Semiconducto Physics, Quantum Electonics & Optoelectonics, 8. V., N. P. 43-49. PACS 7.38.-k High-fequency popeties of systems with difting electons and pola optical phonons S.M. Kukhtauk V. Lashkayov Institute of Semiconducto Physics, 4, pospect Nauk 38 Kyiv, Ukaine E-mail: kukhtauk@isp.kiev.ua Abstact. An analysis of inteaction between difting electons and optical phonons in semiconductos is pesented. Thee physical systems ae studied: thee-dimensional electon gas (3DEG) in bulk mateial; two-dimensional electon gas (DEG) in a quantum well, and two-dimensional electon gas in a quantum well unde a metal electode. The Eule and Poisson equations ae used fo studying the electon subsystem. Inteaction between electons and pola optical phonons ae taken into consideation using a fequency dependence of the dielectic pemittivity. As a esult, the dispesion equations that descibe self-consistent collective oscillations of plasmons and optical phonons ae deduced. We found that inteaction between electons and optical phonons leads to instability of the electon subsystem. The consideed physical systems ae capable to be used as a geneato o amplifie of the electomagnetic adiation in the THz fequency ange. The effect of instability is suppessed if damping of optical phonons and plasma oscillations is essentially stong. Keywods: difting electons, pola optical phonons, dispesion equation, instability. Manuscipt eceived 5.6.7; accepted fo publication 7..8; published online 3.3.8.. Intoduction High-fequency popeties of electon gas induced by plasma and optical phonon oscillations in diffeent semiconducto stuctues wee studied in theoetical [- 9] and expeimental [-] woks. When electons ae acceleated by an electic field in such a manne that thei dift vecity exceeds the sound vecity in semiconducto, a lage numbe of acoustic phonons can be emitted coheently. This socalled Cheenkov acoustoelectical effect was pedicted and demonstated in the 96s in semiconductos [,,, ]. A simila effect fo optical phonons was also pedicted in bulk mateials and expeimentally poven in [] and []. Typical fequencies of optical oscillations of the cystal lattice fo pola semiconductos ae of the ode of THz. Thus, that makes such systems inteesting fo high-fequency applications.. Theoy Fo ou puposes, it is sufficient to analyze the highfequency popeties of systems with difting electons and pola optical phonons using the simple hydodynamical model. Let n ( x,, ν ( x, and ϕ ( x, be the volume concentation, vecity of electons and electostatic potential, espectively. Then, we can wite Eule and continuity equations [3] as folws: ν ν ν e + ( ν ) ν + = ϕ, () t τ m n + div ( nν) =, () t whee e, m ae the chage and effective mass of the electon, espectively; ν denotes the stationay dift ν ν vecity of electons. The tem descibes τ scatteing of the electons by cystal defects, τ denotes the elaxation time. The Poisson equation [4] can be witten as 4πe ϕ = ( n n3d ), (3) 7, V. Lashkayov Institute of Semiconducto Physics, National Academy of Sciences of Ukaine 43
Semiconducto Physics, Quantum Electonics & Optoelectonics, 8. V., N. P. 43-49. whee and n 3 D is the dielectic pemittivity and stationay bulk density of electons. Pesence of optical phonons in pola semiconductos leads to dispesion of the dielectic pemittivity (). The dielectic pemittivity as a function of fequency can be found using the folwing method. A elation between dielectic pemittivity and polaizability α can be witten as: ( + πα) = 4. (4) Thee ae two contibutions in the polaizability α : the atomic polaizability as well as the polaizability bound with the dipole momentum, aising due to lattice + distotion [3]. Let u and u be displacements of two oppositely chaged ion sublattices. The espective dipole momentum is: = ( + ew e u u ). (5) With the model of hamonic oscillato, the equation fo the vecto W is e W + γ gw + tow = E, (6) m whee γ g is esponsible fo damping of the optical vibations, to denotes fequency of tansvese optical vibations of the lattice. Using (4), (5), and (6), we get iγ g ( ) =, (7) iγ whee to g = to is the fequency of ngitudinal optical vibations of the cystal lattice; and denote the optical and static dielectic constants, espectively. Note that dispesion of optical phonons is ignoed in this pape. As, we obtain Liddane-Sachs-Telle elation [3]: = to. (8) The inteval of fequencies = [ ] to, is called in [3] as a esidual adiation zone (Reststahlung). Using (), (), (3), and taking into account (7), we eceive a system of the patial diffeential equations. We study the efeed system of the patial diffeential equations using the methods of the theoy of instability [6, 8]. Namel we conside an unlimited homogeneous dielectic medium and an electon gas chaacteized by the cetain set of magnitudes U (in this aticle, this is the equilibium concentation and dift vecity). At a cetain moment U (, = U + U (,, whee the magnitudes with pimes chaacteize deviations fom the espective steady-state values. Assuming sinusoidal vaiations fo all petubed ik it quantities U e, whee k is the wave vecto, is the adius-vecto; and t denote the fequency and time espectively. Thus, we suppose that the petubation is the wave packet with a limit size, and plane waves ae its sepaate Fouie components. Due popagation, the package speads, and its amplitude (in unstable system with Im ( ) > ) gows up. At the same time, as it is inheent each wave packet, it will move in space [8]. The main poblem of the theoy of instability is to study expation of the package behavio in some fixed egion. Accoding to the theoy of instability [6], it is necessay and sufficient be awae of connection between the fequency and wave vecto to chaacteize behavio of the wave packet. Thus, all poblem is educed to detemination and investigation of the dispesion elations. If the dispesion equation suppose some complex solutions, then this physical system is capable to amplify oscillations [6]. That is the popeties that make such systems inteesting fo high-fequency applications. In the next section, we shall study inteaction between difting electons and optical phonons in thee diffeent physical systems. The main diffeences among them will be given in the next sections. Using methods of the theoy of instabilit we shall pove that these physical systems ae capable to geneate electomagnetic adiation in the THz fequency ange. 3. An analysis of the dispesion equations In this section, we have epesented the esults without taking into account an electon scatteing on the cystal defects (i.e. τ ). An analysis of scatteing contibution to the incement of instability (the imaginay pat of fequency) will be given in Section 4. 3.. Inteactions between thee-dimensional electon gas and optical phonons Let us assume that 3DEG is in bulk pola semiconducto, and has the unpetubed caie density n 3 D and dift vecity ν. Thee is also the electostatic potential ϕ ( x, defined eveywhee inside the stuctue. Afte substitution of petubations in the system of patial diffeential equations, the poblem is educed to solution of a homogeneous system of algebaic equations. Setting the deteminant of these equations to zeo yields the dispesion elation ( Ω ) γ Ω iγω K = Ω pl, (9) Ω iγω 7, V. Lashkayov Institute of Semiconducto Physics, National Academy of Sciences of Ukaine 44
Semiconducto Physics, Quantum Electonics & Optoelectonics, 8. V., N. P. 43-49. whee it is designated: pl Ω pl =, 4πe pl = n m Ω =, 3D kν K =, to γ =,. The value pl is called as the plasma fequency. The equation (9) is quadatic to wave vecto K, so it is simple to find two adicals: K, = Ω ± Ω pl. () It is obvious that in the cetain inteval of fequencies the imaginay pat of the wave vecto becomes nonzeo. If Γ, then this inteval of fequencies coincides with the esidual adiation zone. The dispesion equation (9) is solved numeicall and we find the fequency as a function of the wave vecto. The coesponding pts ae pesented in Fig.. It is seen that the function Ω (K) has a positive imaginay pat, which leads to instability. The hatched line displays the Cheenkov citeion ( Ω < K in ou labels). Accoding to that citeion, systems, of which wave vectos and fequencies ae unde the hatched line, ae capable to amplify oscillations. Thus, accoding to common citeion of instability and amplification of oscillations [6], it is possible to state that the effect of amplification/geneation of optical oscillations is pesented in the consideed physical system. This amplification of optical oscillations by difting 3DEG has been consideed in papes [] and []. Authos used anothe appoaches to solve this poblem. They specified a possibility of optical phonons amplification by dift of chage caies in thee-dimensional pola semiconductos. 3.. Inteaction between two-dimensional electon gas and optical phonons The DEG lies in the plane z =, is infinitely extended paallel to the x axis in both diections, and has an unpetubed two-dimensional caie density (caies pe unit aea) n and dift vecity ν. Thee is also an electostatic potential ϕ ( x, defined eveywhee inside the stuctue, wheeas the density and vecity ae confined to the plane z = and ae functions only of x and t. Two-dimensional electon gas is studied similaly to 3DEG. Thee is ( n n) δ ( z) instead of ( n n3d ) in the Poisson equation (3), whee δ (z) is the Diac deltafunction. That is the main mathematical diffeence among 3D and D situations. Substituting ν ( x,, n ( x,, ϕ ( x, fo ν( x, = ν + νe n ( x, = n + ne ϕ( x, = ϕ ( z) e ikx it ikx it,, ikx it in the system of patial diffeential equations, we get () Fig.. Real (a) and imaginay (b) pats of the fequency as a function of the wave vecto fo 3DEG (at fixed paametes Γ =., γ =.9, τ and Ω pl = ). d ϕ( z) k dz ( kν ) ν ( kν ) e + kϕ = z, m n knν =, 4πen ϕ( z) = δ( z). ( ) = () It is necessay to intoduce two bounday conditions to solve the set of equations. The fist of them is the equiement of continuity of the potential at the point z =. The second one descibes the field jump at the point z =. Besides, the potential must decease as z. Hence, we can find a potential at the point z = : ϕ () z = πen ( ) Φ( k)k. (3) By definition, put, Re( k), Φ ( k) =, Re( k) <, Φ( k) Φ ( k) =. (4) 7, V. Lashkayov Institute of Semiconducto Physics, National Academy of Sciences of Ukaine 45
Semiconducto Physics, Quantum Electonics & Optoelectonics, 8. V., N. P. 43-49.,. Ω Ω iγω K =Ω ν ± Ω iγω ν (7) It is possible to show that in the cetain inteval of fequencies the imaginay pat of the wave vecto is not equal to zeo. Solutions of the dispesion equation ae epesented in Fig.. It is obvious that the function Ω ( K ) has the positive imaginay pat (instability). The hatched line maps the Cheenkov effect. Fig.. Real (a) and imaginay (b) pats of the fequency as a function of the wave vecto fo DEG (at fixed paametes Γ =., γ =.9, τ and ν = ). Then, the potential is substituted in the fist of equations (). Setting the deteminant of the homogeneous system of algebaic equations () to zeo yields the folwing dispesion elation:, (5) ( Ω K ) = ν Φ( K )K whee all the magnitudes designated by the same notations as in equation (9), ν denotes the tem πe n m ν. The equation (5) is quadatic to the wave vecto K. Taking (4) into account, it is simple to find fou analytical solutions fo K: at Re( K ) K, =Ω + ν ± at Re( K ) < Ω + ν (6) 3.3. Inteaction between two-dimensional electon gas and optical phonons unde the metal electode Let us assume that DEG lies in the plane z =, is infinitely extended paallel to the x axes in both diections, and has the unpetubed two-dimensional caie density n and dift vecity ν. The electostatic potential ϕ ( x, is defined eveywhee inside the stuctue, wheeas the density and vecity ae confined to the plane z = and ae functions only of x and t. The symbol h denotes the distance between the electode and DEG. High-fequency popeties of two-dimensional electon gas unde the metal electode ae studied simila to the case with DEG (without any electode). The diffeence consists in bounday conditions. Now, it is necessay to conside the pesence of the metal electode. The fist bounday condition is an equality to zeo of the potential at the point z = h, which is cased by the equipotential suface of the metal. The second of them is the equiement of continuity of the potential at the point z =. The thid condition is pesence of the field jump at the point z =. In addition, the potential must decease as z. Thus, we can find a potential at the point z = : πe ϕ = z = ( ) Φ( k) Φ ( ( k ) kh n e ). (8) k Then, the potential is substituted into the fist of equations (). Setting the deteminant of the homogeneous system of algebaic equations () to zeo yields the folwing dispesion equation: Φ( K ) KS ( Ω K ) = 4 ν ( e ) Φ( K )K, (9) h whee S =. ν It is possible to eceate the dispesion equation (5) as h (as KS >> ), because Φ ( K ) KS ( e ). This can be undestood as folws. If the metal electode is sufficiently fa fom the tanspot channel, then the DEG influence is not appeciable. 7, V. Lashkayov Institute of Semiconducto Physics, National Academy of Sciences of Ukaine 46
Semiconducto Physics, Quantum Electonics & Optoelectonics, 8. V., N. P. 43-49. Fig. 3. Real (a) and imaginay (b) pats of the fequency as a function of the wave vecto fo DEG unde the metal electode (at fixed paametes Γ =., γ =.9, τ, S =. and ν = ). As KS <<, we get ( K ) = 4νS K γ Ω iγω Ω. () The dispesion equations (9) and () contain complex solutions (the numeical solutions of the equation () ae epesented in Fig. 3), theefoe this device is capable to be used as a geneato o amplifie of electomagnetic adiation in the THz fequency ange. Changing the distance between DEG and the electode, it is possible to manipulate these dispesion cuves. 4. A dissipative pocesses and an incement of instability An analysis of scatteing contibution to the incement of instability (an imaginay pat of the fequency) has been made fo all the cases consideed in this pape. It has been shown that dissipative pocesses influence on dispesion cuves and ae anagous fo all the consideed cases. Theefoe, it is sufficient to see any of thee physical systems. The second of them (DEG without a metal electode) is pesented in this section. Tacking scatteing on cystal defects into account, we obtain the dispesion equation:, () ( Ω K ) + it ( Ω K ) = ν Φ( K )K whee T = τ. If we put T =, then the dispesion elation becomes the same as (5). The numeical solutions of the equation () ae illustated in Fig. 4. As we can see in Fig. 4, dissipative pocesses lead to a shift of the imaginay pat of the fequency and/o deceases the incement of instability. Fo the compaison of cases (a) and (b) it is seen: if Γ = T, then the imaginay pat of the fequency has a shift, but cuves do not split. Thus, changing the paametes γ, ν, Γ, and T educes to vaiation of the incement of instability Im ( Ω ( K )). In paticula, at cetain fixed values of these paametes, it is possible to ealize the situation shown in Im Ω K is equal to zeo. Fig. 5a, i.e. maximum of the ( ( )) If Im( Ω ( K )), then the effects of amplification and geneation of optical vibations ae absent. Theefoe, it is necessay to analyze this situation. At some values of paametes γ, ν, Γ, and T, at the cetain point K, the peak of the imaginay pat of the fequency tends to zeo value (as shown in Fig. 5a). If we know K and fix paametes γ and ν, then it is possible to get thee cuves shown in Fig. 5b. These cuves ae built at the fixed paamete γ =.9, and have the folwing indexation. The cuve is built at the fixed paamete ν =.5; cuve is built at fixed ν = ; cuve 3 is built at the fixed ν =.5 (the magnitude ν is continuous, so we can build the pepetual amount of these cuves). Each of these cuves is a geometical place of points, and these points coespond to citical values of the paametes Γ and T. In addition, each of them is the bounday between the egion of the petubation damping and egion coesponding to amplification/geneation of optical phonons. If γ and ν ae fixed, then it is necessay to examine only one of displayed cuves. If magnitudes Γ and T beng to the citical cuve o lie above it, then coesponding petubation has been damp. Suppose fixed paametes Γ and T ae unde the cuve; then effects of amplification and geneation of optical vibations take place. It is clea that any pai of fixed values Γ and T ceate the cetain point at the plane pesented in Fig. 5b. The nge the distance between this point and the citical cuve, the stonge contibution of damping effects (in the damping egion), and geate incement of instability (in the amplification/geneation egion). In paticula, if Γ = T = (this indicates that the dissipative pocesses ae absent in the system), then an incement of instability will be the geatest one. On the othe hand, the stonge the dissipative pocesses, the faste petubation will be damped. 7, V. Lashkayov Institute of Semiconducto Physics, National Academy of Sciences of Ukaine 47
Semiconducto Physics, Quantum Electonics & Optoelectonics, 8. V., N. P. 43-49. ( ( )) Using Fig. 5b, it is possible to make the folwing analysis. Even if electon scatteing on cystal impefections ae inappeciable T, but damping of optical phonons is essentially stong (Γ >> ), then the effects of amplification and geneation will be absent. Even if we could neglect the optical phonons damping Γ, but thee ae many impefections in the cystal, then Im Ω K <, and this leads to damping of petubation. Let values of paametes Γ and T ceate the point at cuve in Fig. 5b. If we incease the density of electons (incease the paamete ν), then Im ( Ω ( K )) >, and this leads to instability. So, it is possible to make conclusions about the possibility of amplification/geneation of optical oscillations in the consideed physical system by using Fig. 5b. 5. Conclusion Fig. 4. Incement of instability as a function of the wave vecto fo DEG (at fixed paametes Γ =., T =. (a) and Γ =., T =. (b)). In this pape, we have deduced and studied the dispesion equations that descibe self-consistent collective oscillations of plasmons and optical phonons. Collective inteaction between chage caies and optical vibations in the cystal lattice leads to eoganization of dispesion cuves in esidual adiation zone and also depends on dimensions of the system. Ou analysis of the dispesion cuves fo 3DEG and optical phonons is pesented. A convective instability and amplification of optical oscillations by difting electons take place in this physical system. Examination of the dispesion law fo DEG and optical phonons is pefomed. Instability and amplification of the optical phonons by difting electons take place in this system. Investigation of the dispesion equation fo DEG unde the metal electode is pesented. An instability and amplifications of optical vibations of the cystal lattice by difting electons take place in this physical system. Changing the distance between DEG and electode gives the possibility to manipulate dispesion cuves. The dissipative pocesses lead to diminution of the instability incement in all the consideed cases. The effect of instability is suppessed if damping of optical phonons and plasma oscillations is essentially stong. Thus, all the consideed physical systems ae capable to be used as a geneato o amplifie of electomagnetic adiation in the THz fequency ange. Acknowledgements The autho is gateful to Pof. V.O. Kochelap fo valuable suggestions and constant attention to this wok. Fig. 5. Incement of instability as a function of the wave vecto fo DEG (at fixed paametes Γ =.73, T =.73 (a)). The measue between the amplification/geneation of the optical vibations egion and damping of the petubations egion (b). Refeences. V.L. Guevich // Sov. Phys. Solid State 4, p. 5 (96). 7, V. Lashkayov Institute of Semiconducto Physics, National Academy of Sciences of Ukaine 48
Semiconducto Physics, Quantum Electonics & Optoelectonics, 8. V., N. P. 43-49.. I.A. Chaban and A.A. Chaban // Sov. Phys. Solid State 6, p. 93 (964). 3. M.I. Dyakonov and M.I. Shu // Phys. Rev. Lett. 7, p. 465 (993). 4. Fank J. Cowne // J. Appl. Phys. 9, No. 3 (997). 5. Fank J. Cowne // J. Appl. Phys. 8, No. 8 (). 6. M.I. Dyakonov and M.I. Shu // J. Appl. Phys. 87, 5 (5). 7. S.M. Komienko, K.W. Kim, V.A. Kochelap, I. Gedoov, M.A. Stoscio // Phys. Rev. B 63, 6538 (). 8. S. Riyopous // Phys. of Plasmas, 774 (5). 9. V. Ryzhii, A. Satou, M.S. Shu // J. Appl. Phys. 93, No., p. 4-45 (3).. W. Liang, K.T. Tsen, Otto F. Sanke S.M. Komienko, K.W. Kim, V.A. Kochelap, Meng-Chyi Wu, Chong-Long Ho, Wen-Jeng Ho, H. Mokoc // Appl. Phys. Lett. 8, No. (3).. W. Liang, K.T. Tsen, Otto F. Sanke S.M. Komienko, K.W. Kim, V.A. Kochelap, Meng-Chyi Wu, Chong-Long Ho, Wen-Jeng Ho, H. Mokoc // J. Phys.: Condens. Matte 8, p. 796 7974 (6).. J. Lusakovski, W. Knap, N. Dyakonova, L. Vaani, J. Meteos, T. Gonzales, Y. Roelens, S. Bollaet, A. Capp and K. Kapiez // J. Appl. Phys. 97, 6437 (5). 3. N.W. Ashcoft, N.D. Memin, Solid State Physics. Moscow, Nauka, 975 (in Russian). 4. V.P. Dagunov, I.G. Neizvestnu V.A. Gidchin, Fundamental of Nanoelectonics. Novosibisk, Nauka, 4 (in Russian). 5. V.V. Mitin, V.A. Kochelap, M.A. Stoscio, Quantum Heteostuctues: Micoeleconics and Optoelectonics. Cambidge Univesity Pess, 999. 6. A.I. Ahieze et al., Collective Oscillations in Plasma. Moscow, Atomizdat, 964 (in Russian). 7. L.D. Landau, E.M. Lifshits, Hydodynamics. Moscow, Nauka, 988 (in Russian). 8. E.M. Lifshits, L.P. Pitaevski Physical Kinetics. Moscow, Nauka, 979 (in Russian). 7, V. Lashkayov Institute of Semiconducto Physics, National Academy of Sciences of Ukaine 49