Monte Carlo Study of a Mesoscopic Capacitor

Transcription

1 Monte Carlo Study of a Mesoscopic Capacitor Takeo KATO Institute for Solid State Physics, University of Tokyo 5--5 Kashiwa-no-ha, Kashiwa, Chiba Abstract In this report, recent theoretical study by the author s group is introduced by focusing on a new topic on dynamic response of a mesoscopic capacitor, which is a device composed of a quantum dot and a lead. In order to perform nonperturbative analyses of mesoscopic systems, we demonstrate the path-integral Monte Carlo (PIMC) simulations equipped with the cluster algorithm. By the PIMC simulation, the charge relaxation resistance is evaluated in the whole range of dot-lead coupling. We predict that the charge relaxation resistance is universal even in the presence of strong Coulomb blockade for the Luttinger parameter K > /, while the Kosterlitz-Thouless transition dramatically influences the universal nature of the charge relaxation resistance for K < /. Introduction Rapid development of nanofabrication techniques has enabled researchers to confine electrons in low-dimensional artificial structures with nanoscale lengths such as quantum wires and quantum dots. These nanostructures have been studied for a few decades in terms of mesoscopic systems [], while they have attracted renewed interest for the purpose of active control of quantum-mechanical states. Among nanostructures experimentally realizable, quantum dot systems provide attractive playgrounds for researchers, since they have several advantages in application to quantum information technologies []; one can manipulate dynamically a quantum bit (qubit) by controlling experimental parameters such as a gate voltage, a source-drain bias, and an ex- Figure : An open quantum dot system. Twodimensional electron gas is confined in the white region by applying the external electronic gates inducing electron depletion (the dark region). ternal magnetic field. Toward future development of rapid manipulation of electronic states in quantum dots, it is highly demanded to reveal dynamic properties of coherent transport phenomena. Despite the long history of mesoscopic physics, it is quite recent that experiments on dynamic properties of coherent electron transport have become possible. One simple example is the measurement of GHz-frequency response of a mesoscopic capacitor, which is composed of a quantum dot and a lead as schematically shown in Fig.. Surprisingly, dynamic resistance through a quantum point contact (QPC) is quantized at h/e, which is independent of transparency of the QPC [3]. In this report, we present our recent numerical study on dynamical properties of a mesoscopic capacitor [5]. In order to investigate the whole range of the lead-dot coupling, we employ the path-integral Monte Carlo (PIMC) method, whose powerful ability has been demonstrated by the author in theoretical study of transport properties through local impurity [6] and quantum dots [7].

2 (a) (b) Point contact Geometrical capacitance scattering theory, and have formulated the relaxation resistance in terms of the scattering matrix. They have proved that R q equals half a resistance quantum h/e for a single-mode electron transmission, irrespective of transmission probability through QPC. This result is contrast to the Landauer-Büttiker formula predicting the dc resistance proportional to the transmission probability. Recently, their prediction on the breakdown of the Kirchhoff s law in phase-coherent conductors has been confirmed experimentally [3]. Figure : (a) The effective circuit of a mesoscopic capacitor, and (b) its microscopic description within the mean-field approximation. Model We first summarize characteristic features of dynamic response in phase-coherent nanoscale devices. Dynamic response of mesoscpic devices is explained in terms of a mesoscopic capacitor, i.e., a quantum analog of a classical RC circuit as shown in Fig. (b); reflection at QPC acts as a resistor R q, while capacitive coupling between the dot and the gate electrode is represented by a capacitor C µ. In this circuit, the admittance is given by the Kirchhoff s law G(ω) = R q (iωc µ ) = iωc µ + ω C µ R q + O(ω 3 ). () The last expression indicates that C µ and R q can be obtained by measurement of the admittance at low frequencies (ω (R q C µ ) ). The resistance R q thus obtained is called a charge relaxation resistance, since it represents coherent charge relaxation induced by the change of the gate-voltage. The quantum-mechanical nature in dynamic response is highlighted by disagreement between the charge relaxation resistance and the dc resistance. In the pioneering work [8], Büttiker et al. have constructed an extension of the Kirchhoff s law based on the quantum Despite the success of the scattering theory, there remains several theoretical problems to be solved. As discussed later, the scattering theory employed by Büttiker et al. is based on the mean-field approximation [8], and therefore cannot be justified in the presence of strong Coulomb interaction. The long-range Coulomb repulsion inside the dot induces a charging energy in the quantum dot, and tends to restrict the electron number to an integer one. This effect is known as the Coulomb blockade effect. When the quantum dot is fabricated in one-dimensional electron systems, additional effect may appear; the short-range Coulomb repulsion significantly changes lowenergy excitation of the system, which is effectively described by the Tomonaga-Luttinger liquid. We note that in the experiments performed so far these effects can be neglected because the quantum dot is fabricated by the edge state of the integer quantum Hall effect, and is coupled to a large gate electrode screening the long-range Coulomb interaction [3]. We, however, point out that strong Coulomb repulsion may affect dynamic response of the mesoscopic capacitor when the dot is fabricated in the edge state of the fractional quantum Hall effect, or when a small gate electrode is attached to the dot. In order to take into account strong Coulomb repulsion between electrons, we utilize the bozonization technique. This approach allows us to study the interaction effects described above in a unified way.

3 . Bosonization For the mesoscopic capacitor shown in Fig., we first assume that the transport property of the system is mainly determined by the onedimensionality near QPC. Then it is not necessary to treat exactly the two-dimensional regions in the quantum dot and the lead, so that we can consider the whole mesoscopic capacitor as a semi-infinite one-dimensional system, which is described by the (non-chiral) Tomonaga-Luttinger model. We consider a D quantum dot system with two QPCs and pinch off one of QPC. The bulk Hamiltonian for an infinite Tomonaga-Luttinger liquid is given by H = dx π [ v K ( ) φ + vk x ( ) ] θ. x () Here the canonically conjugated fields φ and θ satisfy the bosonic commutation relation [φ(x), θ(x )] = (iπ/)sgn(x x ), the Luttinger parameter K describes short-range Coulomb repulsion around QPC, and v v F /K denotes the sound velocity. We define a quantum dot in the range of x < x < x by locating QPCs at x = and x = L with barrier height V and V, respectively. The electron scattering on QPCs is expressed by the bosonic field as [9] H V = i V i D πv F cos[φ(x i ) k F x i ], (3) where D is the bandwidth. We take into account the charging effect in the dot and capacitive coupling between the dot charge Q (e/π)[φ() φ(l)] and the (static) gate voltage V g by adding the Hamiltonian H C = Q C g + QV g, (4) where C g is the geometrical capacitance. It is convenient to formulate the present problem by the path-integral formalism [9]. After integrating out the fields away from x and x, and taking the limit V, we arrive at the effective action S = S kin + S C + S V (5) S kin = ω n φ(ω n ) πkβ ω n e πk ωn / (6) S C = π E C dτ[φ(τ) πn g ] (7) S V = V dτ cos[φ(τ)] (8) where φ(τ) = φ(x, τ) φ(x, τ) + k F L, φ(ω n ) is the Fourier transformation of φ(τ), πv F /L is the level spacing, and E C e /C g is the charging energy.. Mean-field approximation We briefly explain how the result of the scattering theory by Büttiker et al. [8] is reproduced in the present formulation within the Hartree approximation. Assuming that the fluctuation of the dot charge is negligible, the mean-field self-consistent equations are obtained as S MF = S kin + S V + eu g π dτφ(τ), (9) U g V g + φ MF E C + const., () πe where U g is the effective gate voltage felt by charge in the dot, and MF denotes the average over the mean-field action (9). From Eq. (), the change of U g is related to that of V g as e D(ϵ)dU g = C g (du g dv g ), () where D(ϵ) is the density of states in the quantum dot. By using this equation, one can easily reproduce the electrochemical capacitance given in Ref. [8]: C µ Q ( = C ) µ e D(ϵ). () V g C g Here, the electrochemical C µ can be regareded as composition of the geometric capacitance C g and the quantum capacitance C q e D(ϵ) at QPC: /C µ = /C g + /C q. This result justifies the effective classical circuit shown in Fig. (b). These results based on the Hartree approximation are, however, not justified if the charge fluctuation in the dot is large enough. In the rest of this report, we will show that exact treatment of charge fluctuation by the Monte Carlo method clarifies quantum phase transition at zero temperature. 3

4 3 Monte Carlo method For numerical evaluation of the linear admittance, we employ the path-integral Monte Carlo (PIMC) method. Although this method has been utilized so far for various condensedmatter systems such as liquid helium [], its application to mesoscopic systems has started recently. For example, the linear resistance of a single resistance-shunted Josephson junction has been studied by PIMC in Refs. []. For efficient simulation, however, a clever Monte Carlo algorithm was required to overcome reduction of update efficiency at low temperatures. The essential improvement of PIMC for mesoscopic systems has been proposed by Werner and Troyer []. They have assigned Ising variables to the path by using the symmetries of the action, and applied the cluster algorithm to the effective long-range Ising model [3]. Their algorithm has enabled us to access the low-temperature region which is of much interest in many mesoscopic devices. This cluster update has been applied to other systems such as dissipative Josephson junction circuits [4], dissipative double-well systems [5], and so on. We employ this clusterupdate algorithm also to the present model. The procedure of the PIMC simulation for the present model is as follows. Approximating the action by using the discretized path φ j φ(jβ/j) (j =,,, J ) and its Fourier transform φ k j φ je (πi/j)jk πjn g ( + /K E C ), we obtain a discretized action as S = J/ k= σ φ k V τ c k J cos[φ j ], (3) j= where τ c β/j is the short time cutoff. The coefficients are given for k =,,, J/ as σ k = 8k KJ + 4E Cτ c e (π) K k /J τ c π J, and for k = and k = J/ as (4) σ = (E C + /K )τ c π, J (5) σ = + E Cτ c J/ KJ e π K/ τ c π J. (6) The update scheme consists of two procedures as follows. 3. Local update In local update, for each k a value of φk is randomly chosen following a normal distribution e φ k /σ k with variance. Then a new path φ j is obtained from the inverse Fourier transform and is accepted with probability p = {, e (S V new S old) V }. 3. Cluster update We rewrite the effective action in the imaginary-time representation as S kin = j<j κ jj φ j φ j, (7) S C = J π E Cτ c [φ j πn g ], (8) j= j S V = V τ c cos[φ j ], (9) j= where the interaction kernel κ jj κ jj = 4 KJ J/ k= J/+ is defined as k e πi J k(j j ) e (π) K k /J τ c. () We apply the cluster update for this action [, 3]: Each pair of jth and j th sites is connected with bond probability p jj = max{, e κ jj (φ j π/)(φ π/) j }, () and then all the sites φ j are flipped with respect to φ mirror = π/, i.e., φ (new) j = π φ (old) j. 4 Result In this section, we show the numerical results. Through this paper, we fix the parameters as Dτ c = π and E C τ c = τ c /K = (π/). We study both the noninteracting systems (K = ) and systems with short-range Coulomb repulsion (K < ). Here, we focus on the ac response at the charge-degenerate condition N g = / at low temperatures (J ). 4

5 3.5.5 (a) Perturbation K = T = (a) K = /5 /3 / (b) Perturbation K = /3 T =. T =.4 T = (b) J = 4 J = J = Figure 3: Capacitance C µ as a function of V for the temperature J = (solid lines) for (a) K = and (b) K = /3. The dashed line show the predictions of the second-order perturbation with respect to V. Note that the vertical axis in (b) is a logarithmic scale. 4. Capacitance The capacitance can be evaluated in the PIMC simulation by C µ = e β π [ φ φ ], () where φ i φ i/j. Fig. 3 (a) and (b) show the calculated capacitance C µ as a function of the barrier strength V for K = and K = /3, respectively. With increasing V the Coulomb staircase becomes sharper, which results in the increase in C µ Q / V g at the Coulomb peak N = /. The second-order perturbation theory with respect to V, shown as dashed lines, displays an excellent agreement for small V. Especially, it is remarkable that for the case of K = /3, C µ exhibits an abrupt increase at a finite V, signaling quantum phase transition. Indeed, one can see that C µ grows as Figure 4: Extrapolated value R(iω n ) as a function of V at inverse termperature J = in the degenerate case: (a) The result at J = for K =, /3, and /5. (b) The result for K = /3 at J = 5,, and 4. The arrows in the figure (b) indicates the direction of lowering the temperature. /T for large values of V. One can also see that C µ may exceed C g for sufficiently large V. This indicates the breakdown of the scattering theory in this region because it predict C µ = (/C g + /C q ) C g. 4. Relaxation resistance The admittance is formulated by the linear response theory as G(ω) = G(iω n ω + iδ) (3) G(iω n ) = e ω n h π = e h β dτ φ(τ)φ() e iω nτ ω n πβ φ( ω n)φ(ω n ). (4) By the PIMC simulation, we evaluate the admittance by the discretized version of this formula. The relaxation resistance can be esti- 5

6 mated from the frequency-dependence of the impedance in the Matsubara representation: R(iω n ) G(iω n ) ω n C µ, (5) where G(iω n ) and C µ are defined in Eqs. (4) and (), respectively. To obtain the value of R(ω ) = R(iω n ), we plot R(iω n ) as a function of ω n and extrapolate first five points into ω n. In Fig. 4 (a), we plot R() for K =, /3 and /5 as a function of V at the temperature J =. For K =, R() equals h/e irrespective of V, in agreement with the universal charge relaxation resistance [8]. For K = /3 and /5, the same behavior is observed in the weak barrier region, whereas R() abruptly increases beyond a critical value of V. The temperature dependence of R() for K = /3 is shown in Fig 4 (b), which indicates that R() diverges as T (J ) in the strong barrier region. This abrupt change of the relaxation resistance is explained by quantum phase transition of the Kosterlitz-Thouless (KT) type. 4.3 KT transition To reveal the origin of the quantum phase transition, we examine the strong barrier limit using an instanton method which was developed for the Kondo model [6, 7]. At the chargedegeneracy point N = /, the configuration of the bosonic field φ(τ) can be represented in the dilute instanton gas approximation φ(τ) π n j= s j Θ(τ τ j ) + π ( s), (6) where s j = s( ) j, and s denotes the separation between the well minima (Θ is the step function). Inserting Eq. (6) in the full effective action, the partition function becomes β τn Z = t n dτ n n= exp K j k τ dτ n s j s k log τ j τ k τ c dτ, (7) where t denotes the tunneling amplitude between the well minima. One can identify the scaling equations dt dl ds dl = ( ) s t, (8) K = 4s t, (9) which are familiar in the context of a Kosterlitz Thouless transition in the two-dimensional XY model. The transition corresponds to a Kondo type transition associated with the charge pseudo-spin on the dot. Eqs. (8) and (9) determine the tendency of the dot-lead transmission as the temperature is lowered; (t, s ) flows along one of the hyperbolic curves B 4t = const., where B s /(K). For K > /, the tunneling strength always grows upon reducing the temperature, and the system reaches the Kondo fixed point where the dot is strongly coupled to the reservoir. An electron freely tunnels in and out of the dot irrespective of the initial tunneling strength. In particular at K = this implies that the charge relaxation resistance is universal, i.e., R q = h/(e ), as a consequence of the unitary limit of the underlying Kondo model. On the other hand for K < /, there is the possibility that at a critical, sufficiently weak transmission t ( large V ), the RG flow always drives the system into a weak coupling configuration with specified charge. Then the charge fluctuation remains finite, i.e., φ φ (πs/), so that the capacitance diverges as /T at low temperatures. This explains the transition observed in the capacitance for K = /3 (see Fig. 3 (b)). The KT transition plays a crucial role in the relevance of the universal charge relaxation resistance. If t+b >, the system scales to the weak barrier limit, where τ RC = R q C µ is independent of the temperature. If t + B <, on the other hand, the scaling equations (8)-(9) predict s const. and t T B, so that τ RC roughly scales as T (T B + const.), which grows faster than the (thermal) coherence time τ coh /T as temperature is lowered. These observations suggest that if t + B > coherent transport can be realized by lowering temperature to guarantee τ RC < τ coh, while if t + B < electronic coherent transport in the dot breaks down before charge relax- 6

7 .8 (a).6.4. J = (b) Figure 6: Phase diagram of the mesoscopic capacitor at the charge-degenerate point (N = /). J = 5 J = J = Figure 5: (a) Product G(iω n = )R(iω n = ), and (b) the ratio τ RC /τ coh = R(iω n = )C µ T for K = /3 as a function of V for three temperatures. The arrows in the figure (b) indicates the direction of lowering the temperature. ation is finished. In the latter case, the quantum dot effectively acts as a reservoir and consequently the dynamical property of the system is governed by transport through the point contact between the two reservoirs. Therefore the V -dependent low-frequency resistance observed in Fig. 4 (a) reflects the revival of the Landauer-type transport. To see this behavior more clearly, we plot in Fig. 5 (a) the product G(iω n )R(iω n ) for K = /3 as a function of V. In the strong barrier region, G() is finite and equal to [R()], which is a familiar property of transport through QPC. Upon decreasing V, however, G()R() is suppressed since G() decays to zero because of charging up, although R() R q is finite. Moreover, we see that the coherent region G()R() = extends to larger V upon lowering temperature. Finally, we determine the phase boundary of the coherent-incoherent transi- tion by tracing the temperature dependence of the ratio τ RC /τ coh = R()C µ T. The above discussion suggests that there exists a critical backscattering strength V c, below which τ RC /τ coh decays to zero, while it diverges otherwise. From Fig. 5, the critical value is estimated as 4V c τ c 7. Finally, we show the expected phase diagram on the K-V plane. For K > /, the electron transport in the mesoscopic capacitor is always coherent, while for K < / there appears the coherent-incoherent phase transition at a critical value of the potential strength V. 5 Summary Dynamic response of a mesoscopic capacitor in the presence of strong electron interactions has been studied. Unlike the prior theories based on mean-field type approximations, we have employed the bosonization technique to take into account interactions exactly. By the PIMC simulation, we have treated the whole range of dot-lead coupling. Our results show that the relaxation resistance for a dot connected to a Luttinger liquid is universal R = h/(e K) as long as interactions are sufficiently weak. Below K < /, this resistance is governed by the strength of the dotlead coupling: At the charge degeneracy point, there is a critical coupling strength, governed by a KT type phase transition, below which the dot acts as an incoherent reservoir and the 7

8 low-frequency resistance exceeds the universal value. In this incoherent regime, the charge relaxation resistance cannot be defined anymore due to the divergence of the RC time. We note that our result can be observed experimentally in a similar setup in Ref. [3], where the quantum dot is fabricated in the edge state of the integer quantum Hall effect. If a sufficiently high magnetic field T is applied in order to realize the fractional quantum Hall effect at ν = /3, then coherentincoherent transition is expected to be observed in the mesoscopic capacitor made by the edge state. Acknowledgement The author acknowledges the collaborators of this theoretical work (Y. Hamamoto, T. Jonckheere, and T. Martin) for having an opportunity of this fruitful joint research. The author also acknowledges Y. Hamamoto to permit me to quote his result in Ph.D Thesis to this report. This research was partially supported by JSPS and MAE under the Japan- France Integrated Action Program (SAKURA) and by Grant-in-Aid for Young Scientists (B) (No. 74) from the Ministry of Education, Science, Sports and Culture. References [] Electronic Transport in Mesoscopic Systems, S. Datta, (Cambridge University Press, Cambridge, 995). [] Quantum Computation and Quantum Information, M. A. Nielsen and I. L. Chuang, (Cambridge University Press, Cambridge, ). [3] J. Gabelli, G. Feve, J.-M. Berroir, B. Plaçais, A. Cavanna, B. Etienne, Y. Jin, and D. C. Glattli: Science 33 (6) 499. [4] R. Landauer: IBM J. Res. Dev. (957) 33; R. Landauer: Phil. Mag. (97) 863; M. Büttiker: Phys. Rev. Lett. 57 (986) 76. [5] Y. Hamamoto, T. Jonckheere, T. Kato, and T. Martin: Phys. Rev. B 8 () 5335; Y. Hamamoto, Ph.D Thesis (The University of Tokyo, ). [6] Y. Hamamoto, K.-I. Imura, and T. Kato: Phys. Rev. B 77 (8) 654. [7] Y. Hamamoto and T. Kato: Phys. Rev. B 77 (8) 4535; Y. Hamamoto and T. Kato: J. Phys. Conf. Ser. 5 (9). [8] M. Büttiker, H. Thomas, and A. Prêtre: Phys. Lett. A 8 (993) 364; A. Prêtre, H. Thomas, and M. Büttiker: Phys. Rev. B 54 (996) 83; S. E. Nigg, R. López, and M. Büttiker: Phys. Rev. Lett. 97 (6) 684. [9] A. Furusaki and N. Nagaosa: Phys. Rev. B 47 (993) 387. [] D. M. Ceperley: Rev. Mod. Phys. 67 (995) 79. [] C. P. Herrero and A. D. Zaikin: Phys. Rev. B 65 () 456; N. Kimura and T. Kato: Phys. Rev. B 69 (4) 54. [] P. Werner and M. Troyer: Phys. Rev. Lett. 95 (5) 6. [3] R. H. Swendsen and J.-S. Wang: Phys. Rev. Lett. 58 (987) 86; U. Wolff: Phys. Rev. Lett. 6 (989) 36; E. Luijten and H. W. K. Blöte: Int. J. Mod. Phys. C 6 (995) 359. [4] P. Werner, G. Refael, and M. Troyer: J. Stat. Mech. Theory and Experiment 5 (5) P3. [5] T. Matsuo, Y. Natsume, and T. Kato: J. Phys. Soc. Jpn. 75 (6) 3; T. Matsuo, Y. Natsume, and T. Kato: Phys. Rev. B 77 (8) [6] P. W. Anderson, G. Yuval, and D. R. Hamann: Phys. Rev. B (97) [7] A. Furusaki and K. A. Matveev: Phys. Rev. Lett. 88 ()