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Adiabatic electron transport through serially connected quantum annuli
Solid State Conniptions,
Vol. 95, No. 11, pp. 797-800, 1995 Elsevier Science Ltd Printed in Great Britain. 0038~1098/95 $9.50 + .oO
0038-1098(95)00269-3
ADIABATIC
ELECTRON
TRANSPORT THROUGH QUANTUM ANNUL1
SERIALLY CONNECTED
Jian-Xin Zhu and Z.D. Wang Department
of Physics, University of Hong Kong, Pokfulam Road, Hong Kong (Received
25 March 1995 by 2. Gan)
We propose a device which consists of a sequence of serially connected quantum annuli, and derive the transmission via edge channels as a function of Aharonov-Boa flux. For a one-dimensional finite crystal, the relation between the number of connected annuli and resonance peaks in the quantum oscillation is presented. More interestingly, we have also used this device to study the transport properties of a Fibonacci-lattice structure. Keywords: D. electronic transport, A. nanostructures,
WHEN the length scale of a system is mesoscopic [l] (i.e., a few hundred angstrons) and the temperature is in the sub-Kelvin range, the electronic wave function is coherent over the entire system and quanta effects are important. In recent years, with advances of nanotechnology, the phenomenon of Aharonov-Bohm (AB) [2] type oscillations in mesoscopic normal metal rings [3-S] and quantum dots (QDs) [9-111 has been extensively studied, because of its obvious technological significance. The measurement of this effect in singly connected QDs defined in a twodimensional electron gas (2D EG) by means of the split-gate technique, is realized by the formation of magnetic edge states at the boundaries of the 2D EG in the presence of high magnetic fields, which consist of the current-carrying states of each Landau level [12]. The oscillations can be tuned by the magnetic flux penetrating the laterally-confined QDs. Notice that the magnetic length lo = (li~/&)‘/~ decreases with the magnetic field B, and the effective area A enclosed by each edge channel increases. Consequently, the magnetic field contributes to the flux @ = BA in a complicated way. Moreover, due to the decrease of the magnetic length with the magnetic field, the transmission amplitude of electrons through each single barrier connected to the q~nt~ dot definitely varies with the magnetic field, which is not quite clear. In this paper, we study the transport properties in a sequence of quantum annuli with equal coupling to the nearest neighbors (see Fig. 1). 797
quasicrystals.
This device is also defined in a 2D EG. A timeindependent perpendicular magnetic field B is applied to establish an adiabatic transport of electrons in this device. In addition, the hole of each annulus is threaded by an AB flux, the value of which could be controlled independently. The purpose of additional introduction of the AB flux is to preserve the number of occupied states so that we can modulate the phase of electrons by varying this AB flux freely instead of the magnetic field. This setup makes it possible for us to study the strict Aharonov-Bohm oscillations of the conductance. In a sufficiently strong magnetic field, the inter-channel scattering is drastically supressed and we can treat them independently. We concentrate on the AB oscillations and the strength of magnetic field is therefore assumed in such a way that only one bulk Landau level is occupied. For mesoscopic annuli, there will exist one outer and one inner edge channel. The absence of scattering between these two channels makes the inner channel not to be involved in the transport of electrons through this device, which is thus different from the case of metallic rings [3,6]. The only scattering now takes place within a single edge channel at each single barrier, which connects two neighboring annuli. We discuss electron transport within the framework proposed originally by Landauer [13], which expresses transport quantities in terms of stationary scattering properties of the sample. We characterize each single barrier by a scattering matrix S which
ADIABATIC
798
ELECTRON
TRANSPORT
Vol. 95, No. 11
The system transmission coefficient for N quantum annuli (N + 1 barriers) is then written as TN = lMll je2. At this stage, let us calculate the system tran~~on coefficient TN for some typical cases: (i) Fig 1. Schematic geometry of a sequence of serially connected quantum annuli. Electrons carried by the outer edge channel tunneling through a single barrier.
For single quantum annulus (N = l), we have T2
(8) T1 = 1 + (T - 1)2 - 2(T - 1) cos0 ’ with T = itI the transmission of single barriers. When relates the outgoing waves to the incoming waves [ 141, 8 = (2k f l)z, equation (8) gives Tl = 1 regardless of the size of T, which demonstrates the formation of zeros= r t’ dimensional (OD) states by the outer edge channel d ’ along the annulus. Although the model presented here allows for opposite currents (carried by the inner and where r, r’ and t, t’ are the reflection and transmission amplitudes, respectively. The quantities without prime outer edge channel) in the arm of an annuli, these give the reflection and transmission amplitude for currents do not couple to each other. Thus, it is the electrons incident from the left-hand side (LHS) of generalization of OD states in singly connected geometry such as QDs. the barrier and the q~ntities with prime denote those (ii) For the case of an array of periodically for electrons incident from the right-hand side (RHS) of the barrier. Unitary property of the matrix S arranged mesoscopic annuli, if the Al3 fluxes threading the hole of each annulus are the same, we expect required by the current conservation gives that a finite crystal is formed. We plot the numerical Y+I + t+t = 1, (2) results in Fig. 2 for five connected ammli with r+t” + t+r’ = 0. different values of T [16]. It is found that each single (3 peak at positions 0 = (2k + 1)~ for a single annulus For a sy~etric barrier in a magnetic field, timedevelops into a band including five peaks, which reversal symmetry still holds [15], which yields t’ = t, corresponds to the number of annuli, In general, it and I’ = -(t’*)-‘r*t’, where the asterisk “*” denotes can be predicted that for N connected annuli, N the complex conjugation. The waves at the LHS of the resonance peaks will appear periodically. Moreover, barrier can also be expressed in terms of those of the the band width decreases with T which represents the RHS coupling strength between two adjacent annuli. This phenomenon indicates in an alternative way that the tight-binding model is applicable to the narrow band case and the nearly-free-panicle model to the strong with a transfer matrix tz (index 1 for each single coupling case. barrier) given by (iii) Notice that the AB flux penetrating through each annulus can be inde~ndently tailored to meet a tr= (ii; ;;), (5) specific need. The device proposed here is particularly appealing in the study of transport properties in where real t and imaginary r are chosen. When an quasiperiodic structures [17], where the Bloch theorem electron makes one revolution along the outer edge is inapplicable. We focus our interest on the topochannel in each annuhts, the phase of the wave function logical Fibonacci sequence {O#BeAO~. . .} in which changes by 27r$/@s + &, where al is the applied AB the ratio of incommensurate periods 0,/e, is equal to flux on the Ith annulus and &-,is the additional phase the golden mean r = (1 + t/5)/2. Figure 3 shows the accumulated during the trasversal hereafter it is set to transmission through a finite Fibonacci sequence as a be zero). Therefore, we have $$+0 = e-iel/2$~) and function of an alternative AB flux GA. We find that, $1 = e’@~&&). Now one can obtain the transfer several peaks disappear at the position of conduction matrix of the system in a sequence of N quantum annuli, bands compared with the above perfect structures, ~=t~~~t~e~**.~~t~~,, (6) and the irregularity of the detailed AB oscillation in where Fig. 3 partly explains the conductance measurements on a sequence of quantum dots in an indirect way. e,= (7) More interestingly, the envelope of the oscillation exhibits periodicity, which clearly signatures the
(t >
(e-7 $,*).
ADIABATIC
Vol. 00, No. 0
ELECTRON
TRANSPORT
r
2
2
-.
r
(b)
4
6
Fig. 3. System transmission Ts as a function of an alternative AB flux QA/$, through a finite topological Fibonacci sequence with the single barrier transmission T = 0.64. layer of superconducting material to cover the surface of the magnet and no magnetic field can leak out due to the Meissner effect. The electronic devices with their transport properties being controlled by phases will be a goal of nanoelectronics. Acknowledgements - We thank Prof. Biittiker for helpful discussions. This work was supported by the RGC grant of Hong Kong and the CRCG research grant at the University of Hong Kong. REFERENCES
_I 0
1
2
wq Fig. 2. System transmission Ts vs the AB flux @/(a, for a finite 1D crystal composed by five annuli with different values of single barrier transmission: (a) T = 0.16; (b) T = 0.64. difference of quasiperiodic ordered and perfect solids.
structures from 1D dis-
To make the device proposed here in operation is a challenge to device engineering. The main difhculty is that one must make the AB flux penetrate through such a small hole of each mesoscopic annulus and keep the electrons moving along the edge of the device rigorously excluded from this magnetic field. As we know, an AB flux can be supplied by a toroidal magnet. To prevent the flux leaking out of the magnet, it is constructive to make a shielding metal
1.
2. 3.
4. 5.
6.
Y. Imry, in Directions in Condensed Matter Physics (Edited by G. Grinstein & G. Mazenko), World Scientific, Singapore (1986) Mesoscopic Phenomena in So&$ Vol. 1, p. 101; (Edited by B. Altshuler, P.A. Lee & R.A. Webb). Elsevier, Amsterdam (1991). Y. Aharonov & D. Bohm, Phys. Rev. 115,485 (1959). Y. Gefen, Y. Imry & M. Ya. Azbel, Phys. Rev. L&t. 52, 129 (1984); M. Biittiker, Y. Imry & M. Ya. Azbel, Phys. Rev. A30, 1982 (1984); M. Btittiker, Y. Imry, R. Landauer & S. Pinhas, Phys. Rev. B31, 6207 (1985). R.A. Webb, S. Washburn, C.P. Umbach & R.B. Laibowitz, Phys. Rev. Lett. 54,2696 (1985). S. Datta, M.R. Melloch, S. Bandyopadhay 8z M.S. Lundstrom, Appl. Phys. Lett. 48, 487 (1986). D. Takai & K. Ohta, Phys. Rev. B48, 1537 (1993); 48, 14318 (1993); 49, 1844 (1994); 50, 2685 (1994); K. Ohta & D. Takai, Jpn. J. Appl. Phys. 32,4467 (1993).
800 7. 8. 9.
10. 11.
ADIABATIC
ELECTRON
S. Washburn, H. Schmid, D. Kern & R.A. Webb, Phys. Rev. Lett. 59, 1791 (1987). P.G.N. de Vegar, G. Timp, P.M. Mankiewich, R. Behringer & J. Cummingham, Phys. Rev. B40, 3491 (1989). B.J. van Wee!, L.P. Kouwenhoven, C.J.P. Harmans, J.G. Wrlhamson, C.E. Timmering, M.E.I. Broekaart, C.T. Foxon & J.J. Harris, Phys. Rev. Lett. 62,2523 (1989). L.P. Kouwenhoven, F.W. J. Hekking, B.J. van Wees, C.J.P.M. Harmans, C.E. Timmering & CT. Foxon, Plays. Rev. L&t. 65,361 (1990). Jian-Xin Zhu, Bai-Geng Wang, Qi Jiang & Chang-De Gong, ~~~~un. Them. Phys. 21, 113 (1994).
12.
13. 14. 15. 16. 17.
TRANSPORT
Vol. 95, No. 11
In the semiclassical picture, these states correspond to the “whispering gallery” trajectories, grazing the sample’s boundary and creating an effective one-dimensional loop, which is usually named a magnetic edge channel. R. Landauer, IB&$ J. Res. Dev. 1, 223 (1957); Philos. Msg. 21, 863 (1970). Qi Jiang, Shun-Qing Shen & Rui-Bao Tao, Phys. Lett. A152, 101 (1991). M. Biittiker, Phys. Rev. B38,9375 (1988). When the magnetic field is fixed the transmission amplitude through each single barrier can still be tuned by the transverse con~nement. S. Ostlund & R. Pandit, Phys. Rev. B29, 1394 (1984).
Vol. 95, No. 11, pp. 797-800, 1995 Elsevier Science Ltd Printed in Great Britain. 0038~1098/95 $9.50 + .oO
0038-1098(95)00269-3
ADIABATIC
ELECTRON
TRANSPORT THROUGH QUANTUM ANNUL1
SERIALLY CONNECTED
Jian-Xin Zhu and Z.D. Wang Department
of Physics, University of Hong Kong, Pokfulam Road, Hong Kong (Received
25 March 1995 by 2. Gan)
We propose a device which consists of a sequence of serially connected quantum annuli, and derive the transmission via edge channels as a function of Aharonov-Boa flux. For a one-dimensional finite crystal, the relation between the number of connected annuli and resonance peaks in the quantum oscillation is presented. More interestingly, we have also used this device to study the transport properties of a Fibonacci-lattice structure. Keywords: D. electronic transport, A. nanostructures,
WHEN the length scale of a system is mesoscopic [l] (i.e., a few hundred angstrons) and the temperature is in the sub-Kelvin range, the electronic wave function is coherent over the entire system and quanta effects are important. In recent years, with advances of nanotechnology, the phenomenon of Aharonov-Bohm (AB) [2] type oscillations in mesoscopic normal metal rings [3-S] and quantum dots (QDs) [9-111 has been extensively studied, because of its obvious technological significance. The measurement of this effect in singly connected QDs defined in a twodimensional electron gas (2D EG) by means of the split-gate technique, is realized by the formation of magnetic edge states at the boundaries of the 2D EG in the presence of high magnetic fields, which consist of the current-carrying states of each Landau level [12]. The oscillations can be tuned by the magnetic flux penetrating the laterally-confined QDs. Notice that the magnetic length lo = (li~/&)‘/~ decreases with the magnetic field B, and the effective area A enclosed by each edge channel increases. Consequently, the magnetic field contributes to the flux @ = BA in a complicated way. Moreover, due to the decrease of the magnetic length with the magnetic field, the transmission amplitude of electrons through each single barrier connected to the q~nt~ dot definitely varies with the magnetic field, which is not quite clear. In this paper, we study the transport properties in a sequence of quantum annuli with equal coupling to the nearest neighbors (see Fig. 1). 797
quasicrystals.
This device is also defined in a 2D EG. A timeindependent perpendicular magnetic field B is applied to establish an adiabatic transport of electrons in this device. In addition, the hole of each annulus is threaded by an AB flux, the value of which could be controlled independently. The purpose of additional introduction of the AB flux is to preserve the number of occupied states so that we can modulate the phase of electrons by varying this AB flux freely instead of the magnetic field. This setup makes it possible for us to study the strict Aharonov-Bohm oscillations of the conductance. In a sufficiently strong magnetic field, the inter-channel scattering is drastically supressed and we can treat them independently. We concentrate on the AB oscillations and the strength of magnetic field is therefore assumed in such a way that only one bulk Landau level is occupied. For mesoscopic annuli, there will exist one outer and one inner edge channel. The absence of scattering between these two channels makes the inner channel not to be involved in the transport of electrons through this device, which is thus different from the case of metallic rings [3,6]. The only scattering now takes place within a single edge channel at each single barrier, which connects two neighboring annuli. We discuss electron transport within the framework proposed originally by Landauer [13], which expresses transport quantities in terms of stationary scattering properties of the sample. We characterize each single barrier by a scattering matrix S which
ADIABATIC
798
ELECTRON
TRANSPORT
Vol. 95, No. 11
The system transmission coefficient for N quantum annuli (N + 1 barriers) is then written as TN = lMll je2. At this stage, let us calculate the system tran~~on coefficient TN for some typical cases: (i) Fig 1. Schematic geometry of a sequence of serially connected quantum annuli. Electrons carried by the outer edge channel tunneling through a single barrier.
For single quantum annulus (N = l), we have T2
(8) T1 = 1 + (T - 1)2 - 2(T - 1) cos0 ’ with T = itI the transmission of single barriers. When relates the outgoing waves to the incoming waves [ 141, 8 = (2k f l)z, equation (8) gives Tl = 1 regardless of the size of T, which demonstrates the formation of zeros= r t’ dimensional (OD) states by the outer edge channel d ’ along the annulus. Although the model presented here allows for opposite currents (carried by the inner and where r, r’ and t, t’ are the reflection and transmission amplitudes, respectively. The quantities without prime outer edge channel) in the arm of an annuli, these give the reflection and transmission amplitude for currents do not couple to each other. Thus, it is the electrons incident from the left-hand side (LHS) of generalization of OD states in singly connected geometry such as QDs. the barrier and the q~ntities with prime denote those (ii) For the case of an array of periodically for electrons incident from the right-hand side (RHS) of the barrier. Unitary property of the matrix S arranged mesoscopic annuli, if the Al3 fluxes threading the hole of each annulus are the same, we expect required by the current conservation gives that a finite crystal is formed. We plot the numerical Y+I + t+t = 1, (2) results in Fig. 2 for five connected ammli with r+t” + t+r’ = 0. different values of T [16]. It is found that each single (3 peak at positions 0 = (2k + 1)~ for a single annulus For a sy~etric barrier in a magnetic field, timedevelops into a band including five peaks, which reversal symmetry still holds [15], which yields t’ = t, corresponds to the number of annuli, In general, it and I’ = -(t’*)-‘r*t’, where the asterisk “*” denotes can be predicted that for N connected annuli, N the complex conjugation. The waves at the LHS of the resonance peaks will appear periodically. Moreover, barrier can also be expressed in terms of those of the the band width decreases with T which represents the RHS coupling strength between two adjacent annuli. This phenomenon indicates in an alternative way that the tight-binding model is applicable to the narrow band case and the nearly-free-panicle model to the strong with a transfer matrix tz (index 1 for each single coupling case. barrier) given by (iii) Notice that the AB flux penetrating through each annulus can be inde~ndently tailored to meet a tr= (ii; ;;), (5) specific need. The device proposed here is particularly appealing in the study of transport properties in where real t and imaginary r are chosen. When an quasiperiodic structures [17], where the Bloch theorem electron makes one revolution along the outer edge is inapplicable. We focus our interest on the topochannel in each annuhts, the phase of the wave function logical Fibonacci sequence {O#BeAO~. . .} in which changes by 27r$/@s + &, where al is the applied AB the ratio of incommensurate periods 0,/e, is equal to flux on the Ith annulus and &-,is the additional phase the golden mean r = (1 + t/5)/2. Figure 3 shows the accumulated during the trasversal hereafter it is set to transmission through a finite Fibonacci sequence as a be zero). Therefore, we have $$+0 = e-iel/2$~) and function of an alternative AB flux GA. We find that, $1 = e’@~&&). Now one can obtain the transfer several peaks disappear at the position of conduction matrix of the system in a sequence of N quantum annuli, bands compared with the above perfect structures, ~=t~~~t~e~**.~~t~~,, (6) and the irregularity of the detailed AB oscillation in where Fig. 3 partly explains the conductance measurements on a sequence of quantum dots in an indirect way. e,= (7) More interestingly, the envelope of the oscillation exhibits periodicity, which clearly signatures the
(t >
(e-7 $,*).
ADIABATIC
Vol. 00, No. 0
ELECTRON
TRANSPORT
r
2
2
-.
r
(b)
4
6
Fig. 3. System transmission Ts as a function of an alternative AB flux QA/$, through a finite topological Fibonacci sequence with the single barrier transmission T = 0.64. layer of superconducting material to cover the surface of the magnet and no magnetic field can leak out due to the Meissner effect. The electronic devices with their transport properties being controlled by phases will be a goal of nanoelectronics. Acknowledgements - We thank Prof. Biittiker for helpful discussions. This work was supported by the RGC grant of Hong Kong and the CRCG research grant at the University of Hong Kong. REFERENCES
_I 0
1
2
wq Fig. 2. System transmission Ts vs the AB flux @/(a, for a finite 1D crystal composed by five annuli with different values of single barrier transmission: (a) T = 0.16; (b) T = 0.64. difference of quasiperiodic ordered and perfect solids.
structures from 1D dis-
To make the device proposed here in operation is a challenge to device engineering. The main difhculty is that one must make the AB flux penetrate through such a small hole of each mesoscopic annulus and keep the electrons moving along the edge of the device rigorously excluded from this magnetic field. As we know, an AB flux can be supplied by a toroidal magnet. To prevent the flux leaking out of the magnet, it is constructive to make a shielding metal
1.
2. 3.
4. 5.
6.
Y. Imry, in Directions in Condensed Matter Physics (Edited by G. Grinstein & G. Mazenko), World Scientific, Singapore (1986) Mesoscopic Phenomena in So&$ Vol. 1, p. 101; (Edited by B. Altshuler, P.A. Lee & R.A. Webb). Elsevier, Amsterdam (1991). Y. Aharonov & D. Bohm, Phys. Rev. 115,485 (1959). Y. Gefen, Y. Imry & M. Ya. Azbel, Phys. Rev. L&t. 52, 129 (1984); M. Biittiker, Y. Imry & M. Ya. Azbel, Phys. Rev. A30, 1982 (1984); M. Btittiker, Y. Imry, R. Landauer & S. Pinhas, Phys. Rev. B31, 6207 (1985). R.A. Webb, S. Washburn, C.P. Umbach & R.B. Laibowitz, Phys. Rev. Lett. 54,2696 (1985). S. Datta, M.R. Melloch, S. Bandyopadhay 8z M.S. Lundstrom, Appl. Phys. Lett. 48, 487 (1986). D. Takai & K. Ohta, Phys. Rev. B48, 1537 (1993); 48, 14318 (1993); 49, 1844 (1994); 50, 2685 (1994); K. Ohta & D. Takai, Jpn. J. Appl. Phys. 32,4467 (1993).
800 7. 8. 9.
10. 11.
ADIABATIC
ELECTRON
S. Washburn, H. Schmid, D. Kern & R.A. Webb, Phys. Rev. Lett. 59, 1791 (1987). P.G.N. de Vegar, G. Timp, P.M. Mankiewich, R. Behringer & J. Cummingham, Phys. Rev. B40, 3491 (1989). B.J. van Wee!, L.P. Kouwenhoven, C.J.P. Harmans, J.G. Wrlhamson, C.E. Timmering, M.E.I. Broekaart, C.T. Foxon & J.J. Harris, Phys. Rev. Lett. 62,2523 (1989). L.P. Kouwenhoven, F.W. J. Hekking, B.J. van Wees, C.J.P.M. Harmans, C.E. Timmering & CT. Foxon, Plays. Rev. L&t. 65,361 (1990). Jian-Xin Zhu, Bai-Geng Wang, Qi Jiang & Chang-De Gong, ~~~~un. Them. Phys. 21, 113 (1994).
12.
13. 14. 15. 16. 17.
TRANSPORT
Vol. 95, No. 11
In the semiclassical picture, these states correspond to the “whispering gallery” trajectories, grazing the sample’s boundary and creating an effective one-dimensional loop, which is usually named a magnetic edge channel. R. Landauer, IB&$ J. Res. Dev. 1, 223 (1957); Philos. Msg. 21, 863 (1970). Qi Jiang, Shun-Qing Shen & Rui-Bao Tao, Phys. Lett. A152, 101 (1991). M. Biittiker, Phys. Rev. B38,9375 (1988). When the magnetic field is fixed the transmission amplitude through each single barrier can still be tuned by the transverse con~nement. S. Ostlund & R. Pandit, Phys. Rev. B29, 1394 (1984).