Response dynamics of 1D plasmons coupled to nonlocal bulk plasmons of a host semiconductor

Solid State Communications 123 (2002) 357–360 www.elsevier.com/locate/ssc

Response dynamics of 1D plasmons coupled to nonlocal bulk plasmons of a host semiconductor Yu¨ksel Ayaz ¨ niversitesi, Zonguldak 67100, Turkey Fizik Bo¨lu¨mu¨, Zonguldak Karaelmas U Received 2 June 2002; accepted 17 June 2002 by C. Tejedor

Abstract We have carried out an explicit closed-form determination of the inverse dielectric function for a single one-dimensional (1D) quantum wire embedded in a nonlocal 3D bulk semiconductor plasma and analyzed the coupled mode plasmon dispersion relation in the hydrodynamic model. Our analysis shows that the coupled 1D– 3D plasmon is damped due to even small nonlocality of the 3D plasma. Also, there is a new nonlocal low frequency mode which does not exist for either the quantum wire or the nonlocal host in the absence of the other. q 2002 Elsevier Science Ltd. All rights reserved. PACS: 71.45.Gm; 73.21. 2 b; 73.21.Hb Keywords: A. Nanostructures; A. Semiconductors; D. Dielectric response

1. Introduction In recent years, the one-dimensional quantum wire (1DQW) systems have attracted a great deal of attention, and the plasmon spectra of single wire systems as well as of multiple wire systems have been studied theoretically [1 – 13] and experimentally [14 – 16]. There have been studies for quantum wires in dynamic bounded host media [17 – 19]. However, these studies neglect the interaction of the quantum wire plasmons with the nonlocal bulk plasmons of the host medium in which the wire is lodged. It is also important in a variety of problems to consider the role of dynamic nonlocal screening phenomena as well as the role of nonlocal coupled collective excitations. In this paper, we analyze the dynamic nonlocal dielectric response properties of a single quantum wire embedded in a nonlocal bulk semiconductor plasma by carrying out an explicit closed-form determination of the inverse dielectric function Kðr1 ; t1 ; r2 ; t2 Þ within the random phase approximation (RPA). The inverse dielectric function is a major subject of interest due to its significance as a longitudinal potential propagator for the joint system. On one hand, it describes dynamic nonlocal screening phenomena; on the

other hand, its frequency poles define the collective modes resulting from the coupling of the single quantum wire quasi-1D intrasubband plasmons with the bulk plasmon of the host semiconductor. Furthermore, the residues at these poles provide the excitation amplitudes (oscillator strengths) of the coupled collective modes. Considering translational invariance along the x-axis parallel to the quantum wire and time translational invariance, we Fourier transform to a description in terms of a single 1D wavevector qx and frequency v, Kðr1 ; r2 ; t1 2 t2 Þ ! Kðy1 ; z1 ; y2 ; z2 ; qx ; vÞ ! Kðy1 ; z1 ; y2 ; z2 Þ; and we suppress qx ; v. The inversion relation between Kðy1 ; z1 ; y2 ; z2 Þ and the direct dielectric function 1ðy1 ; z1 ; y2 ; z2 Þ satisfies the inverse dielectric function integral equation ð ð dy3 dz3 1ðy1 ; z1 ; y3 ; z3 ÞKðy3 ; z3 ; y2 ; z2 Þ ¼ dðy1 2 y2 Þdðz1 2 z2 Þ:

ð1Þ

Eq. (1) may be expressed in terms of the free electron polarizability a of the composite system under consideration

aðy1 ; z1 ; y3 ; z3 Þ ¼ 1ðy1 ; z1 ; y3 ; z3 Þ 2 dðy1 2 y3 Þdðz1 2 z3 Þ;

E-mail address: [email protected] (Y. Ayaz). 0038-1098/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 8 - 1 0 9 8 ( 0 2 ) 0 0 2 9 4 - 6

ð2Þ

358

Y. Ayaz / Solid State Communications 123 (2002) 357–360

so that the RPA integral equation for the inverse dielectric function takes the form Kðy1 ; z1 ; y2 ; z2 Þ ¼ dðy1 2 y2 Þdðz1 2 z2 Þ ð ð 2 dy3 dz3 aðy1 ; z1 ; y3 ; z3 ÞKðy3 ; z3 ; y2 ; z2 Þ:

ð3Þ

will invoke a cutoff width b in the z-direction with lqx lb p 1 in order to avoid divergencies in a well defined physical problem, due to the singular behavior of K0 function since K0 ðzÞ ! 2lnðz=2Þ for small argument z ), so that the polarizability of the wire may be expressed as ½a1D ðqx ; vÞ ¼ 22e2 R1D ðqx ; vÞ qffiffiffiffiffiffiffiffiffi a1D ðy1 ; z1 ; y3 ; z3 Þ ¼ a1D ðqx ; vÞdðy3 Þdðz3 ÞK0 ðlqx l y21 þ z21 Þ; ð8Þ

2. Joint polarizability of the quantum wire and the nonlocal host medium We consider the quantum wire to be embedded in a nonlocal 3D plasma-like host medium assuming that electrons in the wire are confined in deep potential wells with only the lowest populated subband states in the y- and z-directions. The associated 1D electron plasma is free to move in the x-direction. The free electron polarizability of the quantum wire may be written as ð ð a1D ðy1 ; z1 ; y3 ; z3 Þ ¼ 2 dy2 dz2 v1D ðy1 2 y2 ; z1 2 z2 Þ £ R1D ðy2 ; z2 ; y3 ; z3 Þ:

ð4Þ

Here, v1D ðy1 2 y2 ; z1 2 z2 Þ is the Fourier transform of the 1D Coulomb potential qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v1D ðy1 2 y2 ; z1 2 z2 Þ ¼ 2e2 K0 ðlqx l ðy1 2 y2 Þ2 þ ðz1 2 z2 Þ2 Þ; ð5Þ (K0 ðxÞ is the zeroth order modified Bessel function), and R1D ðy2 ; z2 ; y3 ; z3 Þ ¼ 2iG1D ðy2 ; z2 ; y3 ; z3 ÞG1D ðy3 ; z3 ; y2 ; z2 Þ is the ring diagram density perturbation response function of the wire. Installing the noninteracting (one electron) thermodynamic Green’s function G1D of the wire having only lowest populated subband states xðyÞ in the y-direction and jðzÞ in the z-direction with the corresponding quantized subband energy E1 ¼ EðyÞ þ EðzÞ ; we determine R1D as R1D ðy2 ; z2 ; y3 ; z3 Þ ¼ R1D ðqx ; vÞ½xðy2 Þjðz2 Þxðy3 Þjðz3 Þ2 ;

ð6Þ

"2 q2x =2m1D

refers where (f0 is the Fermi function, and e qx ¼ to the part of the single electron kinetic energy along the xaxis of the wire) R1D ðqx ; vÞ ; R1D

ð dq0 f ðe 0 x 0 qx 2qx þ E1 Þ 2 f0 ðe q0x þ E1 Þ ¼2 ; 2p v þ e q0x 2qx 2 e q0x þ i0þ ð7Þ

and the factor 2 arises from the spin sum over electron states. (The extension to multiple discrete subbands for a quantum wire of finite thickness is straightforward.) Assuming a very thin quantum wire having the lowest subband functions wholly confined in narrow potential wells, we may write lxðyÞl2 ! dðyÞ and ljðzÞl2 ! dðzÞ; (we model the quantum wire to have zero thickness in both the yand z-directions; however, in the subsequent discussions, we

where a1D ðqx ; vÞ represents the wavevector and frequencydependent 1D polarizability function of the wire. In the local cold plasma limit it may be evaluated from Eq. (7), with the result,   2! n qx ; a1D ðqx ; vÞ ; a1D ¼ 22e2 1D m1D v2 ð9Þ "v . "2 qx qF =m1D ; where qF ¼ pn1D =2 is the 1D Fermi wavenumber, and n1D and m1D represent 1D electron density and electron effective mass, respectively. To take into account nonlocality of the dynamic host semiconductor plasma associated with the continuum of extended states for electronic motion above and across the barriers defining the wire, we assume that the 3D medium is adequately represented by an approximately translationally invariant 3D band of extended states, with polarizability,

a3D ðy1 2 y3 ; z1 2 z3 Þ ¼

ð dq ð dq y z iqy ðy1 2y3 Þ iqz ðz1 2z3 Þ e e a3D ðqx ; qy ; qz ; vÞ; ð10Þ 2p 2p

where a3D ðqx ; qy ; qz ; vÞ may be taken as the Lindhard polarizability [20,21] generalized to finite temperature (or as one from a hydrodynamic model such as we will use later). Within the RPA, the noninteracting joint polarizability of the composite nanostructure system is given by the sum of the polarizabilities of the constituent parts (the quantum wire and the nonlocal semiconductor host medium), a ¼ a1D þ a3D ; whence we have qffiffiffiffiffiffiffiffiffi aðy1 ; z1 ; y3 ; z3 Þ ¼ a1D dðy3 Þdðz3 ÞK0 ðlqx l y21 þ z21 Þ þ

ð dq ð dq y z iqy ðy1 2y3 Þ iqz ðz1 2z3 Þ e e a3D ðqx ; qy ; qz ; vÞ: 2p 2p ð11Þ

3. Determination of the inverse dielectric function of the joint system We proceed to solve the RPA integral equation using the

Y. Ayaz / Solid State Communications 123 (2002) 357–360

closed-form as

Fourier transform: Kðqy ; qz ; y2 ; z2 Þ ¼

ð

dy3

ð

dz3 Kðy3 ; z3 ; y2 ; z2 Þ e2iqy y3 e2iqz z3 ;

Kðy1 ; z1 ; y2 ; z2 Þ ¼ K3D ðy1 2 y2 ; z1 2 z2 Þ 2

ð dq ð dq y z Kðqy ; qz ; y2 ; z2 Þ eiqy y3 eiqz z3 : Kðy3 ; z3 ; y2 ; z2 Þ ¼ ð2pÞ ð2pÞ ð12Þ Eq. (12) with Eq. (11) recasts Eq. (3) in the form

ð dq ð dq y z iqy y1 iqz z1 e e a3D ðqy ; qz ÞKðqy ; qz ; y2 ; z2 Þ: 2p 2p ð13Þ

Ð Ð Applying dy1 dz1 expð2iqy y1 Þ expð2iqz z1 Þ across Eq. (13), we form an expression for Kðq0y ; q0z ; y2 ; z2 Þ; with the result, Kðq0y ; q0z ; y2 ; z2 Þ ¼ ð13D ðq0y ; q0z ÞÞ21 " # 0 0 2pa1D £ e2iqy y2 e2iqz z2 2 2 ; z Þ Kð0; 0; y 2 2 ; ðqx þ q02y þ q02z Þ ð14Þ where 13D ðqy ; qz Þ ¼ 1 þ a3D ðqy ; qz Þ is the nonlocal background dielectric function. Substituting Eq. (14) in Eq. (13) we have Kðy1 ; z1 ; y2 ; z2 Þ ¼ K3D ðy1 2 y2 ; z1 2 z2 Þ 2 2pa1D Fðy1 ; z1 ÞKð0; 0; y2 ; z2 Þ;

We use a hydrodynamic model of nonlocal dynamic bulk plasma response to analyze the coupled mode dispersion relation:

ð dq ð dq eiqy ðy1 2y2 Þ eiqz ðz1 2z2 Þ y z ; 2p 2p 13D ðqy ; qz Þ ð16Þ

and Fðy1 ; z1 Þ is given by Fðy1 ; z1 Þ ¼

ð dq ð dq eiqy y1 eiqz z1 y z : 2p 2p 13D ðqy ; qz Þ½q2x þ q2y þ q2z 

ð17Þ

To determine Kð0; 0; y2 ; z2 Þ; we set y1 ¼ 0 and z1 ¼ 0 in Eq. (15), hence we have Kð0; 0; y2 ; z2 Þ ¼

K3D ð2y2 ; 2z2 Þ : ½1 þ 2pa1D Fð0; 0Þ

ð18Þ

Employing Eq. (18) in Eq. (15), we obtain Kðy1 ; z1 ; y2 ; z2 Þ in

u2 ¼ v2 2 b2 q2x ; ð20Þ

2

1=2

where vp ¼ ð4pe n3D =m3D Þ is the classical bulk plasma frequency and we choose the parameter b as b2 ¼ 3v2F =5 (vF ¼ ð"=m3D Þð3p2 n3D Þ1=3 being the electron velocity at the Fermi energy) so that within the RPA it exhibits appropriately the small wavevector dependence of the dispersion relation for the longitudinal oscillations in a Fermi gas. For this model of dielectric function we are able to evaluate Fðy; zÞ in Eq. (17) analytically as follows ðk2 ¼ ½v2 2 v2p =b2 2 q2x Þ : Fðy; zÞ ¼ ðv2 2 v2p Þ21 ½v2 f ðy; zÞ þ v2p gðy; zÞ; where we have defined ð1 dq ð1 dq eiqy y eiqz z y z ; f ðy; zÞ ¼ 2 2 2 21 2p 21 2p ½qx þ qy þ qz 

ð15Þ

where K3D ðy1 2 y2 ; z1 2 z2 Þ is the inverse dielectric function of the 3D nonlocal host

v2p ; u2 2 b2 ðq2y þ q2z Þ

13D ðqy ; qz Þ ¼ 1 2

gðy; zÞ ¼

K3D ðy1 2 y2 ; z1 2 z2 Þ ¼

2pa1D Fðy1 ; z1 Þ K ð2y2 ; 2z2 Þ: ½1 þ 2pa1D Fð0; 0Þ 3D ð19Þ

4. Coupled mode dispersion relation with 3D plasma nonlocality

Kðy1 ; z1 ; y2 ; z2 Þ ¼ dðy1 2 y2 Þdðz1 2 z2 Þ qffiffiffiffiffiffiffiffiffi 2a1D K0 ðlqx l y21 þ z21 ÞKð0; 0; y2 ; z2 Þ 2

359

ð1 21

dqy ð1 dqz eiqy y eiqz z : 2 2p 21 2p ½k 2 q2y 2 q2z 

ð21Þ

ð22aÞ ð22bÞ

The function f ðy; zÞ is readily evaluated to yield (it is related to the 1D Coulomb potential): qffiffiffiffiffiffiffiffiffi 1 K0 ðlqx l y2 þ z2 Þ: f ðy; zÞ ¼ ð23Þ 2p We note that gðy; zÞ; in fact, represents a scalar Helmholtz Green’s function for a line source along the x-axis, satisfying, ! ›2 ›2 2 þ 2 þ k gðy; zÞ ¼ dðyÞdðzÞ: ›y2 ›z We first do the qz -integration. The integrand of Eq. (22b) has poles at qz ¼ p ¼ ^ðk2 2 q2y Þ1=2 : For real qz and p, these poles are on the real qz -axis which make the integral undefined. To ensure convergence, we invoke the causality condition, assuming loss (a small damping) in the system, and let v ! v þ i0þ (or, equivalently, p ! p þ i0þ ). Then the poles are off the real qz -axis and the integral is well

360

Y. Ayaz / Solid State Communications 123 (2002) 357–360

defined. The integral may be evaluated by contour integration as follows: For z . 0; we close the contour in the upper half plane and have 2pi expðipzÞ; and for z , 0 in the lower half plane and 22pi expð2ipzÞ: Combining these two for all values of z, we have ð1=2ipÞ expðiplzlÞ: Thus, gðy; zÞ reduces to gðy; zÞ ¼

1 ð1 dqy eiqy yþiplzl : 2i 21 2p p

Now, the integrand has branch point singularities at qy ¼ ^k: Employing the same considerations as earlier, we let v ! v þ i0þ (or, k ! k þ i0þ ), so the singularities are off the real qy -axis. Then the integral is well defined and represents the zeroth order Hankel function of the first kind [22], qffiffiffiffiffiffiffiffiffi 1 gðy; zÞ ¼ H0ð1Þ ðk y2 þ z2 Þ if k2 . 0: ð24aÞ 4i On the other hand, if t2 ¼ q2x 2 ½v2 2 v2p =b2 . 0; gðy; zÞ should be replaced with qffiffiffiffiffiffiffiffiffi 1 K0 ðt y2 þ z2 Þ if t2 . 0: gðy; zÞ ¼ 2 ð24bÞ 2p Therefore, Fðy; zÞ is given by Eq. (21) jointly with Eqs. (23), (24a) and (24b). Similar considerations yields K3D ðy; zÞ as K3D ðy; zÞ ¼ dðyÞdðzÞ þ ðvp =bÞ2 gðy; zÞ:

ð25Þ

The coupled mode dispersion relation of the joint system is determined by the frequency poles of the inverse dielectric function as given by Eq. (19), namely: 1 þ 2pa1D Fð0; 0Þ ¼ 0:

ð26Þ

The involvement of the Bessel functions in Fðy; zÞ indicates that Fð0; 0Þ has singular character. However, as we have mentioned earlier, we may avoid singular behavior of Fð0; 0Þ by invoking for the wire a cutoff width b in the zdirection. Thus we replace Fð0; 0Þ by Fð0; bÞ; and rewrite Eq. (26) as: 1 þ 2pa1D Fð0; bÞ ¼ 0:

ð27Þ

To examine this, we assume 1D nonlocality to be small ð"v . "2 qx qF =m1D Þ and bqx , v1D ; so we take a1D in the local limit (Eq. (9)); however, with the use of the nonlocal dielectric function for the host, Eq. (27) takes the form ! v2p v21D gð0; bÞ v2 ¼ v2p þ v21D þ 2p ; ð28Þ K0 ðlqx lbÞ v2 where gð0; bÞ is given appropriately by Eqs. (24a) and (24b), and v1D is the 1D intrasubband plasma frequency [4]

v1D ¼ ½2e2 ðn1D =m1D Þq2x K0 ðlqx lbÞ1=2 ;

with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ 2 2 H ðb v21D =b2 2 q2x Þ ð v v Þ ip 0 p 1D v2 ø v2p þ v21D 2 : 2 ðv2p þ v21D Þ K0 ðlqx lbÞ ð30Þ On the other hand, in the low-frequency regime, where v2 , v2p ; Eq. (28) yields the dispersion relation approximately as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 K ðb v2p =b2 þ q2x Þ 0 ð v v Þ p 1D v2 ø 2 : ð31Þ 2 K0 ðlqx lbÞ ðvp þ v1D Þ These results reduce properly to known results for the case where the quantum wire is embedded in a constant dielectric background (we set vp ¼ 0 in Eq. (28)), hence v ¼ v1D ; and for the case where 13D is independent of wavevector (we let b ! 0 in Eq. (28), so H0ð1Þ ðxÞ; K0 ðxÞ ! 0 for x ! 1), hence v2 ¼ v2p þ v21D ; which indicates the hybridization of the single wire 1D plasmon with the bulk plasmon [24]. We also note that (due to vanishing denominator of Fðy; zÞ in the inverse dielectric function, Eq. (19)) the bulk plasmon ðv ¼ vp Þ is always present in the spectrum.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

ð29Þ

which, for small wavevectors, is of logarithmic character, v1D / lqx l½2lnðlqx lbÞ1=2 [23]. Considering weak nonlocality effects (or, accordingly, high-frequency limit), where v2 . v2p þ b2 q2x ; the coupled mode v2 ¼ v2p þ v21D is seen to be damped in accordance

[22] [23] [24]

M.M. Mohan, A. Griffin, Phys. Rev. B 32 (1985) 2030. Q. Li, S. Das Sarma, Phys. Rev. B 40 (1989) 5860. J. Wang, J.P. Leburton, Phys. Rev. B 41 (1990) 7846. A. Gold, A. Ghazali, Phys. Rev. B 41 (1990) 7626. B.S. Mendoza, W.L. Schaich, Phys. Rev. B 43 (1991) 6590. N. Mutluay, B. Tanatar, Phys. Rev. B 55 (1997) 6697. J.S. Thakur, D. Neilson, Phys. Rev. B 56 (1997) 4671. L. Liu, L. Swierkowski, D. Neilson, J. Szymanski, Phy. Rev. B 53 (1996) 7923. S. Das Sarma, A. Madhukar, Surf. Sci. 98 (1980) 563. S. Das Sarma, A. Madhukar, Phys. Rev. B 23 (1981) 805. G.E. Santoro, G. Giuliani, Phys. Rev. B 37 (1988) 937. G.E. Santoro, G. Giuliani, Phys. Rev. B 37 (1988) 8443. A. Gold, Z. Phys. B 86 (1992) 193. T. Demel, et al., Phys. Rev. Lett. 66 (1991) 2657. W. Hansen, et al., Phys. Rev. Lett. 58 (1987) 2586. A.R. Goni, et al., Phys. Rev. Lett. 67 (1991) 3298. L. Wendler, V.G. Grigoryan, Phys. Stat. Sol. (b) 181 (1994) 133. C.R. Bennett, B. Tanatar, Phys. Rev. B 55 (1997) 7165. Y. Ayaz, et al., Microelectr. Engng 51–52 (2000) 219. J. Lindhard, K. Dan, Vidensk. Selsk. Mat. Fys. Medd. 28 (1954) 1. A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York, 1971. P.M. Morse, H. Fesbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953. A.V. Chaplik, M.V. Krashenninnikov, Surf. Sci. 98 (1980) 533. Y. Ayaz, PhD Thesis, Department of Physics and Engineering Physics, Stevens Institute of Technology, USA, 1999.