ARTICLE IN PRESS Physica B 405 (2010) 3497–3500 Contents lists available at ScienceDirect Physica B journal homepage: www.elsevier.com/locate/physb Thickness effects on the Coulomb drag between low density electron layers Nguyen Quoc Khanh n Department of Theoretical Physics, National University in Ho Chi Minh City, 227-Nguyen Van Cu Street, 5th District, Ho Chi Minh City, Vietnam a r t i c l e in fo abstract Article history: Received January 2010 Received in revised form 13 May 2010 Accepted 13 May 2010 Using the Vignale–Singwi approach with the hybrid Hubbard-STLS local-field corrections suggested by Yurtsever et al we have calculated the local-field corrections, quantum-well form factors and Coulomb drag resistivity for different values of well width We have shown that the Coulomb drag resistivity increases with increase in well width when the separation between the wells remained unchanged Our results also indicate that the local-field corrections and inter-layer form factors depend weakly on the well width and the dependence of Coulomb drag resistivity on the well width is determined by the intra-layer quantum-well form factor & 2010 Elsevier B.V All rights reserved Keywords: Two-dimensional electron gas Coulomb drag Electron–electron interaction Introduction In the course of the last decades, there have been extensive theoretical and experimental investigations on the frictional drag in the double-layer electron systems [1–7] Because of the inter-layer Coulomb interaction, the current I1 driven in one layer gives rise to a voltage V2 in the second one, which is kept electrically as an open circuit (I2 ¼ 0) The drag resistivity, defined as rD ¼(w/l)  V2/I1 (where w/l is a geometrical factor) [3], is proportional to the rate of momentum transfer from the driven layer to the drag one Because the inter-layer resistivity is largely determined by the long range Coulomb scattering it provides valuable information on the electron–electron interactions especially when the layer densities are lowered There have been a number of measurements performed at the various regimes of the parameter space of the layer density, temperature, layer thickness, and separation distance between the quantum wells [8,9] At low densities significant differences between experimental results and the random-phase approximation (RPA) based calculations have been found [9] Therefore, several authors have included the correlation effects using appropriate local-field corrections (LFC) and obtained very good agreement with experiments [4–7] Recently, Vazifehshenas and Escourchi [10] have investigated the layer thickness effect on the Coulomb drag rate using the RPA and showed that the Coulomb drag rate increases with increase in well width when the separation between the wells remained unchanged However, it is well-known that the RPA based results are valid only for high-density situation To study the thickness n Fax: + 848 8350096 E-mail address: nqkhanh@phys.hcmuns.edu.vn 0921-4526/$ - see front matter & 2010 Elsevier B.V All rights reserved doi:10.1016/j.physb.2010.05.031 effect at low densities we have to use the beyond-RPA theory Many-body effects beyond the RPA are treated commonly within the self-consistent field approximation of Singwi et al [11] (STLS) Asgari et al [5] pointed out that the LFCs in STLS scheme yield an overestimate of drag resistivity Using the Vignale–Singwi (VS) approach [12] with the LFCs calculated in a hybrid Hubbard-STLS approximation Yurtsever et al [4] have obtained very good agreement with the experimental results of Kellogg et al [9] In this paper, using the method of Yurtsever and co-workers we calculate the LFCs, quantum-well form factors and Coulomb drag resistivity for different values of well width We show that the LFCs and inter-layer form factors depend weakly on layer thickness and the dependence of Coulomb drag resistivity on layer thickness is largely determined by the intra-layer quantumwell form factor Theory We consider a double-quantum-well structure with d as the center-to-center well separation such that there is no tunneling between them and l as the width of the quantum wells The electrons are free to move in the x–y plane parallel to the layers and confined in the z-direction The layers are assumed to be embedded in a uniform neutralizing positive background charge and have equal electron density n, which is related to the Fermi wave vector by n ẳ k2F =2pị We use the dimensionless coupling pffiffiffiffiffiffi constant rs ¼a/aB where, a ¼ 1= np is the average spacing between the electrons and aB ¼ _2 e=ðmà e2 Þ is the effective Bohr radius, e and m* being the background dielectric constant and electron effective mass The bare Coulomb interaction potential, in Fourier space, between the electrons in kth and lth layers is ARTICLE IN PRESS N Quoc Khanh / Physica B 405 (2010) 3497–3500 given by nld(q) ¼ n(q)Fld(ql) Here, n(q)¼2pe2/(eq) and Fkl are infinite quantum-well form factors taking the finite width effects into account which are given by Ref [5] Fkk ðxÞ ẳ Fkl xị ẳ 8p ỵ 3x 32p 1e ị x4p2 ỵx2 ị x2 4p2 ỵ x2 ị2 64p4 sinh2 x=2ị x2 4p2 ỵ x2 ị2 x eÀqd ð1Þ ð2Þ The drag resistivity rD has been obtained in a variety of theoretical models such as the Boltzmann equation, the memory function formalism, and diagrammatic perturbation theory To the lowest order in the dynamically screened effective inter-layer interaction W12(q,o), rD is given as [1–7,13] Z Z _2 rD ¼ 2 dq q3 W12 ðq, oÞ 8p e n1 n2 kB T 0  Imw01 ðq, o,TÞImw02 ðq, o,TÞ sinh ð_o=2kB TÞ 3ị where w0i q, oị, (iẳ1 or 2) is the non-interacting linear response corresponding to the drive and drag layer The key quantity in Eq (3) is the effective inter-layer interaction W12 which has different forms in different approximations In this paper, we use the VS approach [12], which is the generalization to a two-component case of the screened interaction originally developed by Kukkonen and Overhauser [14] In this scheme we have [5] W12 q, oị ẳ v12 qịẵ1G12 qị Àv12 ðqÞG12 ðqÞ Dðq, oÞ Numerical results In this section, we present our numerical calculations for the LFC, quantum-well form factors and drag resistivity of two identical infinite layers of electrons separated by d¼ 28 nm using the theoretical model described above and display the layer thickness effect on the drag resistivity for various values of density and temperature 3.1 The LFCs and quantum-well form factors The results for intra- and inter-layer LFCs calculated in the STLS and Hubbard-STLS scheme at density n¼ 3.1  1010 cm À 2, temperature T¼4 K for different layer thicknesses l are plotted in Figs and 2, respectively We observe that both intra- and interlayer LFCs show weak layer thickness dependence To investigate the effect of the layer thickness on the Coulomb drag resistivity we have calculated the quantum-well form factors at n¼3.1  1010 cm À for l¼ 10 and 18 nm and the results are shown in Fig We find that the inter-layer form factor shows very weak dependence on the layer thickness On the other hand, the layer thickness dependence of the intra-layer form factor is remarkable Therefore, we can conclude that the effects of the layer thickness on the Coulomb drag resistivity stems mainly from the layer thickness dependence of the intra-layer form factor 3.2 The effects of the layer thickness on the Coulomb drag resistivity 4ị where Dq, oị ẳ f1v11 qịẵ1G11 qịw01 q, o,Tịgf1v22 qị fv12 qịẵ1G12 qịg2 w01 q, o,Tịw02 q, o,Tị ẵ1G22 ðqÞw02 ðq, o,TÞg ð5Þ and Gij(q) are intra- and inter-LFCs, which take into account multiple scattering to infinite order between all components of the plasma compared with the RPA where these effects are neglected We note that the VS form of W12 reduces to the RPA when Gij ¼0 It has been shown that at low densities the RPA calculation gives a drag resistivity much lower than the experimental results and one should go beyond the RPA [9] One way to this is through the LFCs to the RPA form of the effective interlayer interaction as shown in Eqs (4) and (5) The simplest form of the LFCs has been obtained in the Hubbard approximation Much widely used LFCs are calculated within the STLS scheme Recent calculations have shown that the Vignale–Singwi form of W12 constructed via the STLS LFCs overestimates the correlation effects and gives a drag resistivity much higher than the experimental results [4,5] The authors of Ref [4] have used the so-called hybrid Hubbard-STLS approach to calculate the LFCs and obtained very good quantitative agreement with the experimental data for the case of equal layer densities In their scheme the intra-layer LFC is calculated in the Hubbard approximation as [4] ! ! ! Z d k q  k u11 ðkÞ ! ! So q k 9,Tị1 qị ẳ 6ị GH 11 2 n ð2pÞ u11 ðqÞ q We have calculated the drag resistivity rD as a function of temperature for matched layer densities n ¼3.1  1010 cm À 2, the well width l ¼18 nm and the inter-layer distance d¼28 nm using different theoretical models The results shown in Fig indicate that the Hubbard-STLS approximation provides the best agreement with the experimental data Therefore, using the Hubbard-STLS approximation we have investigated the dependence of drag resistivity on the well width and the results for different densities at T¼ K are plotted in Fig We observe that the drag resistivity increases with increase in the well width in both the RPA and Hubbard-STLS approximation and the results of the RPA differs remarkably from those of Hubbard-STLS approximation in the whole range of the well width Because the RPA underestimates the experimental results at low densities, we hope that we can use the Hubbard-STLS approximation to 1.0 n= 3.1x10 T=4K 10 cm -2 0.8 Gij(q) 3498 0.6 G 11 G 11 G 12 G 12 , l =18 nm , l =10 nm , l =18 nm , l =10 nm 0.4 0.2 and the inter-layer LFC is assumed to have the form [15] aq G12 qị ẳ p q2 ỵ b2 7ị here So(q,T) is the temperature dependent static structure factor of a non-interacting system and the parameters a and b are determined from the large and small q limits of the inter-layer LFC 0.0 q/q F Fig The intra- and inter-layer STLS LFCs at density n¼ 3.1  1010 cm À and temperature T¼ K for different layer thicknesses l ARTICLE IN PRESS N Quoc Khanh / Physica B 405 (2010) 3497–3500 40 0.10 l = 18 nm l = 10 nm 0.4 30 -2 10 n = 3.1x10 cm l = 18 nm d = 28 nm 0.12 0.5 3499 n = 3.1x10 T=4K 10 cm -2 n = 3.1x10 T=4K 10 cm -2 0.04 0.1 20 10 0.02 0.06 0.2 STLS Hubbard-STLS Hubbard RPA Gij(q) 0.3 ρD (Ω/square) 0.08 6 q/q F q/q F 10 0.00 0.0 Fig The Hubbard-STLS LFCs at density n¼3.1  10 cm and temperature T¼ K for different layer thicknesses l (a) Intra-layer LFC G11(q), (b) inter-layer LFC G12(q) 1.0 F 11 , l = 18 nm F 11 , l = 10 nm F 12 0.8 Fij (ql) cm Fig The drag resistivity rD as a function of temperature for matched layer densities n¼ 3.1  1010 cm À The solid line shows rD within the VS approximation using the Hubbard-STLS LFCs The dotted line uses the Hubbard approximation for G11 and G12 ¼0 The dashed and dash-dotted lines represent calculations within the VS approximation including STLS LFCs and the RPA, respectively The black squares are the experimental data of Ref [9] T=4K 100 n= 3.1x10 T (K) 0.6 10 À2 -2 80 0.4 10 Hubbard-STLS, n= 2.3x10 cm ρD (Ω/square) 10 0.2 Hubbard-STLS, n= 3.1x10 10 -2 -2 cm -2 RPA, n= 2.3x10 cm 60 10 RPA, n= 3.1x10 cm -2 40 0.0 q/q F Fig The quantum-well form factors at density n¼ 3.1  1010 cm À for different layer thicknesses l predict the well width dependence of the Coulomb drag resistivity We have also investigated the effect of well width on the behavior of drag resistivity as a function of temperature and the results are shown in Fig It is seen from the figure that the well width effect on the drag resistivity is more pronounced at high temperatures Discussion and conclusion In spite of the successes obtained in this paper, it is necessary to discuss the validity of our theoretical approach Asgari et al [5] have shown recently that the drag resistivity depends very sensitively on the models of effective inter-layer interactions and LFCs The authors of Ref [7] have employed the approximation scheme proposed by S´vierkowski et al (SSG) [1], taking into account the dynamic correlations in the long-wavelength limit and found that the plasmon contribution and dynamic correlations are very important We have also repeated our calculation using the SSG model combined with Hubbard-STLS LFCs and the obtained results are very close to those given in the previous section These are expected results because the form of W12(q,o) 20 10 12 14 16 18 20 l (nm) Fig The drag resistivity rD as a function of well width at T¼4 K calculated within the RPA and Hubbard-STLS approximation for n¼2.3  1010 and 3.1  1010 cm À (see Eq (4)) within the VS approach is similar to that in the SSG model except for the last term, which is very small in the Hubbard-STLS approximation As the SSG model supplemented by static LFCs from Hubbard-STLS and Fermi hypernetted-chain approach (FHNC) [5] also gives good agreement with experiments, we suppose that this model is appropriate for calculating the drag resistivity and we will discuss only the effects of LFCs From the results of Refs [16,17], it is seen that the LFCs depend remarkably on the frequency and we believe that the static LFCs not give R good results for the drag intensity I(q) defined by rD ¼ I(q)dq The integration over q, however, can lead to reasonable results for the drag resistivity rD due to the cancellation of errors Owing to the weak dependence of LFCs on the layer thickness (see Fig 2) and Ref [7] we believe that this cancellation still holds for other thickness values used in our paper ARTICLE IN PRESS 3500 N Quoc Khanh / Physica B 405 (2010) 3497–3500 thickness We have shown that the LFC depends weakly on the layer thickness in both approximations Using the obtained LFC we have calculated the Coulomb drag resistivity and found that the RPA underestimates while the STLS approximation overestimates experimental results at low densities On the other hand, the Hubbard-STLS approximation within the VS approach yields a very good quantitative agreement with the experimental data Using this scheme, we have found that the drag resistivity increases with increase in well width and this dependence is determined mainly by the intra-layer quantum-well factor We hope that our calculations will be of help in explaining the experimental results on the well width dependence of drag resistivity in the future 40 10 -2 n= 3.1x10 cm 30 ρD (Ω/square) Hubbard-STLS, l = 18 nm Hubbard-STLS, l = 10 nm RPA, l = 18 nm RPA, l = 10 nm 20 10 Acknowledgement 0 We gratefully acknowledge the financial support from the National Foundation for Science and Technology Development T (K) Fig The drag resistivity rD as a function of temperature at n¼ 3.1  1010 cm À calculated within the RPA and Hubbard-STLS approximation for l¼ 10 and 18 nm We now discuss the importance of the layer thickness effects for Coulomb drag investigations First, we note that we have used the infinite quantum-well model to calculate the form factors and LFCs It is well known, however, that the validity of this model depends remarkably on the barrier high and quantum-well width [18,19] and the effect of correct form factor may be crucial in the final results for drag resistivity [1,5] Second, it has been shown that the phonon-mediated drag provides a dominant mechanism in samples with large inter-layer spacing [7,20–22] This means that reducing the well thickness while fixing the center-to-center well separation may cause the role of phonons to be more important Thus, our theoretical results can be used 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Kim, G Vignale, Physica E 34 (2006) 421 A.G Rojo, J Phys.: Condens Matter 11 (1999) R31 ... remarkable Therefore, we can conclude that the effects of the layer thickness on the Coulomb drag resistivity stems mainly from the layer thickness dependence of the intra-layer form factor 3.2 The effects. .. account the dynamic correlations in the long-wavelength limit and found that the plasmon contribution and dynamic correlations are very important We have also repeated our calculation using the SSG... approximations Using the obtained LFC we have calculated the Coulomb drag resistivity and found that the RPA underestimates while the STLS approximation overestimates experimental results at low