. Trivelpiece, in: Principles of Plasma Physics, McGraw-Hill, New York, 1973. [10] S. Ichimaru, Basic Principles of Plasma Physics – a Statistical Approach, Benjamin=Cummings, Reading, MA, 1973. [11] A. Gover, A. Yariv, in: S.F. Jacobs et al. (Eds.), Novel Sources of Coherent Radiation, Addison-Wesley, London, 1978, p. 197. [12] P. Bakshi, J. Cen, K. Kempa, J. Appl. Phys. 64 (1988) 2243. [13] J. Cen, K. Kempa, P. Bakshi, Phys. Rev. B 38 (1988) 10 051. [14] P. Bakshi, J. Cen, K. Kempa, Solid State Commun. 76 (1990) 835. [15] J. Cen, K. Kempa, P. Bakshi, Solid State Commun. 78 (1991) 433. [16] K. Kempa, P. Bakshi, J. Cen, H. Xie, Phys. Rev. B 43 (1991) 9273. [17] H. Xie, K. Kempa, P. Bakshi, J. Appl. Phys. 72 (1992) 4767. [18] K. Kempa, P. Bakshi, H. Xie, W.L. Schaich, Phys. Rev. B 47 (1993) 4532. [19] K. Kempa, P. Bakshi, H. Xie, Phys. Rev. B 48 (1993) 9158. [20] K. Kempa, P. Bakshi, E. Gornik, Phys. Rev. B 54 (1996) 8231. [21] P. Bakshi, K. Kempa, in: J. Clark, P. Panat (Eds.), Condensed Matter Theories, Vol. 12, Nova Science Publishers, New York, 1997, p. 399. [22] M. Vossebuerger, H.G. Roskos, F. Wolter, C. Waschke, H. Kurz, K. Hirakawa, I. Wilke, K. Yamanaka, J. Opt. Soc. Am. B 13 (1996) 1045. [23] K.D. Maranowski, A.C. Gossard, K. Unterrainer, E. Gornik, Appl. Phys. Lett. 69 (1996) 3522, and further experimental work in Vienna. [24] R. Kersting, K. Unterrainer, G. Strasser, H.F. Kau mann, E. Gornik, Phys. Rev. Lett. 79 (1997) 3038. [25] K. Kempa, P. Bakshi, C.G. Du, G. Feng, A. Scorupsky, G. Strasser, C. Rauch, K. Unterrainer, E. Gornik, J. Appl. Phys. 85 (1999) 3708. [26] P. Bakshi, K. Kempa, A. Scorupsky, C.G. Du, G. Feng, R. Zobl, G. Strasser, C. Rauch, Ch. Pacher, K. Unterrainer, E. Gornik, Appl. Phys. Lett. 75 (1999) 1685. [27] K. Kempa, P. Bakshi, J. Engelbrecht, Y. Zhou, Intersubband electron transitions due to electron–electron interactions in quantum well structures, submitted. [28] K. Kempa, P. Bakshi, in these Proceedings (ITQW ’99), Physica E 7 (2000). [29] C.G. Du, P. Bakshi, K. Kempa, Injection and extraction rates for quantum well structures through the transfer matrix at complex energies, to be published.
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[30] E. Merzbacher, Quantum Mechanics, 2nd Edition, Wiley, New York (Chapter 7) 1970. [31] K. Kempa, D.A. Broido, C. Beckwith, J. Cen, Phys. Rev. B 40 (1989) 8385. [32] K.D. Tsuei, E.W. Plummer, A. Libsch, E. Phelke, K. Kempa, P. Bakshi, Surf. Sci. 247 (1991) 302.
[33] P. Bakshi, K. Kempa, Phys. Rev. B 40 (1989) 3433. [34] K.D. Tsuei, E.W. Plummer, A. Liebsch, K. Kempa, P. Bakshi, Phys. Rev. Lett. 64 (1990) 44.