Dispersion law is studied for high-frequency longitudinal plasma waves in

2DEG (the plane z = 0), separated by thin dielectric layer from half-limited 3DEG. It is

shown that drift of 3DEG provokes for special conditions instability of considered

plasma waves

Boiko, I.I.

Semiconductor Physics Quantum Electronics & Optoelectronics

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

 Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 2. P. 148-150.   © 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine  148 PACS 52.35.-g, 71.10.Ca, 73.21.Fg Instability of 2DEG interacting with drifting 3DEG I.I. Boiko V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine 41, prospect Nauky, 03028 Kyiv, Ukraine Phone: (044)236-5422; e-mail: igorboiko@yandex.ru  Abstract. Dispersion law is studied for high-frequency longitudinal plasma waves in 2DEG (the plane z = 0), separated by thin dielectric layer from half-limited 3DEG. It is shown that drift of 3DEG provokes for special conditions instability of considered plasma waves.   Keywords: instability, electron gas, plasma.  Manuscript received 28.02.08; accepted for publication 15.05.08; published online 30.06.08.     1. Introduction The problem of generating the high-frequency electromagnetic waves attracts attention of a great number of investigators. Now various low-dimensional electronic systems are presented as candidates for designing the suitable generators. There is a lot of distinguished variants; all of them have their own advantages and shortages. We propose here to investigate one variant of well-known beam instability in plasma (see for instance [1]), where an activating beam is not actually a beam but plays the significant role of source of energy transferred to plasma waves. Controlling this source gives a possibility to manage the frequency increment of unstable waves.     2. Dielectric susceptibility Investigated in this work is the dielectric susceptibility for the system of charged carriers, which includes two-dimensional electron gas located in the plane z = 0 and separated by an insulating layer from conducting substance occupying the half-space z ≥ l (see Figure). The system like that was considered in [2] and [3], similar systems − in papers [4-6].  We assume here that the dispersion laws for two-dimensional and three-dimensional band carriers are isotropic and parabolic: 222 2/)( mkk ⊥⊥ = hrε  ;  322 2/)( mkk hr =ε .  (1) The dispersion equation for longitudinal high-frequency (collisionless) plasmons has the form 0),(),( =∆+ ⊥⊥ qqS rr ωεωε .     (2) Here (see [2]) ⎥⎦⎤⎢⎣⎡+++=⊥⊥⊥⊥⊥ )tanh(),(1)tanh(),(21),( 21 lqqlqqqS rrrωβωβεεωε ; (3) ∫∞∞−⊥⊥ ∆+= )],([),( 3322qqdqqq z rrωεεπεωβ ;   (4) ;02/)2()()(),(222222 ∫ +−+ −−==∆⊥⊥⊥⊥⊥⊥⊥⊥⊥imqkqqkfkfkdqeqrrhrrrrhrωπωε   (5) ;02/)2()()(),(323332223∫ +−+ −−==∆imqkqqkfkfkdqeqrrhrrrrhrωπωε (6) f2 and f3 are distribution functions for 2DEG and 3DEG. At 21 εε =  and 0),( →⊥qrωβ  (the latter corresponds to application of a metallic field electrode) )]2exp(1/[),( 1 lqqS ⊥⊥ −−= εωε r . At ∞→⊥lq , that is in the case of the infinitely separated three-dimensional electrode, it follows from (3): 2/)(),( 21 εεωε +=⊥qS r  .      Limit our consideration here, for shortness, by non-degenerated 2DEG and 3DEG. In this case, the equilibrium distribution function for 2DEG has the form )/(exp)/2()( 22223202 TT kkknkf −=⊥ πv;    hh //)2( 222/1B22 TT vmTkmk == .  (7)   Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 2. P. 148-150.   © 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine  149   Fig. 1.    In the case of drifting 3DEG, we use the hydrodynamic approximation for the non-equilibrium distribution function: )/()( 3033 hrvv⊥−= umkfkf .   (8) Here ⊥uris drift velocity in },{ yx -plane. Then )/)/(exp[)/2()( 2323332/33 TT kumkknkf hvrv⊥−−π= ;   hh //)2( 332/1B33 TT vmTkmk == . (9)     Substituting the expressions (7) and (9) in Eqs. (5) and (6) and integrating the obtained expression over the wave vectors ⊥kr and kr, we obtain dielectric functions in the form: ;21212),(222222222B2202⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛γγ+λγ+λ−⎟⎟⎠⎞⎜⎜⎝⎛γγ−λγ−λ××π=ωε∆⊥⊥FFqTkneqr(10) .21214),(32323323232B323⎥⎥⎦⎤⎟⎟⎠⎞⎜⎜⎝⎛γγ+λγ+λ−⎟⎟⎠⎞⎜⎜⎝⎛γγ−λ××⎢⎢⎣⎡γ−λπ=ωε∆uuuuFFqTkneqr (11) Here we have used the following designations (see [1] and [2]): TkB/ω=λ h ; Tkuqu B/)( ⊥−ω=λ rrh ;  Tkq 33 /=γ ; Tkq 22 /⊥=γ ;  tdisttssF ∫∞∞− −−−π=0)(exp)(2.   (12) Now turn to the classical limit. Adopting the inequality 2 3,21 γ>>λ>>  , that is 322B 2/ mqTk hh >>ω>>  and  222B 2/ mqTk ⊥>>ω>> hh ,   (13) and considering the case of high frequencies, where 1/;1/ 32 >>ω>>ω ⊥ TT vqvq ,     (14) one obtains from (10) and (11):   2222222 )(2),( ωΩ−=ωπ−=ωε∆ ⊥⊥⊥ qmqneqr ; (15) 223323323 )()(4),(⊥⊥ −ωΩε−=−ωπ−=ωε∆uquqmneq rrrrr . (16) Here  333223222224;2)(mnemqneq επ=Ωπ=Ω ⊥⊥  .  (17) Substituting (16) into (4), we have  ])[(/)(),( 232322 Ω−−ωε−ωε=ωβ ⊥⊥⊥ uquqq rrrrr . (18) 3. Instability  Possible instability of 2DEG for the system shown in the figure was investigated in [3]. In this paper, the role of a field electrode was limited by creation of a node for EM field on the boundary of the electrode. In this case, the dispersion law for two-dimensional plasma waves does not depend on the state of 3DEG. Now, we consider quite another instability. It appears for a limited conductivity of drifting 3DEG.  Introducing the expressions (3), (15) and (18) into (2), we find the dispersion law for two-dimensional plasma waves: 0)()( 22212322222 =+ω−−ω−ω ⊥⊥ QQQQuq rr  (19) Here )tanh( lq⊥=α ;  )/( 2323321 ε+εαΩεα=Q ; )]()(/[)(2 322231232222 ε+εαε+ε+εαεε+εαΩ=Q ; )]()([/)( 3222312132323 ε+εαε+ε+εαεε+εαεΩ=Q .  (20) Consider the following limit: ∞→⊥lq . Then one obtains a usual law for two-dimensional plasma waves:  )/(2)1( 2122222 ε+εΩ==α=ω Q .    (21) More interesting is the opposite limit: 0→⊥lq . Then, as a consequence of the expressions (19) and (20) we have the dispersion law of this form: .0)/()]/(2[)(313232312222=ε+εεΩω−−ε+εΩ−ω−ω ⊥⊥uq rr    (22) One can see that for frequencies )(/2 31222 ε+εΩ<ω   (23) a real solution of the Eq. (22) does not exist.  Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 2. P. 148-150.   © 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine  150 In the case )/(2 31222 ε+εΩ<<ω ,  (24) the equation (22) takes the simple form: 0)(2 3232222 =εΩω+Ω−ω ⊥⊥uq rr .   (25) Solving this equation, we find 222332332/12/1ΩΩε+ΩεΩ±=ω ⊥⊥iuqrr.  (26) Here we see an evident instable branch (Im ω > 0). The form (26) is typical for the so-called beam instability. Note that for the considered case metallic three-dimensional electrode (Ω3→ ∞) used in [3] is not suitable totally. 4. Discussion and conclusion The condition of compatibility of the solution (26) and accepted earlier conditions (13), (14) and (24) has the form of the following set of inequalities: ;)/(2)(]2)(/1[31212332ε+εΩ<<<<ΩεΩ+<<⊥−⊥⊥⊥qquqvq T  13323 ]/2)(1[ −⊥ εΩΩ+<< quv T . (27) It is seen from here that the drift velocity of 3DEG has to exceed significantly the thermal velocities of 2DEG and 3DEG. It follows from the expression (26) that for the given wave vector ⊥q  increment of instability attends a maximal value at the condition 332 2)( εΩ=Ω ⊥q , or 3223 / mnmnq =⊥            (28) Frequency of waves for the given value of ⊥⊥uqrr increases, if the ratio 22233 2/ ΩΩε  goes down.  Consider now another area of frequencies: ⊥⊥<<ω uq rr .  (29) In this case, it follows from (22): 2333122222)()()(2Ωε−ε+εΩ=ω⊥⊥⊥⊥uquqrrrr.    (30) Instability of two-dimensional plasma waves appears at the condition 0)()( 233312 <Ωε−ε+ε⊥⊥uq rr .    (31) The inequality (29) can be represented in the form 02)()( 22233312 <Ω−Ωε<<ε+ε⊥⊥uq rr .     (32) The latter inequality limits from the top both the drift velocity and the wave vector ⊥q . Simultaneously, the condition 1/ 3 >>ω ⊥ Tvq  (see (14)) has to be satisfied.  References 1. E.M. Lifshits and L.P. Pitaevckiy, Physical Kinetics. Nauka, Moscow, 1979 (in Russian).  2. I.I. Boiko, Kinetics of Electron Gas Interacting with Fluctuating Potential. Naukova dumka, Kyiv, 1993 (in Russian). 3. M.I. Dyakonov and M.S. Shur // Phys. Rev. Lett. 71, p. 2465 (1993). 4. J.R. Pierce // J. Appl. Phys. 19, p. 231 (1948). 5. M.V. Nezlin // Uspekhi Fizich. Nauk 11, p. 608 (1971) (in Russian). 6. Z.S.Gribnikov, N.Z. Vagidov and V.V. Mitin // J. Appl. Phys. 88(11), p. 137 (2000).  

Instability of 2DEG interacting with drifting 3DEG

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Instability of 2DEG interacting with drifting 3DEG

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)