Anisotropic g-Factor and Spin-Orbit Field in a Ge Hut Wire Double Quantum Dot
Ting Zhang, He Liu, Fei Gao, Gang Xu, Ke Wang, Xin Zhang, Gang Cao, Ting Wang, Jian-Jun Zhang, Xuedong Hu, Hai-Ou Li, Guo-Ping Guo
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Anisotropic π -Factor and Spin-Orbit Field in a Ge Hut Wire Double Quantum Dot Ting Zhang,
He Liu,
Fei Gao,
3,
Gang Xu,
Ke Wang,
Xin Zhang , Gang Cao,
Ting Wang, Jian-Jun Zhang, Xuedong Hu , Hai-Ou Li, and Guo-Ping Guo CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Institute of Physics and CAS Center for Excellence in Topological Quantum Computation, Chinese Academy of Sciences, Beijing 100190, China Department of Physics, University at Buffalo, SUNY, Buffalo, New York 14260, USA Origin Quantum Computing Company Limited, Hefei, Anhui 230026, China
These authors contributed equally to this work * Corresponding author. Emails: [email protected] (H.-O. L.); [email protected] (G.-P.G.).
Abstract
Holes in nanowires have drawn significant attention in recent years because of the strong spin-orbit interaction, which plays an important role in constructing Majorana zero modes and manipulating spin-orbit qubits. Here, we experimentally investigate the spin-orbit field in a hole double quantum dot in germanium (Ge) hut wires by measuring the leakage current in the Pauli spin blockade regime. By varying the applied magnetic field and interdot detuning, we demonstrate two different spin mixing mechanisms: spin-flip cotunneling to/from the reservoirs, and spin-orbit interaction within the double dot. By rotating the direction of the magnetic field, we observe strong anisotropy of the leakage current at a finite magnetic field induced by the spin-orbit interaction. Through numerical simulation, we extract the full π -tensor and find that the direction of the spin-orbit field is in-plane, and at an azimuth angle of 59 Β° to the nanowire. We attribute our observations to the existence of a strong spin-orbit interaction along the nanowire, which may have originated from the interface inversion asymmetry in our system. Keywords:
Ge hut wires, hole double quantum dot, Pauli spin blockade, spin-orbit interaction, π -tensor Main text
The spin-orbit interaction (SOI) describes the coupling between the motion of a carrier and its spin. When an electron or hole moves in an electric field, it experiences an effective magnetic field that couples to its spin degree of freedom, even in the absence of an external magnetic field . Recently, studies on the SOI have received widespread attention because of its fundamental role in both classical spintronics and in the quantum coherent manipulation of a spin qubit , and more recently in the search for Majorana fermions . The SOI can be exploited for all-electrical control of a spin qubit using electric dipole spin resonance (EDSR) techniques . Without generating a local ac magnetic field or making use of an external source of spin-electric coupling (such as micromagnets), the device design is simplified by using a local ac electric field. In addition, as proposed by theory and confirmed by recent experiments, a strong SOI is required to create Majorana fermions in semiconductor-superconductor heterostructures . Thus, the potential applications of the SOI for quantum information processing will be particularly relevant in the future, especially with respect to fast operation speed , scalability , and topological quantum computation . At the mean field level, SOI can be expressed as an effective magnetic field π΅β SO . Here, the amplitude of π΅β SO reflects the strength of the SOI, which is related to the Rabi frequency in EDSR , and the direction of π΅β SO determines the suitable geometries in experimental setups. For example, the special geometry π΅β β₯ π΅β SO is assumed in proposals for realizing Majorana fermions in nanowires with proximity-induced superconductivity . Furthermore, EDSR is efficient when the externally applied magnetic field π΅β is perpendicular to π΅β SO . However, under such conditions, the degree of mixing between the singlet and triplet states in a double quantum dot also reaches a maximum, which is expected to be proportional to |π΅β SO Γ (π΅β π΅β )| . The unwanted transitions from T + (1,1) and T β (1,1) to S(0,2) induced by the SOI hybridization have an adverse effect on the fidelity of the initialization and readout of spin qubits , and could speed up decoherence . SOI is generally relatively weak for conduction electrons in group IV materials such as Si and Ge. In comparison, the valence-band holes in one-dimensional systems are predicted to have a much stronger SOI called the βdirect Rashba spin-orbit interactionβ (DRSOI) . To establish the feasibility of hole-based spin qubits, and to possibly observe novel phenomena such as the Majorana zero mode, it is imperative to investigate the spin-orbit field (SOF), in terms of both its magnitude and its direction. To date, by analysing the anisotropic effects of the SOI, the SOF direction has been determined in various semiconducting nanostructures such as hole quantum dots (QDs) in the GaAs/AlGaAs heterostructure , silicon MOS , and electron QDs made from InAs nanowires . However, experimental determination of the SOF direction, which is related to the DRSOI in hole nanowires, remains lacking. In this Letter, we extract the anisotropic π -tensor and determine the SOF direction in an electrostatically defined double quantum dot (DQD) in a Ge hut wire (HW) by leakage spectroscopy in the Pauli spin blockade (PSB) region. First, we measure the leakage current as a function of the magnetic field along different directions, which can be explained by a combination of spin-flip cotunneling to the reservoirs and SOI within the DQD. The effective π -factors in-plane can be determined by fitting to the current peak induced by spin-flip cotunneling. Then, the magnetic field is fixed at a value dominated by the SOI. We study the anisotropic effects of the leakage current and extract the direction of π΅β SO , which is in-plane and at a 59 Β° angle to the nanowire, indicating the existence of other SOI mechanisms in addition to the DRSOI. The DQD used in the experiment is fabricated from Ge HWs that are monolithically grown on a Si substrate . Figure 1a shows a scanning electron microscopy (SEM) image of the device comprising five electrodes above a 1- ΞΌm -long nanowire. After a short oxide removal step with buffered hydrofluoric acid, 30-nm-thick palladium contacts are patterned with electron beam lithography. A 20-nm aluminium oxide layer is then deposited as a gate dielectric using atomic layer deposition. Finally, three 30-nm-wide top gates consisting of 3-nm titanium and 25-nm palladium are fabricated between contact pads, spaced at a 40 nm pitch. A 3D schematic of the device is depicted in Figure 1b. By applying voltages to three top gates to create a confinement potential, a DQD is defined along the y -axis (parallel to the Ge HW), as shown in Figure 1c. Transport measurements of the DQD are performed in a dilution refrigerator equipped with a vector magnet at a base temperature of 15 mK. Figure 1. (a) False-coloured scanning electron microscopy (SEM) image of a hole DQD in a Ge HW. (b) 3D schematic of the device, which shows a Ge HW contacted with source and drain electrodes and covered by three top gates (L, M, R). The aluminium oxide layer separating the HW from the gates is not shown. The DQD is formed between two top gates by applying voltages to three grid electrodes. (c) Schematic of the confinement potential created along the nanowire by a gate voltage and the formation of a DQD. Definition of the angles and presentation of the vector for the SOF. (d) Charge stability diagram of the DQD: source-drain current πΌ SD measured as a function of π L and π R with π SD = 2.5 mV . The region marked by the red dashed circle is investigated in detail in the following. The operation of the DQD is demonstrated by the charge stability diagram shown in Figure 1d. We measure the source-drain current πΌ SD as a function of gate voltages π L and π R at a bias voltage π SD = +2.5 mV . Note that only when the energy levels corresponding to transitions of the two QDs both enter the bias window can the current through the DQD be detected, leading to an array of characteristic bias triangles . We focus on the bias triangle highlighted by the red dashed circle at the transition between the (1,1) and (0,2) states in Figure 1d. Here, the first and second numbers refer to the effective hole occupation of the left and right dots, respectively. Figures 2a and 2b display zoom-ins of this triangle for positive and negative source-drain biases. The corresponding line traces along the detuning energy π (green arrow) are plotted in Figure 2c. In comparison, we observe a strong suppression of current at π SD =+2.5 mV , which is attributed to the forbidden transition from T(1,1) to S(1,1) due to spin conservation during hole tunneling (Figure 2d), the characteristic signature of PSB . For π SD = β2.5 mV , a large current is measured in the whole triangle. Figure 2. (a) Zoom-in plot of the measurements in the region marked by the red dashed circle in Figure 1d. The suppression of current πΌ SD at the base of the bias triangle is the signature of PSB. The green arrow indicates the direction of the detuning energy π . (b) Corresponding measurements made in the same region as in panel a but with the opposite bias π SD = β2.5 mV . Reversing the bias results in an enhancement of the baseline current. (c) Comparison of the current from two line cuts along the detuning energy for positive (red triangles) and negative (blue circles) bias voltages. (d) Schematic presentation of PSB for a hole DQD. Transport is blocked for the transition (1,1) βΆ (0,2) due to the Pauli exclusion principle. The blockade can be lifted by spin mixing mechanisms such as hyperfine interaction with the nuclear spin bath of the host lattice , spin-flip cotunneling , a π -factor difference in the DQD and the SOI . The dominant spin mixing mechanism can be investigated based on the leakage current in the PSB region. Figure 3 shows a measurement of the leakage current as a function of detuning energy π and magnetic field π΅β applied in different directions. Here, π is tuned by sweeping π L and π R simultaneously along the green arrow in Figure 2a, and the top panels of Figure 3 depict the direction of the applied magnetic field π΅β . Figure 3a shows the measured leakage current πΌ SD as a function of detuning energy π and magnetic field π΅ π§ . The leakage current πΌ SD in the PSB region increases monotonically as π΅ π§ increases. A line cut along π = 0 is plotted (red open circles) in Figure 3d. The PSB is lifted at a finitely large magnetic field, and the current profile shows a broad dip, reflecting the presence of a spin-nonconserving transport mechanism. In Ge HWs, the strong SOI leads to hybridization of the T(1,1) triplet and S(0,2) singlet, which allows the previously forbidden T(1,1) βΆ S(0,2) transition to be lifted at a finite magnetic field . In the simple physical picture, spins are oriented along the direction of the SOF π΅β SO , so an external magnetic field π΅β applied perpendicular to π΅β SO causes the spin to precess around π΅β . This rotates the spin and enables spin-flip tunnelling to lift the PSB, leading to the leakage current spectroscopy shown in Figure 3a. However, when the external magnetic field rotates perpendicularly and parallelly to the nanowire in the plane, we find a different field-dependent behaviour of the leakage current, as shown in Figures 3b and 3c. From the line cut at π = 0 (Figures 3e and 3f), the current profile is composed of a peak at approximately |π΅β | = 0 and a broad dip at a relatively large magnetic field, which is induced by two different spin mixing mechanisms. For the broad dip in the leakage current line shape, we find that the current increment induced by the SOI with the in-plane magnetic field is much smaller than that with the out-of-plane magnetic field. Although no saturation of leakage current is observed as before , we can demonstrate that the widths of the dip are dramatically different in different directions. We attribute this phenomenon to the anisotropy of the π -factor in our hole system and the change in the angle between π΅β and π΅β SO , which will be discussed in detail below. Furthermore, because the width of the peak is approximately 200 mT, we rule out the hyperfine interaction and consider spin-flip cotunneling to be the dominant mechanism for lifting of the PSB at zero magnetic field. When the holes in the DQD form a spin-triplet state T(1,1), one of the spins can flip through inelastic cotunneling between the dot and its nearest lead, leading to an enhanced leakage current. The spin-flip rates due to cotunneling from the spin-polarized triplet states are exponentially suppressed when the Zeeman energy is large compared to the thermal broadening of the hole states in the leads . Therefore, the leakage current induced by spin-flip cotunneling has a maximum at B=0 and falls to zero at a finite magnetic field, giving rise to a peak in the leakage current line shape. Figure 3. (a-c)
Leakage current πΌ SD through the DQD in the PSB regime measured as a function of the detuning energy π and magnetic field π΅β . Panels a, b, and c are for the measurements (colour plots) with π΅ π§ , π΅ π¦ , and π΅ π₯ , respectively, as shown in the schematics depicted in the top panels. The nanowire is in the x-y plane (substrate plane) along the y-axis. π is tuned by sweeping π L and π R simultaneously along the green arrow in Figure 2a. (d-f) Leakage current measured at π = 0 (line cuts of the measurement data along zero detuning energy indicated by the yellow bars in panels a, b, c) as a function of the magnetic field for the three different directions. The black solid lines represent the best fits to the theory of eq (1). Theoretically, the leakage current πΌ leak through a DQD in the PSB regime at a finite magnetic field |π΅β | can be approximated by πΌ leak = β π B π΅3sinh πβπBπ΅πBπ + πΌ
SO0 π΅ π΅ +π΅ C2 + πΌ B . (1) The first term on the right-hand side represents the spin-flip cotunneling-induced leakage current, with π = βπ [{Ξ l Ξβ } + {Ξ r (Ξ β 2π M β 2ππ SD )β } ], where π is the hole charge, π β is the effective π -factor, π B is the Bohr magneton, β is Plankβs constant, π B is the Boltzmann constant, T is the hole temperature, Ξ l,(r) is the tunnel coupling with the left (right) reservoir, Ξ is the depth of the two-hole level, and π M is the mutual charging energy . The second term on the right-hand side describes the leakage current resulting from the SOI, which depicts a Lorentzian-shaped dip with a width π΅ C . The last term πΌ B on the right-hand side is an overall background current caused by the other spin mixing mechanisms, such as a difference in the π -factor between the two dots. Here, we note that because spin-flip cotunneling lifts the PSB at small magnetic fields while the SOI lifts the PSB at relatively large magnetic fields, the leakage current induced by the interplay of these two mechanisms is expected to be small and is thus ignored in eq (1). The black solid lines in Figures 3d-3f show the fitting to experimental data of the leakage current at π = 0 with eq (1). Clearly, the theory agrees well with the experiment. From the fitting to the current peak induced by spin-flip cotunneling, we can extract the values of the effective π -factor π π₯β = π π¦β = 1.2 Β± 0.2 with a hole temperature T = 40 mK. The obtained π -factors for a magnetic field applied in-plane are consistent with those previously extracted for a single QD . By fitting the data in Figures 3d-3f in three different directions and focusing on the SOI part of eq (1), we can extract the values of π΅ C , with π΅ Cπ₯ = 9 T , π΅ Cπ¦ = 6.5 T and π΅ Cπ§ = 1.7 T . The values of π΅ C are so large that they exceed the limit of the magnetic field we can reach, which is why we cannot observe saturation of the leakage current induced by the SOI at a finite field. Here, π΅ C is related not only to the effective π -factor but also to the angle between πΜ β π΅β and π΅β SO , and it can be expressed as π΅ C = π΅ C0 π β β sinπΌ , (2) where π΅ C0 is a constant, π β is the effective π -factor along the external magnetic field and πΌ is the angle between π΅β SO and πΜ β π΅β ; here, πΜ is the π -tensor. The π -factor in Ge HW has been demonstrated to be extremely anisotropic, with π β₯ > π β₯ , leading to
10 / 20 a much smaller π΅ C along the z -axis, which is consistent with our experimental result. To systematically investigate the anisotropy of the PSB lifting and determine the SOF direction, a fixed magnetic field is rotated in the x-y , x-z, and y-z planes while the current through the DQD is monitored in the PSB regime. Figure 4 shows the leakage current as a function of detuning energy π and rotation angle in three orthogonal planes for a fixed magnetic field of 0.8 T, where spin-flip cotunneling is completely suppressed and the SOI dominates. Combining eq (1) and eq (2), the measured leakage current πΌ leak induced by the SOI at finite magnetic field B can be expressed as πΌ leak = πΌ SO0 π΅ π΅ +( π΅C0πββ sinπΌ ) + πΌ Bβ² . (3) Based on the π -factor in the plane we obtained above, the π -tensor for holes in our DQD is well approximated by πΜ β (1.2 0 00 1.2 00 0 π π§ ) , (4) where π π§ is the π -factor along the z -direction. For simplicity, we assume that the off-diagonal elements in eq (4) are smaller than the diagonal elements and take them as zero. Hence, the effective π -factor along the external magnetic field can be expressed by π β (π, π) = βπΜ β (π, π)πΜ β πΜπΜ(π, π) , (5) where πΜ β (π, π) = (sinπcosπ, sinπsinπ, cosπ) (6) is a unit vector pointing along the magnetic field direction. The angle πΌ between π΅β SO and πΜ β π΅β can also be written explicitly in terms of the polar angle π and azimuth angle π of π΅β with πΜ and the polar angle π and azimuth angle π of π΅β SO . Figures 4d-4f show the leakage current measured at π = 0 , which changes periodically with the angle of the magnetic field. From the theoretical expression of eq (3), the extreme anisotropy in the leakage current induced by the SOI in Figure 4 can be understood as arising from two processes: First, when the magnetic field is rotated
11 / 20 out-of-plane, the Zeeman splitting between triplets increases dramatically due to the highly anisotropic heavy-hole π -factor in Ge HWs , which causes the anisotropic leakage current in Figures 4a and 4b. In our system, the effective π -factor reaches its maximum (minimum) when the field points perpendicular (parallel) to the substrate of the nanowires, leading to a minimum (maximum) of π΅ C and a maximum (minimum) of πΌ leak , as shown in eq (3), which agrees very well with our experimental results in Figure 4d and 4e. Second, when the magnetic field is varied in-plane (Figure 4c), the relative orientation of π΅β with respect to π΅β SO changes, which affects the degree of hybridization of T(1,1) and S(1,1) . Especially when the angle between π΅β and π΅β SO reaches a minimum (i.e., π΅β is aligned with the projection of π΅β SO in-plane), the spin-flip tunnelling induced by the SOI is suppressed the most, leading to a minimum leakage current in Figure 4f. Figure 4.
Anisotropic lifting of PSB. (a) Leakage current πΌ SD through the DQD as a function of the detuning energy π and the angle of the magnetic field rotated in the x-
12 / 20 z plane, as depicted in the top panel. π is tuned by sweeping π L and π R simultaneously along the green arrow in Figure 2a, and the magnetic field is fixed to 0.8 T. π is the polar angle between π΅β and the z -axis in the x-z plane. (b) The same as in panel a but for π΅β rotated in the y-z plane. πΏ is the polar angle between π΅β and the z -axis in the y-z plane. (c) The same as in panel a but for π΅β rotated in the x-y plane. π is the azimuth angle between π΅β and the x -axis in the x-y plane. (d-f) Leakage current πΌ SD measured at π = 0 (line cuts of the measurement data along zero detuning energy indicated by the yellow bars in panels a, b, c) as a function of the angle of the magnetic field rotated in three orthogonal planes. The red solid lines represent the best fits to the theory of eq (3). To accurately determine the direction of the SOF π΅β SO , we simultaneously perform numerical fitting of the leakage current measured in three orthogonal planes with eq (3). The red solid lines in Figures 4d-4f are the results of such fits, which show excellent agreement with the experiment. From these fits, we can extract the SOF direction, which corresponds to (π , π ) = (90Β° Β± 10Β°, 31Β° Β± 5Β°) , and the value of π π§ = 3.9 Β±0.1 . Compared to other one-dimensional systems such as Ge/Si core/shell nanowires, a crucial feature of the Ge HWs studied here is their triangular cross-section . The lack of inversion symmetry of the cross-section can lead to a large intrinsic SOI that is dependent on the orientation of the wire even without external electric fields . Because the wires grow horizontally along either the [100] or [010] direction , the intrinsic SOF points in the x -direction according to the theoretical model in ref 63. Notably, the interface inversion asymmetry can also contribute to the SOI . A Dresselhaus SOI may exist in our system due to the anisotropy of the chemical bonds at material interfaces . Including the spin-orbit terms we mentioned above, the DRSOI, interface SOI and intrinsic SOI induced by the asymmetry of the cross-section, the SOI Hamiltonian in our system can be described by π» SO = πΌπ π¦ π π₯ + π½π π¦ π π¦ . (7)
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Here, πΌ is the Rashba interaction coefficient, which accounts for the effect of the DRSOI and intrinsic SOI, π½ is the Dresselhaus interaction coefficient, which corresponds to the interface SOI, π π¦ is the hole wave operator along the wire, and π π₯ and π π¦ are the Pauli spin matrices. Hence, the direction of the SOF in our system can be expressed as (πΌ, π½, 0) , and we can extract the ratio between the Rashba and Dresselhaus coefficients in our system π½ πΌβ = tan 31Β° β 0.6 . Because the DRSOI is related to the electric field , the direction of the SOF π΅β SO can be adjusted by the gate voltages we apply. In conclusion, we have measured the leakage current through a DQD in Ge HWs in the PSB regime. We observe and distinguish two different PSB lifting mechanisms: spin-flip cotunneling and SOI, which lead to an increase in the leakage current. By varying the magnetic field orientation, we demonstrate the anisotropic behaviour of the leakage current induced by the SOI and explain the experimental results with a combination of the anisotropic hole π -factor and the angle between π΅β SO and πΜ β π΅β . Based on the anisotropic properties of the leakage current, the direction of the SOF π΅β SO is determined and found to be in-plane with an azimuth angle of to the nanowire, indicating a large interface SOI along the nanowire. The results we obtained may have important implications in the operation of spin-orbit qubits and the detection of Majorana fermions. Acknowledgements
This work was supported by the National Key Research and Development Program of China (Grant No.2016YFA0301700), the National Natural Science Foundation of China (Grants No. 12074368, 61674132, 11674300, and 11625419), the Strategic Priority Research Program of the CAS (Grant No. XDB24030601), the Anhui initiative in Quantum Information Technologies (Grants No. AHY080000), X. H. acknowledge financial support by U.S. ARO through No. W911NF1710257, and this work was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication.
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