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Collective excitations and inelastic Coulomb scattering rate of coupled Q1D electron gases in semiconductor quantum wires
Physica E 7 (2000) 541–544
www.elsevier.nl/locate/physe
Collective excitations and inelastic Coulomb scattering rate of coupled Q1D electron gases in semiconductor quantum wires Guo-Qiang Hai ∗ , Marcos R.S. Tavares Istituto de FÃsica de S˜ao Carlos, Universidade de S˜ao Paulo, 13560-970 S˜ao Carlos, SP, Brazil
Abstract We show that a weak resonant tunneling between two quantum wires leads to a nonlinear wave-vector q dependence of the acoustic plasmon mode at small q. Besides, an extra weak intersubband mode appears. As a consequence of the tunneling e ects on the acoustic plasmon mode dispersion, the inelastic Coulomb scattering rate of an injected electron is modi ed c 2000 Published by Elsevier Science B.V. All rights reserved. signi cantly. Keywords: Electron–electron interaction; Quantum wires; Tunneling e ects; Electron lifetime
The plasmons of coupled low-dimensional electron gas systems provide a valuable platform to study the electronic many-body e ects. In coupled double one-dimensional (1D) electron quantum wires, similar to coupled two-dimensional electron systems, optical and acoustic plasmon modes were found. They were interpreted, respectively, as in-phase and out-of-phase oscillations of the electron charge density in the two wires. Theoretical studies [1–9] have been done on the plasmon dispersions, electron–electron correlation, far-infrared absorption, Coulomb drag, and tunneling e ects in these systems. Experimentally, far-infrared spectroscopy and Raman scattering were used to detect the collective excitations [10,11]. Very recently, it was shown that a weak resonant tunneling in the ∗ Corresponding author. Tel.: +55-162-739-863; +55-162-739-827. E-mail address: [email protected] (G.-Q. Hai)
fax:
coupled two 1D electron gases leads to a plasmon gap in the acoustic mode at zero wavevector [1,2]. In this work, we report a theoretical study of the e ects of tunneling on the collective excitation spectrum and the inelastic Coulomb scattering rate in coupled quasi-1D electron gases. Tunneling between quantum wires can modify the collective behaviour of the electron systems in several aspects. Interwire charge transfer and intersubband scattering become possible through the tunneling. As a consequence, new plasmon modes and coupling between di erent modes appear. We expect that the tunneling will mainly a ect the acoustic plasmon mode because its polarization eld is localized in the space between the two wires where the tunneling occurs. Our numerical results show that a weak resonant tunneling between the quantum wires modi es mainly the acoustic modes at small wavevector which is characteristic of an intersubband plasmon mode. Besides, a weak
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 7 9 - 3
542
G.-Q. Hai, M.R.S. Tavares / Physica E 7 (2000) 541–544
intersubband mode appears at lower frequency. Furthermore, we calculate the inelastic Coulomb scattering rate of an injected electron due to the plasmon excitations. This scattering rate is inversely proportional to the lifetime of the electron and plays an important role in relaxation processes of a “hot electron” in the conduction band at low temperature. Its lifetime provides valuable information on the interactions between the electron and the di erent excitations. We consider a two-dimensional system in the xy-plane subjected to an additional con nement in the y-direction which forms two quantum wires parallel to each other in the x-direction. The con nement potential in the y-direction is taken to be of square well type of height Vb and widths W1 and W2 representing the rst and the second wire, respectively. The potential barrier between the two wires is of width Wb . The subband energies En and the wave functions n (y) are obtained from the numerical solution of the one-dimensional Schrodinger equation in the y-direction. The interpretation of the index n depends on tunneling between the two wires. When there is no tunneling, n is the wire index. On the opposite, when the wires are in resonant tunneling condition, n is the subband index. We restrict ourselves to the case where W1 = W2 = W . In this case, in the absence of tunneling, the two quantum wires are of the same subband energy E2 = E1 . When tunneling is present, the two wires are in resonant tunneling and there is a gap SAS = E2 − E1 between the two subbands. The dispersions of the plasmon modes are obtained by zeros of the determinant of the dielectric matrix det |(!; q)| = 0 within the random-phase approximation (RPA). The RPA has been proved to be a successful approximation in studying the collective charge excitations of Q1D electron gas by virtue of the vanishing of all vertex corrections to the 1D irreducible polarizability [1,2]. The dynamical dielectric function is given by nn0 ; mm0 (!; q) = nm n0 m0 − Vnn0 ; mm0 (q)nn0 (q; !), where nm is the Kronecker function, Vnn0 ; mm0 (q) the bare electron–electron Coulomb interaction potential, and nn0 (!; q) the 1D polarizability. In the absence of tunneling between the wires, we have intrawire Coulomb interaction potential VA (q) = V11; 11 (q) = V22; 22 (q) and the interwire potential VC (q) = V11; 22 (q) = V22; 11 (q). If the two wires have the same electron density, one has
11 (!; q) = 22 (!; q) = 0 (!; q) and 12 (!; q) = 21 (!; q) = 0. The optical (!+ ) and acoustic plasmon (!− ) modes are decoupled and determined by the equations 1 − 2U± 0 = 0;
(1)
where U± = (VA ± VC )=2. It is clear that the potential U+ and U− are related to the optical (in-phase) and acoustic (out-of-phase) plasmon modes, respectively. For two symmetric wires in weak resonant tunneling, the plasmon modes are determined by equations 1 − U+ (11 + 22 ) = 0
(2)
and 1 − U− (12 + 21 ) = 0:
(3)
The solution of Eq. (2) gives rise to the dispersion of the optical mode while Eq. (3) yields that of the acoustic modes. It is clear that the acoustic modes determined by Eq. (3) are of intersubband characteristic resulted from the tunneling e ects. Fig. 1(a) shows the plasmon dispersions of the coupled GaAs=Al0:3 Ga0:7 As (Vb = 228 meV) quan and Wb = 70 A tum wires of W1 = W2 = W = 150 A with total electron density Ne = 106 cm−1 . We nd SAS = 0:14 meV which indicates a very weak tunneling between the two quantum wires. The numerical results, with tunneling e ects, of the optical (!+ ) and acoustic (!− ) modes are presented by the solid and dashed curves, respectively. The plasmon modes without tunneling are plotted in the thin-dotted curves. We observe that, in resonant tunneling, the acoustic mode losses its linear q-dependence characteristic at small q replaced by two intersubband-like modes. For !SAS , the higher frequency one approaches the acoustic mode of absence of tunneling. Notice that, due to the symmetry of the con nement system, the optical (intrasubband-like) and acoustic (intersubband-like) modes do not couple with each other. To reveal the relative importance of the di erent plasmon modes, we performed a numerical calculation of the oscillator strength de ned by {|9(det||)=9!|!=!± }−1 as shown in Fig. 1(b). The higher frequency acoustic plasmon mode is of nite oscillator strength at q = 0. But the lower one has a very small oscillator strength, as shown in the inset, and is unimportant.
G.-Q. Hai, M.R.S. Tavares / Physica E 7 (2000) 541–544
Fig. 1. (a) Plasmon dispersions and (b) the corresponding oscillator strength in two coupled GaAs=Al0:3 Ga0:7 As (Vb = 228 meV) separated by a barrier of quantum wires (W1 = W2 = 150 A) The total electron density Ne = 106 cm−1 . The width Wb = 70 A. shadow area presents the quasiparticle excitation regions.
Now, we are going to show the e ects of a weak tunneling on the inelastic Coulomb scattering rate n (k) of an injected electron in the subband n with momentum k. The inelastic Coulomb scattering rate was obtained by the imaginary part of the electron self-energy within the GW approximation [12,13]. Fig. 2 gives (a) the scattering rate and (b) the lifetime of an injected electron in subband n = 1 (thick-solid) and n = 2 (thick-dashed curve) two quantum wires in resonant tunneling. and Wb = 70 A of W = 150 A Those in the absence of tunneling is presented in the thin-solid curve. The lower and higher scattering peaks result from the emission of the acoustic and optical plasmons, respectively. The divergent scattering rate at the onset of the optical plasmon scattering is similar to the scattering rate in a one-subband single quantum wire. At lower electron momentum k, the scattering due to the acoustic plasmon mode is not divergent at the onset when the tunneling is absent. This is a signature of the linear wavevector q-dependence
543
Fig. 2. (a) The inelastic Coulomb scattering rate and (b) the lifetime for an injected electron in the bi-wire system. The thick (thin) curves indicate the results in the presence (absence) of tunneling.
of the acoustic mode. We observe that a weak resonant tunneling a ects the scattering rate, especially, the acoustic plasmon scattering. The tunneling introduces intersubband scattering channel and modi es strongly the mechanism of the acoustic plasmon emission. In this case, the acoustic plasmon mode is the non-linear behaviour also with a nite oscillator strength at q → 0 resulting in a small divergency in the scattering rate. The acoustic plasmon scattering for the injected electron in the lowest subband is enhanced signi cantly and a quite strong scattering peak appears. In the second subband, tunneling introduces a small divergency in the scattering rate and shifts the scattering threshold to the lower k. The divergency of the acoustic plasmon scattering re ects its intersubband characteristics and non-linear q-dependence at small wavevector. In summary, we have studied the e ects of a weak tunneling on the collective excitations in two coupled quantum wires. In contrast with Refs. [1,2], we give full dispersion relations of the plasmon modes in
544
G.-Q. Hai, M.R.S. Tavares / Physica E 7 (2000) 541–544
weak resonant tunneling condition. Furthermore, we show that two scattering peaks appear in the inelastic Coulomb scattering spectrum in coupled wires due to the optical and acoustic plasmon modes, respectively. The scattering of the optical plasmons is divergent at the onset of the scattering. However, the acoustic plasmon mode does not produce such a divergency when it is linear q-dependent at small q. A weak resonant tunneling between the wires leads to a non-linear behavior of the acoustic mode at small q and enhances signi cantly the acoustic plasmon scattering for an injected electron in the lowest subband. Acknowledgements This work was supported by CNPq and FAPESP, Brazil.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
S.D. Sarma, E.H. Hwang, Phys. Rev. B 59 (1999) 10 730. Q.P. Li, S. Das Sarma, Phys. Rev. B 43 (1991) 11 768. J.S. Thakur, D. Neilson, Phys. Rev. B 59 (1997) 4671. T.V. Shahbazyan, S.E. Ulloa, Phys. Rev. B 54 (1996) 16 749. B. Tanatar, Solid State Commun. 99 (1996) 1. A. Cold, Philos. Mag. Lett. 66 (1992) 163. Z. Wu, P.P. Ruden, J. Appl. Phys. 74 (1993) 6234. Z. Wu, P.P. Ruden, J. Appl. Phys. 71 (1992) 1318. C. Steinebach, D. Heitmann, V. Gudmundsson, Phys. Rev. B 56 (1997) 6742. T. Demel, D. Heitmann, P. Grambow, K. Ploog, Phys. Rev. B 38 (1988) 12 732. C. Schuller et al., Phys. Rev. B 54 (1996) R17 304. M.R.S. Tavares, G.-Q. Hai, to be published. B.Y.-K. Hu, S. Das Sarma, Phys. Rev. B 48 (1993) 5469.
www.elsevier.nl/locate/physe
Collective excitations and inelastic Coulomb scattering rate of coupled Q1D electron gases in semiconductor quantum wires Guo-Qiang Hai ∗ , Marcos R.S. Tavares Istituto de FÃsica de S˜ao Carlos, Universidade de S˜ao Paulo, 13560-970 S˜ao Carlos, SP, Brazil
Abstract We show that a weak resonant tunneling between two quantum wires leads to a nonlinear wave-vector q dependence of the acoustic plasmon mode at small q. Besides, an extra weak intersubband mode appears. As a consequence of the tunneling e ects on the acoustic plasmon mode dispersion, the inelastic Coulomb scattering rate of an injected electron is modi ed c 2000 Published by Elsevier Science B.V. All rights reserved. signi cantly. Keywords: Electron–electron interaction; Quantum wires; Tunneling e ects; Electron lifetime
The plasmons of coupled low-dimensional electron gas systems provide a valuable platform to study the electronic many-body e ects. In coupled double one-dimensional (1D) electron quantum wires, similar to coupled two-dimensional electron systems, optical and acoustic plasmon modes were found. They were interpreted, respectively, as in-phase and out-of-phase oscillations of the electron charge density in the two wires. Theoretical studies [1–9] have been done on the plasmon dispersions, electron–electron correlation, far-infrared absorption, Coulomb drag, and tunneling e ects in these systems. Experimentally, far-infrared spectroscopy and Raman scattering were used to detect the collective excitations [10,11]. Very recently, it was shown that a weak resonant tunneling in the ∗ Corresponding author. Tel.: +55-162-739-863; +55-162-739-827. E-mail address: [email protected] (G.-Q. Hai)
fax:
coupled two 1D electron gases leads to a plasmon gap in the acoustic mode at zero wavevector [1,2]. In this work, we report a theoretical study of the e ects of tunneling on the collective excitation spectrum and the inelastic Coulomb scattering rate in coupled quasi-1D electron gases. Tunneling between quantum wires can modify the collective behaviour of the electron systems in several aspects. Interwire charge transfer and intersubband scattering become possible through the tunneling. As a consequence, new plasmon modes and coupling between di erent modes appear. We expect that the tunneling will mainly a ect the acoustic plasmon mode because its polarization eld is localized in the space between the two wires where the tunneling occurs. Our numerical results show that a weak resonant tunneling between the quantum wires modi es mainly the acoustic modes at small wavevector which is characteristic of an intersubband plasmon mode. Besides, a weak
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 7 9 - 3
542
G.-Q. Hai, M.R.S. Tavares / Physica E 7 (2000) 541–544
intersubband mode appears at lower frequency. Furthermore, we calculate the inelastic Coulomb scattering rate of an injected electron due to the plasmon excitations. This scattering rate is inversely proportional to the lifetime of the electron and plays an important role in relaxation processes of a “hot electron” in the conduction band at low temperature. Its lifetime provides valuable information on the interactions between the electron and the di erent excitations. We consider a two-dimensional system in the xy-plane subjected to an additional con nement in the y-direction which forms two quantum wires parallel to each other in the x-direction. The con nement potential in the y-direction is taken to be of square well type of height Vb and widths W1 and W2 representing the rst and the second wire, respectively. The potential barrier between the two wires is of width Wb . The subband energies En and the wave functions n (y) are obtained from the numerical solution of the one-dimensional Schrodinger equation in the y-direction. The interpretation of the index n depends on tunneling between the two wires. When there is no tunneling, n is the wire index. On the opposite, when the wires are in resonant tunneling condition, n is the subband index. We restrict ourselves to the case where W1 = W2 = W . In this case, in the absence of tunneling, the two quantum wires are of the same subband energy E2 = E1 . When tunneling is present, the two wires are in resonant tunneling and there is a gap SAS = E2 − E1 between the two subbands. The dispersions of the plasmon modes are obtained by zeros of the determinant of the dielectric matrix det |(!; q)| = 0 within the random-phase approximation (RPA). The RPA has been proved to be a successful approximation in studying the collective charge excitations of Q1D electron gas by virtue of the vanishing of all vertex corrections to the 1D irreducible polarizability [1,2]. The dynamical dielectric function is given by nn0 ; mm0 (!; q) = nm n0 m0 − Vnn0 ; mm0 (q)nn0 (q; !), where nm is the Kronecker function, Vnn0 ; mm0 (q) the bare electron–electron Coulomb interaction potential, and nn0 (!; q) the 1D polarizability. In the absence of tunneling between the wires, we have intrawire Coulomb interaction potential VA (q) = V11; 11 (q) = V22; 22 (q) and the interwire potential VC (q) = V11; 22 (q) = V22; 11 (q). If the two wires have the same electron density, one has
11 (!; q) = 22 (!; q) = 0 (!; q) and 12 (!; q) = 21 (!; q) = 0. The optical (!+ ) and acoustic plasmon (!− ) modes are decoupled and determined by the equations 1 − 2U± 0 = 0;
(1)
where U± = (VA ± VC )=2. It is clear that the potential U+ and U− are related to the optical (in-phase) and acoustic (out-of-phase) plasmon modes, respectively. For two symmetric wires in weak resonant tunneling, the plasmon modes are determined by equations 1 − U+ (11 + 22 ) = 0
(2)
and 1 − U− (12 + 21 ) = 0:
(3)
The solution of Eq. (2) gives rise to the dispersion of the optical mode while Eq. (3) yields that of the acoustic modes. It is clear that the acoustic modes determined by Eq. (3) are of intersubband characteristic resulted from the tunneling e ects. Fig. 1(a) shows the plasmon dispersions of the coupled GaAs=Al0:3 Ga0:7 As (Vb = 228 meV) quan and Wb = 70 A tum wires of W1 = W2 = W = 150 A with total electron density Ne = 106 cm−1 . We nd SAS = 0:14 meV which indicates a very weak tunneling between the two quantum wires. The numerical results, with tunneling e ects, of the optical (!+ ) and acoustic (!− ) modes are presented by the solid and dashed curves, respectively. The plasmon modes without tunneling are plotted in the thin-dotted curves. We observe that, in resonant tunneling, the acoustic mode losses its linear q-dependence characteristic at small q replaced by two intersubband-like modes. For !SAS , the higher frequency one approaches the acoustic mode of absence of tunneling. Notice that, due to the symmetry of the con nement system, the optical (intrasubband-like) and acoustic (intersubband-like) modes do not couple with each other. To reveal the relative importance of the di erent plasmon modes, we performed a numerical calculation of the oscillator strength de ned by {|9(det||)=9!|!=!± }−1 as shown in Fig. 1(b). The higher frequency acoustic plasmon mode is of nite oscillator strength at q = 0. But the lower one has a very small oscillator strength, as shown in the inset, and is unimportant.
G.-Q. Hai, M.R.S. Tavares / Physica E 7 (2000) 541–544
Fig. 1. (a) Plasmon dispersions and (b) the corresponding oscillator strength in two coupled GaAs=Al0:3 Ga0:7 As (Vb = 228 meV) separated by a barrier of quantum wires (W1 = W2 = 150 A) The total electron density Ne = 106 cm−1 . The width Wb = 70 A. shadow area presents the quasiparticle excitation regions.
Now, we are going to show the e ects of a weak tunneling on the inelastic Coulomb scattering rate n (k) of an injected electron in the subband n with momentum k. The inelastic Coulomb scattering rate was obtained by the imaginary part of the electron self-energy within the GW approximation [12,13]. Fig. 2 gives (a) the scattering rate and (b) the lifetime of an injected electron in subband n = 1 (thick-solid) and n = 2 (thick-dashed curve) two quantum wires in resonant tunneling. and Wb = 70 A of W = 150 A Those in the absence of tunneling is presented in the thin-solid curve. The lower and higher scattering peaks result from the emission of the acoustic and optical plasmons, respectively. The divergent scattering rate at the onset of the optical plasmon scattering is similar to the scattering rate in a one-subband single quantum wire. At lower electron momentum k, the scattering due to the acoustic plasmon mode is not divergent at the onset when the tunneling is absent. This is a signature of the linear wavevector q-dependence
543
Fig. 2. (a) The inelastic Coulomb scattering rate and (b) the lifetime for an injected electron in the bi-wire system. The thick (thin) curves indicate the results in the presence (absence) of tunneling.
of the acoustic mode. We observe that a weak resonant tunneling a ects the scattering rate, especially, the acoustic plasmon scattering. The tunneling introduces intersubband scattering channel and modi es strongly the mechanism of the acoustic plasmon emission. In this case, the acoustic plasmon mode is the non-linear behaviour also with a nite oscillator strength at q → 0 resulting in a small divergency in the scattering rate. The acoustic plasmon scattering for the injected electron in the lowest subband is enhanced signi cantly and a quite strong scattering peak appears. In the second subband, tunneling introduces a small divergency in the scattering rate and shifts the scattering threshold to the lower k. The divergency of the acoustic plasmon scattering re ects its intersubband characteristics and non-linear q-dependence at small wavevector. In summary, we have studied the e ects of a weak tunneling on the collective excitations in two coupled quantum wires. In contrast with Refs. [1,2], we give full dispersion relations of the plasmon modes in
544
G.-Q. Hai, M.R.S. Tavares / Physica E 7 (2000) 541–544
weak resonant tunneling condition. Furthermore, we show that two scattering peaks appear in the inelastic Coulomb scattering spectrum in coupled wires due to the optical and acoustic plasmon modes, respectively. The scattering of the optical plasmons is divergent at the onset of the scattering. However, the acoustic plasmon mode does not produce such a divergency when it is linear q-dependent at small q. A weak resonant tunneling between the wires leads to a non-linear behavior of the acoustic mode at small q and enhances signi cantly the acoustic plasmon scattering for an injected electron in the lowest subband. Acknowledgements This work was supported by CNPq and FAPESP, Brazil.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
S.D. Sarma, E.H. Hwang, Phys. Rev. B 59 (1999) 10 730. Q.P. Li, S. Das Sarma, Phys. Rev. B 43 (1991) 11 768. J.S. Thakur, D. Neilson, Phys. Rev. B 59 (1997) 4671. T.V. Shahbazyan, S.E. Ulloa, Phys. Rev. B 54 (1996) 16 749. B. Tanatar, Solid State Commun. 99 (1996) 1. A. Cold, Philos. Mag. Lett. 66 (1992) 163. Z. Wu, P.P. Ruden, J. Appl. Phys. 74 (1993) 6234. Z. Wu, P.P. Ruden, J. Appl. Phys. 71 (1992) 1318. C. Steinebach, D. Heitmann, V. Gudmundsson, Phys. Rev. B 56 (1997) 6742. T. Demel, D. Heitmann, P. Grambow, K. Ploog, Phys. Rev. B 38 (1988) 12 732. C. Schuller et al., Phys. Rev. B 54 (1996) R17 304. M.R.S. Tavares, G.-Q. Hai, to be published. B.Y.-K. Hu, S. Das Sarma, Phys. Rev. B 48 (1993) 5469.