中觀系統的量子傳輸---(1)中觀常態結構(2)中觀超導結構

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(4) Quantum transport in mesoscopic systems: [I] Mesoscopic normal structures [II] Mesoscopic superconducting structures .

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(16) time-modulated potential on the supercurrent in a SNS junction. The essential feature involves the interplay between the inelastic scattering and the Andreev reflection. In this study, the oscillating potential is introduced in the normal region of the junction and the current-phase relation (CPR) is calculated. Here the phase φ is the phase difference between the two super-conducting electrodes. The originally discrete Andreev levels become leaky when acted upon by the time-modulated potential, and do not contribute to the supercurrent directly. Indirectly, however, these Andreev levels contribute to the suppercurrent by trapping incoming quasi-particles temporarily through the induced emission of nhω. Τhe contribution from each quasi-particle shows peak and dip structures which depend both on the energy of the incident quasi-particle and the values of φ. Τhe CPR involves integrating, for a given φ, the contribution from quasiparticles of all possible incident energies, and thus the trapping processes have significant effects on the CPR. We have analyzed in detail these trapping features in the CPR. We have also studied the finite-temperature CPR in a ballistic double SNS junction. It is found that the phase φ2 of the middle superconductor is multivalued and it plays a crucial role in shaping the features of the CPR. In contrast with the zero-temperature CPR, which has one branch, and with a φ−period of 4π, the finite-temperature CPR for an asymmetric junction has two branches and each has a φ−period of 2π. On the other hand, the finite-temperature CPR for a symmetric junction has two branches, one with a φ−period of 4π and the other with a φ−period of 2π. Only part of the finite-temperature CPRs can be identified with the zero-temperature CPRs, and the physical reason for the differences is deduced.. CPR

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(20) ! "#$%&'() ) Abstract We have studied the quantum transport in mesoscopic stystems, including [I] normal and [II] superconducting structures. For the mesoscopic normal structures, we have studied the situation when a finite-range longitudinally polarized time-dependent electric field acts upon a narrow constriction (NC). Our finding is that the dc conductance G exhibits suppressed features. These features are recognized as the quasi-bound-state (QBS) features associated with electrons making transitions to the vicinity of a subband bottom, of which the density of states is singular. However, these features, which are valley-like instead of dip-like, are different from the G characteristics when constrictions are acted upon by a time-modulated potential. In addition, the electric field is found to cause an effective static potential barrier which barrier height is proportional to the square of the electric field and to the inverse square of the frequency. The origin of this potential barrier is traced to the square of the vector potential and its role in giving rise to the interesting field-sensitive QBS features is analyzed in full detail. We have also studied a mesoscopic ring threaded by a magnetic flux that is changing linearly in time. For the case when the ring has an elastic scatterer, and no incoherent processes, the dc current is found to be zero. For the case when the ring has incoherent processes and no elastic scatterers, the dc current is calculated. Furthermore , we find the surprising results that the contribution to the dc current is dominated not by electrons in the vicinity of the Fermi level but by lower energy electrons. A physical understanding is obtained for these findings. For the mesoscopic superconducting struct-ures, we have studied the effects of a. Keywords: Quantum transport, mesoscopic normal junction, mesoscopic superconducting junction, time-dependent, quasi-bound state, magnetic flux, Andreev level tunneling.. 2.

(21) Motivations and goals. equation of motion concept in the semiconductors was proposed to describe the [I] Normal structures: situation. Accordingly, the dc current was argued to be zero [12]. But later studies have A. A longitudinally polarized time-dependent electric field acting upon applied a perturbative method to the same a NC problem and concluded that the dc current is The influence of time-modulated fields on not zero [15]. This discrepancy is due to the the quantum transport has received extensive lacking of a fully quantum mechanical interest recently due to the potential time-dependent calculation that goes beyond importance in future technological applications. finite-order perturbation. In this work we have The systems recently considered are primarily developed a nonperturbative method for the mesoscopic systems, such as narrow system. constrictions (NCs)[1-9]. The time-modulated [II] Superconducting structures: fields can give rise to either inter-subband or intra-subband transitions. When the dimension A. A time-modulated potential acting of the time-modulated region is less than the upon a SNS junction Due to the progress of microtechnology, incoherent length the scatterings are coherent the nanostructures consisting of both normal inelastic scatterings. To understand the full effect of these time-modulated fields, we need metal (N) and superconductor (S) have caused interests in the study of to study systematically, and in great detail, renewed superconducting junctions. Numerous efforts, each of these scatterings. When a finite-range time-modulated both theoretical and experimental, are devoted field acts upon a NC, the inelastic scatterings to explore such super-conducting mesoscopic involved are the intra-subband scatterings. systems [16], where not only the Cooper pairs Quasi-bound-state (QBS) features, which were in S regions but also the quasi-particles in N not widely recognized, except for the work of regions are coherent. The interplay between Bagwell and Lake [10] who considered a these two kinds of coherence induces peculiar time-dependent potential with a delta profile, transport properties in the mesoscopic regime were found also in the case when a finite-range [17-20]. Quantum transport in the presence of time-modulated potential acts upon a NC that has a uniform widths.[8] The energy of this time-modulated fields in normal structures has QBS is below, but close to, the band bottom of been used by us [7-9] and by other researchers the one-dimensional structure. These QBS are [1-10, 21-22]. Recently, there is interest in associated with the singular density of states exploring the effect of an electromagnetic (DOS) at each subband bottom. These features wave on the Josephson current through a are found also in NC that have varying widths mesoscopic SNS constriction [23]. The [9]. Then it is legitimate to investigate how a electromagnetic wave was supposed to induce longitudinally polarized time-dependent electric transitions between the Andreev levels formed in the normal region. A resonant field influences the conductance G of a NC. approximation has been used in the paper [23]. B. Mesoscopic 1D ring with a This motivates us to implement the exact time-dependent magnetic flux time-dependent scattering method that we have A small conducting 1-D ring threaded by a developed for normal structures into this magnetic flux have been of interest to problem. The reason, based on our experience physicists because it provides a paradigm in mesoscopic transport, is that multiple allowing issues of fundamental importance to inelastic scattering should be important even be tested experi-mentally [11-13]. In the case when resonant conditions are not met. We when the magnetic flux is changing linearly want to find the possible manifestations of the with time, and the coherent ring has an elastic QBS features in a SNS junction scatterer, a physical picture equivalent to the. 3.

(22) time-modulated region is very long so that transmission via direct tunneling is totally suppressed, leading to the large G suppression. For the assisted transmission, the electron must tunnel into the time-modulated region first before it can absorb the needed energy. When ∆XV < ∆X, the minimum energy needed is ∆X. As X increases from N to N+∆XV, the electron can tunnel deeper into the time-modulated region, so that the extent it get assisted is increased. Subsequently, G increases monotonically. The value of G saturates near N+∆XV, showing that the saturated value of G, which increases with E0, is a measure of the effectiveness of the assisted process. Similar understanding applies to the case when the electron can make transition to the QBS in the effective barrier region by emitting ∆X. These field-sensitive QBS features are interesting and is potentially important for applications such as detecting photons in the THz regime.. B. Finite-temperature effect on the CPR of a ballistic double SNS junction Recently we have proposed a current-conserving condition to determine the current-phase relation (CPR) of a mesoscopic, and ballistic, double SNS junction [24-26]. Interesting results such as the cut-off features and the Andreev level tunneling features are found. In this work, we explore such features in the case of finite temperatures.. Results and discussions [I] Normal structures A. A longitudinally polarized time-dependent electric field acting upon a NC In this work, our main finding is that the conductance G exhibits two types of suppressed features the valley-like structures in the plateau regions and the suppressed features near each integral values of X. Here X represents a rescaled energy of an electron, and the integral value of X is the number of propagating channels. The widths ∆XV of both types of the suppressed features are the same in the G versus X curve. Both of the suppressed features are sensitive to the field amplitude Ε 0 and the field. B. Mesoscopic 1D ring time-dependent magnetic flux. with. a. The magnetic flux is changing linearly in time. For the case when the ring has an elastic scatterer, and no incoherent processes, we have solved the Schrodinger equation for the wavefunction. It is shown explicitly that the dc current is zero. This result holds for any rate of change of the magnetic flux, including the regime when the semi-classical argument, the effective equation of motion argument, cannot be applied. For the case when the ring has incoherent processes and no elastic scatterers, the wavefunction is obtained exactly, using a model incoherent scatterer first proposed by Buttiker.. The advantage of the model is that the electrons that suffer incoherent are injected back into the system so that the current is conserved in the incoherent processes. We obtain an expression for the dc current. The analytic results allow us to show explicitly that the dc current is contributed by electrons in the lower energies rather than by those close to the Fermi energy.. frequency ω. In particular, an explicit expression, ∆ΧV = E20 /(2ω 2∆ε ) , for the widths of these suppressed features is obtained. These findings for ∆XV suggest that the widths for both of the suppressed features must have been caused by the same physical factor. After careful analysis, a physical picture for the features in G is obtained and is summarized in the following. As the longitudinallypolarized time-modulated electric field acts upon the constriction, an effective potential ∆XV is induced in the time-modulated region, thus setting up an effective potential barrier. The effective potential barrier causes a transmitting N-th subband electron, with incident energy N<X<N+∆XV, to transmit via direct tunnelling, or to transmit via assisted transmission by absorbing m∆X, where ∆X corresponds to an energy of hω. The. [II] Superconducting structures: A. A time-modulated potential acting upon a SNS junction. 4.

(23) of the ac field is large enough, there is observable shift in the Andreev levels. It is found that the most important contribution comes from the trapping processes that occur in particles at incident energies close to the energy gap. We expect this trapping mechanism to remain important in general ac properties in mesoscopic superconducting structures.. In this study, we have shown that the transmitting quasi-particles can be trapped in the normal region when it can make transition to an Andreev level by giving away nhω. However, also because of the time-modulated potential in the normal region, the quasi-particle cannot be trapped forever. Since the Andreev levels are current-carrying states, this quasi-trapping processes show up in the CPR of the junction. The structures in the CPR that associated with these processes are identified and explained. In a SNS junction, without impurities and ac fields, there is bound Andreev levels below the gap potential due to Andreev reflections at both S-N interfaces. In such case, the lifetime of discrete Andreev levels is infinite, and both discrete bound states and continuous scattering states can contribute to supercurrent [27-28]. But the lifetime of discrete levels becomes finite due to the oscillating potential, so that the current contribution from these discrete levels would not exist after long enough time. The continuous scattering current becomes the only component so that Iφ = Isca(φ). In a dc-biased SNS junction, the discrete Andreev levels are also leaky, and the scattering current is the only current too [29-31]. Although the discrete bound current is absent in the influence of the time-modulated potential, quasi-particles in the scattering states can be trapped near an original Andreev level by emitting one or several photons, and give rise to additional current. A quasi-particle at energy |E|>∆ incident from one of the S sides may emanate in sidebands at energies E+nhω, due to the ac field. We find sets of dips and peaks in the current that are contributed from one of such quasi-particles, which energy is nhω higher than the original Andreev levels Eb. When the strength of the ac field is small, the dips and peaks happen just at Eb+ nhω. The time-modulated potential can induce the quasiparticles to emit nhω and be trapped temporarily in such a discrete Andreev level which is a carry current state. The current contributed from the same set of bound state decreases gradually with n. When the strength. B. Finite-temperature effect on the CPR of a ballistic double SNS junction Our study shows that the finite-temperature CPR is interestingly different between a symmetric and an asymmetric mesoscopic ballistic double SNS structure. The difference is due to both the Andreev level tunneling and the special role played by the phase of the middle superconductor. These results suggest that the quantum transport in superconductor superlattice could have very interesting behavior.. Self-evaluation of project results In this project, we have proposed a new time-dependent mode-matching method for solving the quantum transport in mesoscopic systems acted upon by external time-dependent fields, including both normal and super-conducting systems. New understandings have been obtained and various important processes identified. Part of these results have been presented in the 1997 annual meeting of the Physical Society of the Republic of China [32]. One paper has been submitted for publication in an international journal, and three papers are in preparation [33]. The issues studied are of current interest to the mesoscopic communities and the results obtained should have impacted their studies.. Hekking and Y.V. Nazarov, Phys. Rev. B 44, 11506 (1991). \3^!! Q. Hu, Appl. Phys. Lett. 62, 837 (1993). \4^!! S. Feng and Q. Hu, Phys. Rev. B 48, 5354 \2^!! F.. 5.

(24) (1993). Wyss et al., Appl. Phys. Lett. 63, 1522 (1993). \6^!! T.J.B.M. Janssen et al,, J. Phys. Condens. Matter 6, L163 (1994). \7^!! L.Y. Gorelik et al., Phys. Rev. Lett. 73, 2260 (1994). \8^!! C.S. Chu and C.S. Tang, Solid State Commun. 97, 119 (1996). \9^!! C.S. Tang and C.S. Chu, Phys. Rev. B 53, 4838 (1996). \:^!! C.S. Tang and C.S. Chu, Physica B 254, 178 (1998). \21^!!P.F. Bagwell and R.K. Lake, Phys. Rev. B 46, 15329 (1992). \22^!!M. Buttiker et al., Phys. Lett. 96A, 365 (1983). \23^!!R. Landauer, Phys. Rev. B 33, 6497 (1986). \24^!!Y. Gefen et al., Phys. Rev. Lett. 52, 129 (1984). \25^!!Y. Gefen et al., Phys. Rev. Lett. 59, 1752 (1987). \26^!!D. Lenstra et al., Phys. Rev. Lett.57,1623 (1986). \27^!!See the review by T.M. Klapwijk, Physica B 197, 481 (1994), and the conference papers in 203, 201 (1994). \28^!!B.L. Al'tshuler and B.Z. Spivak, Sov. Phys. JETP 65, 343 (1987). \29^!!C.W.J. Beenakker, Phys. Rev. Lett. 67, 3836 (1991). \2:^!!C.W.J. Beenakker and H. van Houten, Phys. Rev. Lett. 66, 3056 (1991). \31^!!A. Furusaki et al., Phys. Rev. Lett. 67, 132 (1991). \32^!! F.A. MaaØ and L.Y. Gorelik, Phys. Rev. B 53, 15885 (1996). \33^!!M.Wagner, Phys. Rev. B 49, 16544 (1994). \34^!!L.Y. Gorelik et al., Phys. Rev. Lett. 75, 1162 (1995). \35^!!V.C.Y. Chang and C.S. Chu, Phys. Rev. B 55, 6004 (1997). \36^!!V.C.Y. Chang and C.S. Chu, Physica B, 252, 249 (1998). \37^!!V.C.Y. Chang and C.S. Chu, Solid State Commun. 107, 433 (1998). \38^!!B.J. van Wees et al., Phys. Rev. B 44, 470 (1991). \39^!!M. Hurd et al., Phys. Rev. B 49, 15258 (1994); ibid. 51, 3745 (1995). \3:^!!T.M. Klapwijk et al., Physica B 109&110, 1657 (1982). \41^!!M. Octavio et al., Phys. Rev. B 27, 6739 (1983). \42^!!D. Averin et al., Phys. Rev. Lett. 75, 1831 (1995).. The 1998 annual meeting of the Physical Society of the Republic of China. \5^!! R.A.. \43^!Conference. [C1] C.S. Tang and C.S. Chu, “Quantum transport in narrow constrictions in a time-dependent electric field: longitudinally or transversely polarized.” Oral session: D'd3. [C2] V.C.Y. Chang and C.S. Chu, “Critical current of a ballistic asymmetric double superconductor normal-metal superconductor junction.” Oral session: Ad5. \44^!Published. and submitted papers:. [P1] C.S. Tang and C.S. Chu, 1998, “Quantum transport in the presence of a finite-range longitudinally polarized time-dependent field,” (submitted). [P2] C.S. Chu and M.J. Liu, 1998, “DC current induced by a linearly time-dependent magnetic flux in a mesoscopic ring with an incoherent scatterer,” (in preparation). [P3] H.C. Liang and C.S. Chu, 1998, “Suppercurrent in the presence of a time-modulated potential inside a SNS junction,” (in preparation). [P4] V.C.Y. Chang and C.S. Chu, 1998, “Finite-temperature effect on the current-phase relation of a ballistic double superconductor normal-metal superconductor junction,” (in preparation).. . papers:. 6.

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