Interplay of Coulomb Blockade and Luttinger-Liquid Physics in Disordered 1D InAs Nanowires with Strong Spin–Orbit Coupling

R. Hevroni, V. Shelukhin, M. Karpovski, M. Goldstein, E. Sela, A. Palevski and Hadas Shtrikman

Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel Aviv, 69978, Israel

Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, 76100, Israel

1 Introduction

Ballistic 1D nanowires (NWs) with strong spin–orbit coupling are theoretically predicted¹ to exhibit nonmonotonic (up and down) conductance steps of size G₀ = e²/h as the electron density is varied by the gate voltage V_G. Although many attempts have been made to measure these conductance steps, they have not been observed yet in either InAs or InSb NWs. This indicates that disorder plays an essential role, preventing the motion of the electrons between the contacts from being ballistic. It is well known that in 1D systems, electron–electron interactions, described by the Luttinger-liquid (LL) model, amplify the role of disorder significantly, causing the conductance to vanish at zero temperature even for very weak disorder.² Experimentally, however, the effects of the interactions in NWs with strong spin–orbital scattering have not yet been reported.

In this chapter we report on experimental studies of the Coulomb blockade in disordered InAs NW at low temperatures. We demonstrate that sequential tunneling is strongly affected by electron–electron interactions. The analysis of the temperature dependence of the conductance and of the lineshape of the sequential tunneling in the Coulomb blockade regime within the framework of the existing theories allows us to deduce the corresponding LL parameter g. We show that in our NWs the effective LL parameter reaches a value less than 1/2, leading to a decrease in the Coulomb blockade peak-to-valley difference as the temperature is reduced. To the best of our knowledge, this phenomenon, predicted by the LL model, has never been experimentally observed before. While there were a number of experimental papers^{3 4} in which the Coulomb peaks decreased with decreasing temperature, this behavior was sporadic, that is, did not occur for consecutive peaks. Thus, these previous results do not follow the predictions of the LL theory but are rather consistent with stochastic Coulomb blockade,⁵ while the opposite is true for our results, as we discuss below.

2 Sample preparation and the experimental setup

Our InAs NWs, approximately 2 µm long and 50 nm in diameter, were grown by Au-assisted vapor–liquid–solid MBE on a 2 in. SiO₂/Si substrate. A ∼1 nm gold layer was evaporated in situ in a chamber attached to the MBE growth chamber after degassing the substrate at 600 °C. The substrate was heated to 550 °C after being transferred to the growth chamber to form gold droplets, then cooled down to the growth temperature of 450 °C. Indium and As₄ were evaporated at a V/III ratio of 100. The NWs were studied by SEM and TEM and were found to have a uniform morphology with no tapering and a pure wurtzite structure with a negligible number of stacking faults.⁶ The NWs were deposited randomly from an ethanol suspension onto 300 nm-thick SiO₂ thermally grown on a p⁺-Si substrate, to be used as a back gate. The NWs were then mapped with respect to alignment marks using optical microscopy and e-beam lithography and evaporation were used to deposit Ti/Al (5/100 nm) contact leads. A short dip in an ammonium polysulfide solution was used for removing the oxide from the InAs NWs surface prior to contact deposition.⁷ The leads were separated by ∼650 nm (see Fig. 1).

Generally, InAs NWs are highly sensitive to surface impurities and other imperfections (such as surface steps and dangling bonds) since the conductance electrons are near the surface. Hence, impurities resulting from sample fabrication and the external environment, as well as the substrate on which the sample is placed, induce disorder potential barriers.

Conductance was measured by a four-terminal method using a low-noise analog lock-in amplifier. The current was passed between two probes (I₊ and I₋ in Fig. 1), while the voltage was measured by two different probes (V₊ and V₋ in Fig. 1). It should be noted that I and V probes are connected to the NW at the same point, so that the contact resistance is always included in the conductance measured. The measurements were done in a cryogenic system in the 1.7–4.2 K temperature range.

3 Experimental results

Figure 2 shows the measured conductance G as a function of gate voltage V_G of an InAs NW. The as-grown NWs are conducting and the gate voltage bias required to pinch off the conductance is V_G = −0.35 V. A series of distinct conductance peaks is clearly observed, with a typical spacing of δV_G ∼ 25 mV. At lower temperatures, the conductance peaks become sharper, but the peak conductance values are reduced. This behavior indicates the occurrence of a Coulomb blockade, similar to quantum dots.

The peak spacing in our device is shown in Fig. 3. It is well known that the conductance peaks in quantum dots are equally spaced if the energy level spacing Δ between single electron states in the dot is negligible relative to the charging energy E_C. In the opposite limit, the conductance peaks are irregularly spaced,⁸ which is the case in our NW sample since the distance between the peaks varies by over 50%. In this regime every peak corresponding to an odd number of electrons in the dot should be separated from the previous one by a roughly constant value proportional to E_C, whereas the next peak should be separated by value proportional to (E_C + Δ) that varies from level to level. Indeed, we see that every second peak of the first 10 peaks has a gate voltage spacing of δV_G = 25 mV, with the exception of the distance between the fifth and sixth peaks that is slightly lower (∼21 mV). This δV_G value should thus correspond to the charging energy, and yields a value of C_G = 6.4 × 10⁻¹⁸ F for the gate capacitance. Since the geometry of the sample and its dimensions are known, we can estimate the size L_QD of the quantum dot from the expressions for the capacitance of a cylinder in the vicinity of a conducting plate, C_G = 2πL_QD/ln(4d/D_NW), where is the dielectric constant, D_NW is the NW diameter, and d is the SiO₂ thickness. Substituting the values of the sample dimensions, the average dielectric constant of ⁴He and SiO₂ ( = 2.5₀) and the estimated value of the capacitance give L_QD ≈ 200 nm.

We see that L_QD is smaller than the NW length (L = 2 µm) by an order of magnitude, and smaller by more than a factor of 3 compared to the 650 nm separation between the voltage leads. Thus it is legitimate to assume that the QD is formed as a puddle of 1D electrons separated by two barriers on both sides, lying somewhere in the NW segment between the leads. In addition, we see that the segments of the NW to the right and left of the dot are long enough so that their single-particle level spacing and charging energies are well below the temperatures reached in our experiment, allowing us to describe them as infinite 1D leads. In such a case it is reasonable to carry out our data analysis in the framework of the theory of tunneling between two 1D NWs through a quantum dot.

Resonant and sequential tunneling were well studied theoretically and experimentally in the past for both interacting and noninteracting 1D electrons, see, for example, Ref. 2 and references therein. In our system the peak widths are found to scale linearly with the temperature T. We thus try to explain our results using Furusaki's expression⁹ for the conductance due to sequential electron tunneling in a QD connected to LL leads. The lineshape of a single conductance peak as a function of the energy E (distance from the peak) is then

1

where A is a constant related to the asymmetry and height of the barriers defining the dot, the factor γ(T) ∼ T^1/g−1 accounts for the renormalization of the tunneling rates by the LL effects, and Γ(z) is the gamma function. Note that the temperature variation of G at the peak then becomes

2

In all the above expressions, g is the effective LL interaction parameter; g = 1 for a noninteracting NW and decreases (g < 1) with increasing repulsive interactions. It is a combination of the charge and spin interaction parameters, as we discuss below.

The experimental data in Fig. 2 shows that both the height and the width of the conductance peaks decrease as temperature is reduced. Thus, the interaction parameter g should be smaller than 1/2. Our experimental data in Fig. 4 shows that indeed the temperature dependence of the height of each peak can be well described by the power law, Eq. (2), from which we can deduce the value of g for each peak. For the first two peaks, we find g = 0.38 ± 0.03.

In order to verify that the lineshape of the Coulomb blockade peaks as a function of V_G can be described by Eq. (1), we fitted the gate voltage dependence of our data to a sum of terms (one for each peak) of the form of Eq. (1), with E = α(V_G − V₀), where V₀ is the gate voltage value at the peak and α is the ratio between the gate capacitance and the total capacitance of the dot. Parameters α and A are used as fitting parameters, and for g we plug in the value extracted from the data analysis presented in Fig. 4. The result for the first three peaks is shown in Fig. 5.

We find that, as expected, α = 0.1 ± 0.005 does not vary between the peaks and/or as function of temperature, indicating that Eq. (1) indeed gives a consistent description of our data set.

We have performed a similar fitting procedure for peaks #17–19 (see Figs 6 and 7). We find g ∼ 0.5. It is indeed expected that the interaction constant should increase as the Fermi energy E_F increases, since g depends on the ratio between the Coulomb energy U and the Fermi energy in the NW.

As we pointed out earlier, the reduction of the conductance peaks at low temperatures has been observed in previous experiments^{3 4} but with markedly different results. In previous experiments, the decrease was sporadic, occurring only for nonconsecutive peaks, and thus cannot be accounted for by the LL picture, but rather indicates a stochastic Coulomb blockade.⁵ In contrast, in our system the peak reduction occurs in a similar way for several consecutive peaks.

Now we address the question of why in our InAs NWs the effective LL parameter g is smaller than 1/2 at low filling (so that G_MAX decreases with decreasing temperature), while other experimental studies of 1D quantum wires, for example, carbon nanotubes,¹⁰ GaAs wires formed at the cleaved edge overgrowth of a GaAs/AlGaAs heterostructure,¹¹ or V-groove GaAs NWs,¹² all exhibit effective LL parameters higher than 1/2 (so that G_MAX increases with decreasing temperature). We believe that there are two main reasons that contribute to the lower value of the LL parameter in our InAs NWs.

The first is related to the environment of the quantum wires. Both types of GaAs wires reported in the literature^{11 12} were created within 2DEG structures embedded well inside a semiconductor material with a large dielectric constant (AlGaAs and GaAs). In contrast, while the InAs NWs have a similar dielectric constant to GaAs, they are placed on SiO₂ surface, so the surrounding materials, namely liquid ⁴He and SiO₂, possess much smaller dielectric constants. These reduced dielectric constants enhance the effect of Coulomb interaction in our system as compared to the GaAs wires reported before.

The second reason for observing a smaller LL parameter in our system is related to an inherent property of InAs – the strong spin–orbit coupling that breaks the spin rotation symmetry. The effective LL parameter g is related to the interaction parameters in the charge and spin channels, g_C and g_S, respectively, by⁹

3

In GaAs, the spin–orbit coupling is very small; therefore, spin-rotation symmetry dictates that g_S = 1. Thus, g < 1/2 can only be obtained if the interaction in the charge sector is extremely strong, g_C < 1/4. In carbon nanotubes, the additional valley degeneracy results in 1/g = 1/(4g_C) + 3/4, so reaching g < 1/2 requires an even stricter condition g_C < 1/5. On the other hand, in InAs spin rotation symmetry is broken, allowing for g_S < 1 and making it easier to reach g < 1/2.

4 Conclusion

We believe that the combination of a lack of orbital degeneracy due to strong spin–orbit coupling and of the low effective dielectric constant makes our InAs NW a unique system where strong effective interactions, g < 1/2, can be achieved, and thus a decrease in Coulomb blockade peak heights with decreasing temperature can be observed.

Acknowledgments

We are thankful to Ronit Popovitz-Biro for professional TEM study of the InAs NWs. We gratefully acknowledge support by the ISF BIKURA program and GIF (MG); ISF and Marie Curie CIG grants (ES); as well as ISF grant #532/12 and IMOST grants #3-11173 (AP and HS) & #3-8668 (HS).

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