NANO LETTERS
Flux Quantization Effects in InN Nanowires
2008 Vol. 8, No. 9 2834-2838
Thomas Richter,† Christian Blo¨mers,† Hans Lu¨th,† Raffaella Calarco,† Michael Indlekofer,‡ Michel Marso,† and Thomas Scha¨pers*,† Institute of Bio- and Nanosystems (IBN-1) and JARA-FIT Ju¨lich-Aachen Research Alliance, Forschungszentrum Ju¨lich, 52425 Ju¨lich, Germany, and Studienbereich Informationstechnologie and Elektrotechnik, Wiesbaden UniVersity of Applied Sciences, Am Bru¨ckweg 26, 65428 Ru¨sselsheim, Germany Received May 19, 2008; Revised Manuscript Received July 25, 2008
ABSTRACT InN nanowires, grown by plasma-enhanced molecular beam epitaxy, were investigated by means of magnetotransport. By performing temperaturedependent transport measurements and current measurements on a large number of nanowires of different dimensions, it is proven that the carrier transport mainly takes place in a tube-like surface electron gas. Measurements on three representative nanowires under an axially oriented magnetic field revealed pronounced magnetoconductance oscillations with a periodicity corresponding to a single magnetic flux quantum. The periodicity is explained by the effect of the magnetic flux penetrating the coherent circular quantum states in the InN nanowires, rather than by Aharonov-Bohm type interferences. The occurrence of the single magnetic flux quantum periodicity is attributed to the magnetic flux dependence of phase-coherent circular states with different angular momentum quantum numbers forming the one-dimensional transport channels. These phase coherent states can exist because of the almost ideal crystalline properties of the InN nanowires prepared by self-assembled growth.
Semiconductor nanowires deserve ever growing attention and increasing interest of the scientific community both because of their potential to build future electronic systems and because of intriguing new physical effects which can lead to new functionalities.1-7 In contrast to most III-V semiconductors, where surface depletion layers usually lead to carrier depletion in thinner wires, the narrow gap semiconductor InN offers a new playground for physical phenomena because of an unusual topology of the current carrying regions in wire structures.8 Because of Fermi-level pinning at the surface in the conduction band region,9,10 InN nanowires exhibit a highly conducting quasi two-dimensional electron gas (2DEG) on their surface, which causes nearly metallic conductivity even at low temperatures (Figure 1). This tubelike conductance channel will lead to interesting quantization phenomena as long as coherent electron wave functions around the wire can form.11 A magnetic field oriented parallel to the nanowire produces a flux through the current carrying tube-like structure and electronic transport along the wire will show quantization phenomena. Our investigations were motivated by the assumption that the condition of coherent quantum states in an InN nanowire might be fulfilled, since nanowire growth allows for nearly perfect crystallinity even on mismatched substrates due to * Corresponding author. E-mail:
[email protected]. † Institute of Bio- and Nanosystems (IBN-1) and JARA-FIT Ju ¨ lichAachen Research Alliance. ‡ Wiesbaden University of Applied Sciences. 10.1021/nl8014389 CCC: $40.75 Published on Web 08/20/2008
2008 American Chemical Society
Figure 1. (a) Electron beam micrograph of an InN nanowire contacted by Ti/Au pads. (b) Schematics of an InN nanowire of length L and diameter d. The magnetic field B is oriented parallel to the wire axis. On the right side, the band profile is sketched, with EF the Fermi energy and E0 the energy of the lowest state. (c) Schematics of the energy levels (El - Ez)/E0 as a function of the normalized magnetic flux Φ/Φ0, with Ez ) p2kz2/2m* and E0 ) p2/2m*r02. For the sake of simplicity, we assumed only two energy levels (El)0, El)(1) occupied at B ) 0. For the actual samples, the number of occupied energy levels is considerably larger, for example, 26 for wire A. The circles located at EF indicate the points where the conductance is changed by (2e2/h.
relaxation of the nanostructures during growth.12 In contrast to lithographically prepared nanostructures, the surface of nanowires, because of their self-assembled growth under ultra high vacuum conditions, also might be essentially free of surface defects. InN nanowires produced by self-assembled growth thus represent ideal systems for the observation of
Table 1. Geometrical Dimensions and Resistance at 0.5 K of the Investigated InN Nanowires sample length L (nm) diameter d (nm) resistance at 0.5 K (Ω) wireA wireB wireC
37 ( 3 41 ( 3 58 ( 3
420 220 205
3620 1580 562
ballistic transport and quantum effects on a 10 to 100 nm scale. These structures might be of high interest for future phase-based quantum electronics.13-15 The present paper demonstrates transport through coherentquantum states in InN-nanowires at low temperatures. Quantum corrections to the conductance are observed which are due to circular coherent electronic states within the cylinder-like 2DEG on the nanowire surface. These states give rise to magnetic field induced conductance oscillations similar to those which have been seen in the magnetoconductance of multiwall carbon nanotubes.16 The InN nanowires investigated in this study were grown by plasma-assisted MBE under N-rich conditions on Si(111). The resulting nominally undoped wires have a diameter between 30 and 200 nm and a length of up to 1 µm. The growth of InN nanowires as well as their structural, morphological, and optical properties have been discussed in detail elsewhere.12,17 To characterize and analyze the transport through those nanostructures, the wires were removed from the original growth substrate in an ultrasonic bath in an acetone solution. The resulting nanowire acetone solution is then dropped on a SiO2 (100 nm thick) coated Si (100) wafer prestructured with marker arrays. By means of these markers, the exact position of the randomly disposed wires can be determined using scanning electron microscopy. Finally, the wires were contacted individually with nonannealed Ti/Au electrodes patterned by electron beam lithography. The prepared contacts result to be ohmic. A scanning electron beam micrograph of a contacted InN is depicted in Figure 1a. Transport measurements at large magnetic fields were performed on a set of more than 10 single nanowire devices. Subsequently, three representative wires, wires A, B, and C, are investigated in detail. Their dimensions, that is, diameter d and contact separations L, are summarized in Table 1. The low-temperature measurements were carried out in a He-4 flow cryostat (4-300 K) and in a He-3 cryostat (0.5-30 K). The latter was equipped with a 10 T magnet. The orientation of the magnetic field was chosen to be parallel to the wire axis. The nanowire resistance was measured with a lock-in technique using a bias current of 30 nA. For the interpretation of the observed magnetoconductance oscillations, it is essential to first consider the transport properties at zero magnetic field in detail. The observed temperature dependence of the resistance (Figure 2) is typical for a metallic sample, with a linear regime (phonon scattering) at temperatures above 200 K, a curved dependence for lower temperature with a tendency to saturation (defect scattering and contact resistance) below 10 K. The data in the linear regime are fitted by R(T ) ) R0[1 + R(T - T0)]
(1)
with T0 ) 300 K, R0 ) 439 Ω, and R ) 6.84 × 10 Ω/K. The R value resembles literature data of 4 × 10-4 Ω/K.18 -4
Nano Lett., Vol. 8, No. 9, 2008
Figure 2. Resistance of an InN nanowire as a function of temperature. The solid line represents the linear increase of the resistance at higher temperatures. The inset shows the conductance g ) G ) L normalized to a length of 1 µm of 40 InN nanowires with different diameters at room temperature. The solid lines represent the predicted conductance vs diameter dependence g ∝ dβ for bulk (β ) 2) and surface (β ) 1) conductivity, respectively. The dashed line corresponds to the fitted curve with β ) 1.15.
This metallic behavior of the resistance can only be explained by a high degree of degeneration in the bulk conduction band (EF > EC) and/or by a strong accumulation space charge layer on the column surface. Thus, it is necessary to clarify whether the carriers in InN nanowires are located in the bulk crystal and/or at the surface. In order to tackle this problem, the normalized conductance of nanowires with different diameters: g ) G × L, with G the wire conductance and L its length, was measured at room temperature. Figure 2 (inset) shows a dependence g ∝ dβ, with β ) 1.15 being slightly above linear. A linear dependence is expected for conductance through a surface 2DEG, the surface of which increases as πd, while degenerate bulk conductance increases with diameter as πd2/4, the cross section area of the column. Possible effects of the contact resistance in this analysis were taken into account under the assumption of solely bulk conductance by plotting the measured resistance versus the wire length divided by the cross section. The obtained contact resistance values then range between 200 and 350 Ω (4 × 10-8 Ω·cm2 e Fcontact e 8.5 × 10-8 Ω·cm2); that is, they are lower by a factor of 3 than values reported in the literature,19 however for Pd/Ti/ Pt/Au rather than in the present case for Ti/Au contacts. The experimental transport data with the observed conductance/diameter slope of β ) 1.15, therefore, suggest a major conductance contribution through a cylinder-like metallic surface accumulation layer with some additional contribution through a degenerate bulk region, with the Fermi-level located in the conduction band. This picture is indeed in good agreement with results from high resolution electron energy loss spectroscopy (HREELS) on InN surfaces. Mahboob et al.9 found that on InN surfaces the Fermi-level EF is pinned at 0.9 eV above the conduction band edge, thus establishing a metallic surface 2DEG, the carrier concentration of which is not dependent on bulk doping nor temperature. Our assumption of an accumulation space charge layer is in agreement to theoretical results of Van der Walle and Segev,10 since no post growth processes 2835
were performed in our experiments. For electron concentrations around 1019 cm-3, as determined for the present nanowires,12 the assumption of a degenerate volume, as in the HREELS data,9 seems reasonable. We therefore conclude that the major contribution to the conductivity is due to a cylinder-like 2DEG with metallic conduction properties and a minor contribution to the degenerate bulk region (Figure 1b). For the following description of the magnetotransport properties, we therefore concentrate on the conductance through the metallic cylinder-like 2DEG on the nanowire surface. Important additional information is based on recent temperature dependent studies of the phase-coherence length lφ performed on the same InN nanowires, which have been used in the present experiments.20 From the analysis of the characteristic fluctuation patterns in magnetoconductance phase-coherence lengths of electrons in the InN nanowires were derived to be about 400 nm at temperatures below 1.5 K. Note that this distance is about the length of the wires studied in this investigation. From the comparison of the phase-coherence length and the elastic mean free path of electrons in the InN nanowires, it is possible to estimate the type of transport and if coherent states along and around the wire may exist thus giving rise to quantization phenomena. The elastic mean free path le can be estimated from simple Drude formalism using le ) νF as Fermi-velocity and τe as mean elastic scattering time: le )
pkF Fne2
(2)
Here, kF ) m*νF/p is the Fermi wave vector with m* as the effective mass, F as the specific resistivity, and n as the threedimensional (3D) electron concentration. From the pinning position of the Fermi-level at the surface EF - EC ) 0.9 eV,9 the Fermi wave vector kF is estimated to be 13.6 × 106 cm-1, and the Fermi wavelength λF is estimated to be about 4.6 nm. Neglecting the effect of the bulk conductivity, a twodimensional (2D) electron concentration of n2D ≈ 3 × 1013 cm-2 within the surface 2DEG is estimated. Referred to the bulk, this amounts to a fictitious 3D electron concentration n of 2.4 × 1019 cm-3. This value fits nicely to average carrier concentrations which have been determined from photoluminescence data on the present InN nanowires.12 For a series of different nanowires the specific resistivity F was determined to vary around 5 × 10-4 Ω·cm at room temperature. From the above data, a mean free elastic path le is estimated according to (eq 2) to be at least 50 nm at low temperature. With lφ in the order of 400 nm, that is, the typical length of the nanowires, the electronic transport along the columns probably occurs in the diffusive-ballistic transition regime with fixed phase relations, while perpendicular across the column ballistic transport might be possible. We therefore can assume that around the wire circumference coherent electronic eigenstates can form within the surface 2DEG at low temperature, through which transport along the column might occur. In Figure 3, the magnetoconductance correction ∆G is plotted in units of e2/h for various temperatures. The conductance correction was determined from the original resistance measurements by subtracting the typical contact 2836
Figure 3. Magnetoconductance correction of wire A in units of e2/h at different temperatures ranging from 0.5 to 30 K.
resistance and the slowly varying background contribution. The magnetoconductance reveals a clear oscillatory behavior, in which the oscillation amplitude decreases with increasing temperature. Remarkably, even at temperatures as high as 30 K, the oscillations are clearly resolved. As can be seen in Figure 3, the oscillation amplitude is in the order of e2/h indicating that the carrier transport occurs in one-dimensional transport channels.21,22 By performing a fast Fourier transform (FFT) of ∆G, the corresponding frequency spectrum is obtained. For wire A, the most pronounced peak is found at 0.28 T-1, while a second peak appears at around 0.50 T-1 (Figure 4d). Note that at this temperature the height of the second peak is almost a factor of 2 lower than the first one. A similar behavior is seen for wire B. In order to confirm that these frequencies do indeed govern the oscillations in the magnetoconductance, for wire A, a back-transform of the FFT spectrum was performed, restricting to a small frequency range around the first and second peak, respectively (Figure 4d, inset). Obviously, the general shape of the experimental curve acquired for wire A is almost completely reproduced by adding the oscillations corresponding to the first and second peak of the FFT spectrum. The first peak found in the fast Fourier spectrum can directly be related to an oscillation with a period corresponding to a single magnetic flux quantum, given by Φ0 ) h/e. The frequency in the FFT spectrum is thus given by fFFT ) S/Φ0, with S ) πd2/4 the cross section of the nanowire. Using electron microscopy, we determined a diameter of d ) 37 ( 3 nm for wire A (Table 1). As can be inferred from the corresponding frequency range indicated in Figure 4d, the first peak at 0.28 T-1 is located within the frequency interval determined by the cross section corresponding to a diameter in the range of 37 ( 3 nm. The second peak at 0.50 T-1 falls within the interval belonging to a Φ0/2 flux periodicity. In order to discuss our experimental observations, namely, the predominant occurrence of the Φ0 periodicity, we will briefly introduce the theoretical framework which explains the carrier transport in a cylindrical-shaped conductor penetrated by an axially oriented magnetic field. The quasiballistic spiral-type motion of an electron along the 2DEG cylinder can be split into a free motion along z (direction of Nano Lett., Vol. 8, No. 9, 2008
Figure 4. (a-c) Magnetoconductance and (d-f) corresponding fast Fourier transform (FFT) of wire A, B, and C, respectively. All measurements were taken at 0.5 K. The gray vertical lines in (d,e) indicate the frequency range for the expected location of first and second peak for the experimentally determined wire diameters. (d,inset) Magnetoconductance of wire A at 0.5 K (lower curve). Back transform of the fast Fourier spectra of the first (1st) and second (2nd) peak shown in Figure 4d. The upper curve shows the sum of the back-transform corresponding to the first and second peak. The curves are offset for clarity.
cylinder axis) from one contact to the other and a circular motion within the surface 2DEG. The latter states enclose the magnetic flux Φ which is produced by the external magnetic field Bz along z. Neglecting spin effects, the Hamiltonian for this motion then reads Hk )
2 1 p2 ∂2 1 k - eB r 2 L z 2 z0 2m* ∂z2 2m * r02
(
)
(3)
with k L z the angular momentum operator in z direction for the circular motion and r0 the radius of the circular motion (approximately wire radius). With a product-ansatz for the electronic wave function Ψ(r0, φ)χ(z) and πr02Bz ) Φ as the magnetic flux through the 2DEG cylinder, we find the energy eigenvalues of eq 3 as El )
p2kz2
(
p2 Φ + l/ / 2 Φ 2m 2m r0 0
)
2
(4)
where l is the angular momentum quantum number for the circular motion around the wire and Φ0 ) h/e the magnetic flux quantum. The first term in eq 4 is the kinetic energy for ballistic motion (approximated) along z. The second term describes, as a function of magnetic flux Φ, a series of parabolas, each shifted against the other one by a flux quantum Φ0 (Figure 1c). Since l is the angular momentum (|z) quantum number, each parabola belongs to a specific angular momentum state. According to the Landauer-Bu¨ttiker model,23,24 the electronic transport occurs in one-dimensional channels, with each channel defined by the states with angular momentum l and the free motion along the z direction. (Figure 1c). Only those channels being located below the Fermi energy EF take part in the transport; that is, the conductance is determined by the number of states with angular momentum l located below EF. Each time a parabola crosses EF while Φ/Φ0 is increased, the conductance is changed by (2e. This situation is illustrated in Figure 1c. Here, for the sake of clarity, we only plotted a few levels being situated below EF. Because of the periodicity of the energies El (eq. 4) in Φ with a periodicity interval Φ0, Nano Lett., Vol. 8, No. 9, 2008
conductance variations with a period of Φ0 are expected. This periodicity is exactly found in the experiment. The fact that conductance variations are slightly smaller than 2e2/h can be attributed to backscattering, which was neglected in the model discussed above. For EF ) 0.9 eV, we estimate the number of occupied energy levels to be approximately 26. Except for l ) 0, at B ) 0 the energy states, El are 4-fold-degenerate, due to the spin degeneracy and due to the degeneracy of momentum states (l. It is interesting to note that a typical wire circumference of 126 nm (d ) 2r0 ) 40 nm) contains about 26 wavelengths λF ≈ 4.6 nm of electronic waves with energy EF. This enlightens from another angle of view the quantisation of the electronic states within the cylindrical surface 2DEG, through which electronic transport along the wire occurs. In the ideal case at zero magnetic field, each onedimensional channel contributes with 4e2/h to the conductance, owing to the 4-fold degeneracy. For 26 levels below EF, one ends up at a resistance of approximately 260 Ω. A comparison with the measured resistance of 3620 Ω, as given in Table 1, reveals that the latter value is considerably larger. We attribute this to backscattering along the wire, connected to the relatively small elastic mean free path of 50 nm.23,24 In a rigorous theoretical treatment of quantum transport in ballistic cylinder-like wires, Tserkovnyak and Halperin11 indeed calculated a quantum correction to the conductance of ∞
∆G )
∑g
s
s)-∞
(
exp
isBzπr02 Φ0
)
(5)
with gs as the spectral weight of the sth harmonic which contributes to the overall variation of the conductance G. The strongest contribution with s ) 1 in eq 5 exhibits the flux repetition frequency of a single flux quantum Φ0 as in eq 4 and as is observed in the present experiment. The second less intense frequency band in Figure 4 (wire A and B) might, therefore, be due to the second harmonic in eq 5 and/or to Al’tshuler-Aronov-Spivak oscillations which are super2837
imposed to the transport through coherent angular momentum states. The period Φ0 appears to be the same as in an Aharonov-Bohm experiment,25 but note that the present interpretation of the magnetoconductance oscillations is based on the periodicity of the energy eigenvalues induced by coherent angular momentum states rather than on interference between two separately propagating partial waves. The strong temperature-dependence of the conductance oscillations (Figure 3) also indicates that the magnetoconductance oscillations can indeed be attributed to phasecoherent transport. From previous transport measurements on InN nanowires in the diffusive regime, we found that the phase-coherence length is reduced by about a factor of 2 if the temperature is increased from 1 to 30 K.20 Further support of the interpretation provided above is given by measurements of two additional nanowires (wires B and C) with different geometrical dimensions compared with wire A (Table 1). The corresponding measurements of the magnetoconductance ∆G are shown in Figure 4b,c. Wire B has a diameter of 41 ( 3 nm, which is only slightly larger than the one of the previously discussed wire A. As can be seen in the fast Fourier spectrum (Figure 4e), the peak corresponding to a Φ0 periodicity is found at 0.27 T-1 being located at the lower boundary of the frequency interval expected from the geometrical dimensions. Interestingly, the peak corresponding to a periodicity of Φ0/2 is by a factor of 6 smaller than the first peak belonging to the Φ0 periodicity. This phenomenon might be due to the fact that wire B is considerably shorter than wire A, leading to a suppression of localization-related transport mechanisms, that is, Al’tshuler-Aronov-Spivak oscillations.26 As can be seen in Figure 4b, wire B shows a relatively large oscillation amplitude in the order of 2e2/h. The large amplitude might be due to the fact that this wire is only half as long as wire A, so that the back-scattering in the one-dimensional transport channel is reduced. This interpretation is also supported by the relatively small resistance found for this wire (cf. Table 1). In contrast to wire A and B, the magnetoconductivity pattern of wire C is relatively irregular (Figure 4c). This is confirmed by the FFT spectrum (Figure 4f), where in addition to the peak at 0.6 T-1 related to the Φ0 periodicity, a larger number of additional peaks is found. Owing to the considerably larger diameter of 58 ( 3 nm, the peak belonging to the Φ0 periodicity is found at a higher frequency. For this wire, no distinct peak corresponding to Φ0/2 is resolved. We attribute the substructure in the low-frequency range to universal conductance fluctuations,27,28 due to phase coherent transport in the presence of disorder.20,29 The enhanced degree of disorder compared with the other two wires might be due to the fact that for wire C the diameter is considerably larger, and thus lattice relaxation of the wire grown on a highly mismatched substrate is incomplete. In summary, the magnetoconductance of InN nanowires was studied. By means of temperature-dependent resistance measurements and by comparing the transport through a large
2838
number of nanowires of different size, we could confirm that the carrier transport mainly takes place in a tube-like surface two-dimensional electron gas. For InN nanowires with diameters below 60 nm, magnetoconductance oscillations with a very pronounced single flux quantum periodicity were found. We attribute the occurrence of the Φ0 periodicity to transport through coherent angular momentum states. Our investigations clearly demonstrate a new type of phasecoherent transport, which might be useful for the design of future phase-based quantum electronics. Acknowledgment. The authors thank Radoslav Nemeth for fruitful discussions on the quantum transport in nanocolumns. References (1) Thelander, C.; Agarwal, P.; Brongersma, S.; Eymery, J.; Feiner, L.; Forchel, A.; Scheffler, M.; Riess, W.; Ohlsson, B.; Go¨sele, U.; Samuelson, L. Mater. Today 2006, 9, 28–35. (2) Lu, W.; Lieber, C. M. J. Phys. D: Appl. Phys. 2006, 39, R387–406. (3) Ikejiri, K.; Noborisaka, J.; Hara, S.; Motohisa, J.; Fukui, T. J. Cryst. Growth 2007, 298, 616–619. (4) van Dam, J. A.; Nazarov, Y. V.; Bakkers, E. P. A. M.; Franceschi, S. D.; Kouwenhoven, L. P. Nature 2006, 442, 667. (5) Pauzauskie, P.; Yang, P. Mater. Today 2006, 9, 36–45. (6) Wensorra, J.; Indlekofer, K. M.; Lepsa, M. I.; Fo¨rster, A.; Lu¨th, H. Nano Lett. 2005, 5, 2470–2475. (7) Calarco, R.; Marso, M.; Richter, T.; Aykanat, A. I.; Meijers, R.; van der Hart, Stoica, T.; Lu¨th, H. Nano Lett. 2005, 5, 981–984. (8) Calarco, R.; Marso, M. Appl. Phys. A: Mater. Sci. Process. 2007, 87, 499–503. (9) Mahboob, I.; Veal, T. D.; McConville, C. F.; Lu, H.; Schaff, W. J. Phys. ReV. Lett. 2004, 92, 036804. (10) Van der Walle, C. G.; Segev, D. J. Appl. Phys. 2007, 101, 081704. (11) Tserkovnyak, Y.; Halperin, B. I. Phys. ReV. B 2006, 74, 245327. (12) Stoica, T.; Meijers, R. J.; Calarco, R.; Richter, T.; Sutter, E.; Lu¨th, H. Nano Lett. 2006, 6, 1541–1547. (13) Appenzeller, J.; Schroer, Ch.; Scha¨pers, Th.; Hart, A. v. d.; Fo¨rster, A.; Lengeler, B.; Lu¨th, H. Phys. ReV. B 1996, 53, 9959–9963. (14) Liang, W.; Bockrath, M.; Bozovic, D.; Hafner, J. H.; Tinkham, M.; Park, H. Nature 2001, 411, 665–669. (15) Tserkovnyak, Y.; Halperin, B. I.; Auslaender, O. M.; Yacoby, A. Phys. ReV. Lett. 2002, 89, 136805. (16) Bachththold, A.; Strunk, C.; Salvetat, J.-P.; Bonard, J.-M.; Forro´, L.; Nussbaumer, T.; Scho¨nenberger, C. Nature 1999, 397, 673–675. (17) Stoica, T.; Meijers, R.; Calarco, R.; Richter, T.; Lu¨th, H. J. Cryst. Growth 2006, 290, 241–247. (18) Wang, W.-M.; Chen, L.-C.; Chen, K.-H.; Ren, F.; Pearton, S. J.; Chang, C.-Y.; Chi, G.-C. J. Electron. Mater. 2006, 35, 738–743. (19) Chang, C.-Y.; Chi, G.-C.; Wang, Wei-Ming; Chen, L.-C.; Chen, K.H.; Ren, F.; Pearton, S. J. Appl. Phys. Lett. 2005, 87, 093112. (20) Blo¨mers, Ch.; Scha¨pers, Th.; Richter, T.; Calarco, R.; Lu¨th, H.; Marso, M. Appl. Phys. Lett. 2008, 92, 132101–132101. (21) Beenakker, C. W. J.; van Houten, H., Semiconductor Heterostructures and Nanostructures Solid State Physics; Ehrenreich, H., Turnbull, D., Eds.; Academic: New York, 1991; Vol. 44, p 1. (22) Datta, S. Electronic transport in mesoscopic systems; Cambridge University Press, Cambridge, 1995. (23) Landauer, R. IBM J. Res. DeV. 1957, 1, 223. (24) Bu¨ttiker, M. Phys. ReV. Lett. 1986, 57, 1761. (25) Aharonov, Y.; Bohm, D. Phys. ReV. 1959, 115, 485–491. (26) Al’tshuler, B. L.; Aronov, A. G.; Spivak, B. Z. Pis’ma Zh. Eksp. Teo. Fiz. 1981, 33, 101-03 [JETP Lett. 1981, 33, 94]. (27) Al’tshuler, B. L. JETP Lett. . 1985, 41, 648 [Pis’ma Zh. Eksp. Teo. Fiz. 1985, 41, 530]. (28) Lee, P. A.; Stone, A. D. Phys. ReV. Lett. 1985, 55, 1622. (29) Blo¨mers, Ch.; Scha¨pers, Th.; Richter, T.; Calarco, R.; Lu¨th, H.; Marso, M. Phys. ReV. B 2008, 77, 201301.
NL8014389
Nano Lett., Vol. 8, No. 9, 2008
