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Physica E 35 (2006) 332–337 www.elsevier.com/locate/physe
Quantum nanostructures of paraelectric PbTe G. Grabeckia,, J. Wro´bela, P. Zagrajeka, K. Fronca, M. Aleszkiewicza, T. Dietla,b,c, E. Papisd, E. Kamin´skad, A. Piotrowskad, G. Springholze, G. Bauere a
Institute of Physics, Polish Academy of Sciences, al. Lotniko´w 32/46, PL-02-668 Warszawa, Poland b Institute of Theoretical Physics, Warsaw University, Warszawa, Poland c ERATO Semiconductor Spintronics Project, Japan Science and Technology Agency, Japan d Institute of Electron Technology, al. Lotniko´w 32/46, PL-02-668 Warszawa, Poland e Institut fu¨r Halbleiterphysik, Johannes Kepler Universita¨t Linz, A-4040 Linz, Austria Available online 27 October 2006
Abstract This article provides a review of our results on nanostructurization of lead telluride, PbTe. This IV–VI group narrow-gap semiconductor exhibits paraelectric behaviour leading to a huge dielectric constant 41000 at helium temperatures. Because the Coulomb potential fluctuations produced by charged defects are strongly suppressed in PbTe nanostructures, one can reach the quantum ballistic regime at significantly relaxed conditions in comparison with other systems. In particular, we observe precise zero-field conductance quantization in the wires made of modulation doped PbTe/PbEuTe quantum wells where the heavily doped layer is separated from the conducting channel only by a 2 nm thick spacer layer. The second important property is the very large Zeeman splitting. It reaches 4 meV/T. Accordingly, significant spin splitting of the conductance plateaux is observed already at fields below 1 T. Therefore, the system is attractive for the construction of local spin filters. We show that the presence of metal layers does not impair the quantum ballistic properties. Furthermore, we have developed a new method of tuning the PbTe nanostructures, using laterally placed metallic electrodes. We have found that this method is more effective than previous schemes using used p–n junctions and it provides better stability of the nanostructures. r 2006 Elsevier B.V. All rights reserved. PACS: 73.23.Ad; 72.25.Dc; 73.40.Lq Keywords: Nanostructures; Ballistic transport; Spin polarized current; PbTe
1. Introduction Present state-of-the-art epitaxial growth and processing techniques enable one to fabricate semiconductor nanostructures whose dimensions are comparable to the de Broglie wavelength of band carriers. Typical examples are narrow point contacts whose conductance becomes quantized in 2e2/h units [1], as well as quantum dots revealing discrete energy levels, in close analogy to atomic levels [2]. Recently, much research effort has been devoted to quantum nanostructures in which controllable quantum states necessary for the hardware basis of quantum information and communication technologies are formed Corresponding author. Tel.: +48 22 843 53 24; fax: +48 22 843 09 26.
E-mail address:
[email protected] (G. Grabecki). 1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.08.022
[3,4]. In the present work we review our works on lead telluride-based nanostructuctures, PbTe. This IV–VI group compound is a narrow gap semiconductor with the rocksalt structure. It has a multivalley band structure with anisotropic conduction band valleys at the L point of the Brillouin zone. It possesses excellent semiconducting properties [5], namely high carrier mobility and controllability of the electron concentration. We will show that this material also offers several unique features which may be very useful from the point of view of quantum nanostructures [6–9]. The first and most important property of PbTe is its paraelectric character leading to a huge dielectric constant, ¼ 1350 at 4.2 K [10]. It is well known that one of the fundamental limits for nanostructuring is the discrete character of the electric charge. One of its consequences
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is the distortion of the nanostructure potentials by random potential fluctuations produced either by unintentional defects introduced during the processing or heteroepitaxial growth or by artificially incorporated doping impurities. The latter cannot be completely avoided as they provide the free carriers necessary for the device operation. Modulation doping, in which the doped region is spatially separated from the free carriers, has been utilized as a means to reduce the problem; however, even in this case, one still observes distinct effects caused by the potential fluctuations produced by the long-range tails of Coulomb potentials of the remote ionized dopants [11]. We have demonstrated [9] that in PbTe nanostructures the fluctuation amplitude is strongly suppressed, even in the presence of a significant concentration of charged impurities in the close vicinity of the device. In particular, we observe pronounced onedimensional (1D) quantization in a modulation doped PbTe wire separated from the heavily doped layer by only 2 nm. This is possible because the huge dielectric constant strongly diminishes contributions from single charged centres to the total potential, which as a result is smoothened. In other words, in PbTe the limits of nanostructuring imposed by granularity of the electric charge are significantly relaxed. The second property is the very small Coulomb charging energy. In particular, we have not observed any indications of Coulomb blockade in PbTe quantum dots [8]. Potentially, PbTe would offer the possibility of tuning the Coulomb charging energy independently of the device size. Such property is necessary for realization of the quantum entanglement in the quantum dots [12]. The third property results from the narrow-gap character of PbTe. For appropriate crystallographic orientations, PbTe is characterized by a very large magnitude of the Lande´ factor of electrons, g 60. Accordingly, the quantized conductance shows pronounced spin splitting of the plateaux already in magnetic fields below 1 T [7]. It is well known that if the Fermi energy is adjusted to the spin gap of the lowest 1D subband, the quantum point contact becomes a spin-dependent barrier. Therefore, it can be exploited to develop devices generating spin polarized current at a small spatial scale [13]. Importantly, in the case of PbTe, the necessary magnetic field is so low that it could be generated locally, e.g. by means of deposited micromagnets [14]. The fourth property of PbTe may also be useful for spin electronics. Namely, PbTe forms only very small barriers if connected with various metals. In particular, one expects that junctions between PbTe and superconductors would be of high transparency, enabling observations of Andreev reflection mediated transport [15]. It is well known that Andreev reflection is very sensitive to the spin polarization [16]. Placing the superconducting nanoelectrodes near the spin filter would allow for measurements of the spin polarization degree of the emitted current. Another interesting possibility is the fabrication of an electrically
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controlled source of entangled electron pairs as proposed in Ref. [17]. However, one disadvantage of the small metal–semiconductor barriers is that it is not possible to use the ‘‘splitgate’’ technique [1]. Instead, we have used etched trenches for the definition nanostructures and naturally occurring interface p–n junction as a gate electrode for tuning their properties [8]. However, such a gate does not provide good stability because of a large number of dislocations in the interface region [9]. Therefore, in the present work we demonstrate another gating method using an in-plane metallic gate electrode. In the following sections, we review the fabrication methods and show several experimental results illustrating these unique properties of PbTe nanostructures. 2. Initial multilayer fabrication and properties The multilayers used for fabrication of the nanostructures were grown by molecular beam epitaxy (MBE) onto (1 1 1) BaF2 substrates by using the protocols described in detail in Ref. [18]. As shown schematically in Fig. 1, in the structures, a PbTe quantum well resides between Pb0.92Eu0.08Te barriers. We have studied quantum wells with widths 12, 25 and 50 nm. For an Eu composition of 8%, the barrier is as high as 235 meV. Due to the (1 1 1) growth direction, the fourfold L-valley degeneracy of the conduction band in PbTe is lifted, so that the ground-state 2D subband is formed in a single valley with the long axis parallel to the [1 1 1] growth direction. Due to this orientation, the 2D electrons occupying the ground state are characterized by a very small effective mass, ¼ 0.021m0 [19]. In order to introduce electrons into the quantum well, modulation doping with Bi (NDE3 1018 cm3) was used with an undoped 2 nm wide Pb0.92Eu0.08Te spacer layer separating the quantum well and the doping layer. Standard transport measurements reveal total electron densities n2D from 11012 to 1 1013 cm2 and mobilities not exceeding 105 cm2/V s in the PbTe quantum wells at T ¼ 4:2 K. This corresponds to the electron mean free paths l e p3 mm.
Fig. 1. Schematic view of the initial multilayer used for nanostructurization of PbTe.
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It has to be stressed that in the quantum wells, the electron mobility is strongly reduced with respect to record values for the bulk-like PbTe epilayers of up to 2 106 cm2/ V s [18] despite the application of the modulation doping. This results mainly from the strong alloy scattering [20] at the PbTe/Pb0.92Eu0.08Te interfaces. Additionally, the system contains a significant number of threading dislocations formed during the initial growth on the BaF2 surface as a result of 4.2% lattice mismatch. These dislocations act as acceptors and account for the presence of a thin p+ interfacial layer. It is worth noting that this layer has been used as a gate electrode for tuning the nanostructures. The application of a thick buffer layer reduces the dislocation density, but it remains still significant in the quantum well region (at least at the level of 107 cm2) [21]. Furthermore, the difference in thermal expansion coefficients between the whole layer structure and the BaF2 substrate produces thermal strains of the order of 0.16% when the structure is cooled down to cryogenic temperatures [18]. Finally, due to the non-stochiometric composition, the background defect density would be as high as 1017 cm3. Obviously, any of the enlisted mechanisms would preclude the observation of conductance quantization in a standard material. However, as we show below, this is not the case for PbTe due to its paraelectric nature. 3. Nanostructure fabrication In our previous works [6–9], PbTe nanostructures were patterned by means of electron beam lithography and wet chemical etching. In Fig. 2, we show atomic force microscopy (AFM) images of several PbTe nanostructures: (a) Hall device, (b) quantum point contact and (c) threeterminal device. They were obtained in a single lithography step. For tuning their properties, we used biasing of the interface p–n junction [8]. On the other hand, Fig. 2(d) represents a PbTe wire with an in-plane metallic gate
Fig. 3. AFM profile taken across the PbTe wire (the same as in Fig. 2(d)). Thick vertical arrows mark the position of the wire. A pair of horizontal axes indicate position of the PbTe quantum well.
obtained in a two-step lithographic process [22]. In the first step, the trenches were defined in the same way as for the previous devices. It should be noted here that a direct evaporation of the metal on top does not work because the metal would short circuit to the underlying PbTe quantum well. Therefore, we have to remove the underlying material by etching the gate area down to the buffer layer. The etching depth of about 300 and 100 nm indium layer was evaporated into the etched region. Finally, the lift-off procedure was carried out. Because the etching process produces an undercut below the resist layer, the evaporated indium layer does not contact with the quantum well. Fig. 3 shows an AFM profile taken across the gate and the PbTe wire. Because the quantum well is positioned about 100 nm below the top surface, the indium layer lies about 200 nm below it. It is worth noting that the same method is applied for making In–PbTe interfaces for studying Andreev reflection mediated transport [23]. However, in that case, the etching process is stopped earlier, at the quantum well level, to provide an electrical contact between indium and PbTe. 4. Transport properties of PbTe nanostructures We start our review of experimental results on PbTe nanostructures with classical ballistic effects [24]. They are observed if the device size is much larger than the electron Fermi wavelength. A canonical example is the measurement of the bend resistance in a four-probe configuration performed on the Hall cross. For PbTe device shown in Fig. 2(a), we have obtained the result presented in Fig. 4. The occurrence of a deep negative peak can be easily understood in terms of the Landauer–Buttiker formalism [25]. The multiprobe resistance is described in terms of transmission coefficients Tij. They denote probabilities that an electron injected at probe j leaves the device at probe i: h (1) R21;34 ¼ ðT 32 T 41 T 31 T 42 Þ=D, 2e2
Fig. 2. AFM topography images of PbTe nanostructures: (a) Hall device, (b) a device containing 1 mm long quantum point contact (indicated by arrow), (c) three terminal device and (d) PbTe quantum wire equipped with a lateral gate.
where D does not depend on Tij (see also Fig. 2(a) for the probe assignment). The current flows through two adjacent contacts 1–2, and the voltage is measured on the remaining pair of contacts 3–4. Negative values of the resistance
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Fig. 4. Four-terminal resistance measured on the Hall device at different temperatures shown in Fig. 2(a). Inset represents the corresponding Hall resistance.
indicate that the second term of Eq. (1), describing straight transmissions between opposite probes, is dominating. This indicates ballistic motion of the electrons across the device at a length scale of about 3 mm. The effect is strongest at helium temperatures, but it persists up to 50 K. However, if the magnetic field is applied, the electron trajectories are bent by the Lorentz force towards the side probes and the positive value of the resistance is restored. The negative resistance is observed for fields smaller than 70.25 T corresponding to a cyclotron radius lc larger than 0.65 mm. This distance coincides well with the minimal width of the arms of the Hall cross. In the same field range the Hall resistance (shown in inset to Fig. 2) exhibits an anomaly. This too is a consequence of electron guiding into the side probes [24]. Observation of the quantum ballistic effects requires devices of dimensions comparable to the electron wavelength. A canonical example of such effects is conductance quantization in quantum point contacts [1]. In the case of PbTe, we have used the device shown in Fig. 2(b). It consists of a 1 mm long and 0.5 mm wide, deeply etched point contact. For tuning purposes, we have used the interface p–n junctions [8]. The conductance measured at millikelvin temperatures is shown in Fig. 5. Despite the very thin spacer layer and presence of background defects, the electron transmission is almost hundred percent. This is possible because of the suppression of nanoscale Coulomb potential fluctuations by the huge static dielectric constant. Therefore, PbTe nanostructures are robust against charged defects in their vicinity, in contrast to any other known systems. The only influence of the charged defects is to shift the threshold voltage [9]. The above result is important because it demonstrates the usefulness of PbTe for studies of quantum ballistic phenomena.
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Fig. 5. Zero-field conductance quantization in PbTe point contact (indicated by arrow in Fig. 2(b)).
Fig. 6. Resistance of the three-terminal device (of the geometry shown in Fig. 2(c)) as a function of gate voltage. The subsequent curves were taken for different values of the magnetic field. Note different vertical scales for two panels corresponding to opposite field directions.
Three-terminal nanostructure junctions with the geometry shown in Fig. 2(c) have been suggested as basic components for realization of solid-state electron entanglement [17]. Because such studies are planned with PbTe devices, we have checked their basic transport properties. The arms denoted 2 and 3 are much thinner than the arm 1 and, to induce conductance through them, one needs to apply a positive gate voltage of at least +0.15 V. Fig. 6 plots the three-terminal resistance, when the current is passed between 1 and 2 and the voltage is measured between 1 and 3. Measurements were performed as a function of Vg at different magnetic fields. The data below the threshold Vg are out of phase and thus not valid. There is a very strong asymmetry with respect to the magnetic field direction (left and right panels). It may be understood
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in terms of the magnetic field dependent transmission between the probes. According to the analysis given in Ref. [26], the measured voltage U13 is proportional to U 13 /
T 32 . T 31 þ T 32
(2)
At high magnetic fields, the electrons injected from a lead are forced to go down the adjacent left or right lead, dependently on the field direction. Therefore, T32 changes between 0 and 1 if the field direction is changed. Note that for any field direction either T31 or T32 equals 1. For the negative B-field direction, we observe pronounced quantum Hall plateaux. The results obtained for the positive field direction in some sense corresponds to the longitudinal resistance. However, it contains some fielddependent contribution from the large contact areas. The next results have been obtained on the newly developed sample with the in-plane metallic gate, shown in Fig. 2(d). In this case, a PbTe quantum well with a thickness of 25 nm has been used. At V g ¼ 0 the wire was fully depleted, therefore one had to apply positive Vg to induce electrons into it. Using the interface, p–n junction would lead to occurrence of the leak due to forward biasing of the junction. In consequence, large conductance hystereses were observed when Vg was swept up and down. Furthermore, the sample properties gradually deteriorated with time. Applying positive Vg to the in-plane gate and simultaneously grounding the p+ interfacial layer avoided these inconveniences because the p–n junction was kept at the reverse bias. Then the wire conductance has been stable and we can perform time-consuming conductance measurements as a function of multiple parameters. Fig. 7 shows the nonlinear differential conductance as a function of the DC source–drain bias voltage, Vsd. The gate voltage increases by 4.6 mV by each Vsd sweep. Plateaux appear as an accumulation of traces. They are also indicated by arrows. Close to V sd ¼ 0 we observe usual
Fig. 8. Shadow contour plot of transconductance dG/dVg as a function of gate voltage and perpendicular magnetic field. The dark stripes represent thresholds of the subsequent 1D levels. Arrows denote the spin polarization of the levels.
plateaux corresponding to the conductance quantization. However, in the absence of the magnetic field, their height is lowered with respect to the quantized values n 2e2 =h. We suggest that the transmission is lowered in the slightly curved wire (Fig. 2(d)). At large Vsd values, we observe additional ‘‘half-plateaux’’. This nonlinear effect appears when the number of conducting subbands for the two directions of transport differs by 1 [27]. If the magnetic field is applied, the quantization becomes accurate and a pronounced spin-resolved plateau of the bottom 1D level is observed. For higher Vg we observe complex behaviour due to contribution of 1D levels originating from excited 2D subbands of the starting PbTe quantum well. This is possible because the energy of the first excited subband in the 25 nm wide well is estimated to be 6 meV. In a similar manner, we have mapped the conductance as a function of gate voltage and magnetic field. The result is shown in Fig. 8. The black shadow stripes indicate the occupation thresholds of the subsequent 1D subbands. Because the subband energy is a monotonic function of the gate voltage, the stripes reflect behaviour of the energy levels as a function of B. Similarly to the previously published data [7], we observe very large spin splitting. For the bottom 1D level, the upper spin branch crosses the first excited 1D level at about 2.5 T. This result confirms spinfiltering capabilities of PbTe. Surprisingly, for Vg40.30 V, one cannot resolve higher 1D subbands (left upper corner of Fig. 8). This is probably caused by 1D level broadening due to the contribution of subbands formed from oblique valleys. 5. Summary
Fig. 7. Traces of differential conductance measured on the wire represented in Fig. 2(d) vs. the applied source–drain voltage Vsd, taken during a slow increase of the gate voltage, by +4.6 mV per each trace.
PbTe opens new and interesting possibilities for studying nanostructure physics. This material possesses unique features not met in other semiconductors. The most
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important is paraelectric behaviour, which causes a huge dielectric constant and suppresses the nanoscale Coulomb potential fluctuations. Therefore, PbTe offers relatively easy access to studies of the quantum ballistic phenomena. PbTe is characterized by a very large Zeeman splitting which is desirable for spin-filtering devices. Additionally, high transparency of PbTe interfaces with superconducting metals would be useful for constructing electric detectors of the spin currents. Finally, the development of a new method of tuning based on in-plane metallic gates strongly enhances the usefulness of this system for future research. Acknowledgements This work has been supported by ERATO Semiconductor Spintronics project of Japan Science and Technology Agency. References [1] B.J. van Wees, H. van Houten, C.W.J. Beenakker, J.G. Williamson, L.P. Kouvenhoven, D. van der Marel, C.T. Foxon, Phys. Rev. Lett. 60 (1988) 848; D.A. Wharam, T.J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J.E.F. Frost, D.G. Hasko, D.C. Peacock, D.A. Ritchie, G.A.C. Jones, J. Phys. C 21 (1988) L209. [2] S. Tarucha, D.G. Austing, T. Honda, R.J. van der Hage, L.P. Kouwenhoven, Phys. Rev. Lett. 77 (1996) 3613. [3] D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120. [4] R. Hanson, L.H. Willems van Beveren, I.T. Vink, J.M. Elzerman, W.J.M. Naber, F.H.L. Koppens, L.P. Kouwenhoven, L.M.K. Vandersypen, Phys. Rev. Lett. 94 (2005) 196802. [5] For review of PbTe properties see D. Khokhlov (Ed.), Lead Chalcogenides: Physics and Applications, Taylor & Francis, New York, London, 2003. [6] G. Grabecki, J. Wrobel, T. Dietl, K. Byczuk, E. Papis, E. Kaminska, A. Piotrowska, G. Springholz, M. Pinczolits, G. Bauer, Phys. Rev. B 60 (1999) R5133. [7] G. Grabecki, J. Wrobel, T. Dietl, E. Papis, E. Kaminska, A. Piotrowska, G. Springholz, G. Bauer, Physica E 13 (2002) 649.
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