Measurements of anisotropic g-factors for electrons in InSb nanowire quantum dots
Jingwei Mu, Shaoyun Huang, Ji-Yin Wang, Guang-Yao Huang, Xuming Wang, H. Q. Xu
1 Measurements of anisotropic g-factors for electrons in InSb nanowire quantum dots
Jingwei Mu , Shaoyun Huang , Ji-Yin Wang , Guang-Yao Huang , Xuming Wang , and H. Q. Xu , , * Beijing Key Laboratory of Quantum Devices, Key Laboratory for the Physics and Chemistry of Nanodevices, and Department of Electronics, Peking University, Beijing 100871, China Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China Beijing Academy of Quantum Information Sciences, Beijing 100193, China * To whom correspondence should be addressed. Email: [email protected] (September 25, 2020)
Abstract
We have measured the Zeeman splitting of quantum levels in few-electron quantum dots (QDs) formed in narrow bandgap InSb nanowires via the Schottky barriers at the contacts under application of different spatially orientated magnetic fields. The effective g-factor tensor extracted from the measurements is strongly anisotropic and level-dependent, which can be attributed to the presence of strong spin-orbit interaction (SOI) and asymmetric quantum confinement potentials in the QDs. We have demonstrated a successful determination of the principal values and the principal axis orientations of the g-factor tensors in an InSb nanowire QD by the measurements under rotations of a magnetic field in the three orthogonal planes. We also examine the magnetic-field evolution of the excitation spectra in an InSb nanowire QD and extract a SOI strength of ∆ 𝑠𝑠𝑠𝑠 ∼ μ eV from an avoided level crossing between a ground state and its neighboring first excited state in the QD.
1. Introduction
Semiconductor InSb nanowires are among the emerging, key materials for developments of solid-state based quantum device and quantum information technology. These nanomaterials have been employed in building semiconductor quantum dot (QD) spin qubits with fast operations of spin states by electrical means [1-3] and in constructing semiconductor-superconductor hybrid structures with potential applications towards topological quantum computing [4,5]. Strong spin-orbit interaction (SOI) and large g-factor have been exploited as the key physics properties of the nanomaterials in these applications [2-5]. Bulk InSb possesses an electron g-factor of −51 [6].
In low-dimensional InSb systems, such as InSb nanowire QDs, the magnitude of the electron g-factor is found to remain large but vary from level to level [7]. Due to a complex confinement potential profile and coupling of the spin degree of freedom of an electron to its orbital motion, the g-factor of electrons in a low-dimensional InSb quantum structure is expected to be strongly anisotropic. There have been studies showing that the g-factors in nanostructures, such as metal copper nanoparticles [8,9], semiconductor InAs QDs [10-17], Si QDs [18-20], InP nanowire QDs [21], Ge-Si core/shell nanowire QDs [22], InAs/InAlGaAs self-assembled QDs [23], and p-type GaAs/AlGaAs QDs [24], are anisotropic and/or level-dependent. Especially, with regards to semiconductor nanowire quantum structures, large orbital contributions to the electron g-factor have recently been theoretically identified [25] and experimentally demonstrated [17]. Some works on InSb nanowire QDs have also been reported [7,26,27,28]. However, full, solid experimental measurements of the anisotropic properties of the g-factor in InSb nanowire QDs have still not yet been carried out, although it is highly anticipated that the information obtained from such measurements is crucial for controlling of spin-orbit qubits [29,30] and topological Majorana zero modes [31] created in InSb nanowire quantum structures.
In this article, we report on low-temperature transport measurements of the Zeeman splitting of spin-1/2 electrons in single InSb nanowire QDs. The InSb nanowires are grown via metal organic vapor phase epitaxy and each QD is defined between the two metal contacts in an InSb nanowire via the naturally formed Schottky barriers. We extract the g-factors from the measured Zeeman splitting of spin-1/2 electrons at quantum levels of the QDs in the few-electron regime. We show the measurements performed for InSb nanowire QDs under the magnetic fields applied along the three axes of a coordinate system, in which one axis is set along the nanowire axis, and demonstrate that the g-factor of a spin-1/2 electron in these QDs is strongly anisotropic and level-dependent. The measurements are also performed for an InSb nanowire QD under 360 o -rotations of an applied magnetic field in three orthogonal planes. These measurements allow us to determine the principal values and principal axes of the g-factor tensors of spin-1/2 electrons at quantum levels of the QD and thus provide a complete experimental demonstration for anisotropy and level dependence of the g-factor in the InSb nanowire QDs. To extract the strength magnitude of SOI in the InSb nanowire QDs, we also perform the measurements for the excitation spectra of an InSb nanowire QD as a function of the magnetic field applied perpendicular to the nanowire. From the anti-crossing of two neighboring quantum levels occupied by electrons with opposite spins in the excitation spectra, a SOI strength of ∆ 𝑠𝑠𝑠𝑠 ∼
180 μeV is extracted.
2. Methods
The InSb nanowires employed in this work for device fabrication are grown on top of InAs nanowires by metal-organic vapor-phase epitaxy on an InAs (111)B substrate [32]. Previous studies showed that these InSb nanowires are in zincblende phase, grown along the [111] crystallographic direction and free from stacking faults [32]. The nanowires are transferred by a dry method onto a heavily n-doped Si substrate, capped with a 200-nm-thick layer of SiO on top, with predefined Ti/Au bonding pads and markers. The Si substrate and the capped SiO layer serve as a global back gate and a gate dielectric to the nanowires, respectively. Electron-beam lithography is used to define the source and drain contact areas on the nanowires. Then, in order to remove surface oxides and obtain clean metal-semiconductor interfaces, we etch the contact areas in a low concentration (~2%) ammonium polysulfide [(NH ) S x ] solution at 40 ℃ for 1 min [33]. After the etching process, contact electrodes are fabricated by deposition of 5-nm-thick titanium and 90-nm-thick gold via electron-beam evaporation and lift off. In each fabricated device, a QD is defined in the InSb nanowire segment between the two contacts by naturally formed Schottky barriers [7,26]. Figure 1(a) shows a false-colored scanning-electron microscope (SEM) image of a representative fabricated device and figure 1(b) is a schematic cross-sectional view of the device structure and the measurement circuit setup. The fabricated QD devices are studied by transport measurements in a dilution refrigerator equipped with a vector magnet. In this work, we will report the results of measurements for two representative InSb nanowire QD devices, which will be labeled as devices A and B. Device A is made from an InSb nanowire of ∼
85 nm in diameter with a contact spacing of ∼
150 nm and device B is made from an InSb nanowire of ∼
75 nm in diameter with the same ( ∼
150 nm) contact spacing. In the measurements, the back gate [cf. figure 1(b)] is used to control the chemical potential in the QDs and the magnetic field is applied in a direction with respect to the coordinate system with the X axis pointing along the nanowire axis as shown in figure 1(b).
3. Results and discussion
Figure 1(c) shows the differential conductance dI ds /dV ds measured for device A as a function of source-drain bias voltage V ds and back-gate voltage V bg (charge stability diagram) at zero magnetic field. Typical quantum transport characteristics of a single QD in the few-electron regime are observed in the measurements. The regular diamond-shaped low-conductance regions are the regions in which the number of electrons in the QD is well defined and the electron transport through the QD is blockade by Coulomb repulsion (Coulomb blockade effect). Since no more Coulomb diamond structure is observable at back-gate voltages V bg lower than −2.55 V, the QD is empty of electrons at V bg below − n in figure 1(c)]. The alternative changes in Coulomb diamond size seen in figure 1(c) are the traces of spin degeneracy and level quantization in the QD [34,35]. In the first Coulomb blockade region, the QD is occupied by one electron. For a second electron of the opposite spin to enter and then pass through the QD, an excess energy is required to overcome single electron charging energy E C . Thus, from the vertical span of the first Coulomb diamond, a charging energy of E C ~ C Σ ~ C g ~ C Σ gives a back- gate lever arm factor of α (= C g / C Σ ) ~ one more electron to the QD needs to overcome an electron addition energy which includes both single electron charging energy E C and level quantization energy Δε [7]. Thus, by assuming that E C remains unchanged, we can extract, from the vertical span of the second Coulomb diamond (a measure for the addition energy of the third electron to the QD), a level quantization energy of Δε ~ ↑ doublet ground state | 𝐷𝐷 ↑ ⟩ to the two-electron singlet ground state | 𝑆𝑆⟩ and the high-energy high conductance line involves transitions of quantum states from the spin- ↑ doublet ground state | 𝐷𝐷 ↑ ⟩ to two-electron excited states | 𝑆𝑆 ∗ ⟩ and | 𝑇𝑇 ∗ ⟩ involving both the first and the second quantum level [36]. Thus, the energy difference between the two-electron singlet ground state and excited states can be extracted out from the half of the vertical distance in bias voltage between the two high-conductance lines, which yields Δε ~ Figure 2 shows the magnetic field evolutions of the first and second quantum levels of the QD in device A. Figure 2(a) displays the current I ds measured for device A at V ds = 0.2 mV as a function of V bg and magnetic field B X applied along the X direction. Here, the spin states of subsequently filled electrons in the level are marked by arrows. As B X is increased from zero, the current peaks corresponding to the spin- ↑ and spin - ↓ states move apart. After converting the peak positions in energy using the back-gate lever arm factor determined above, an effective g-factor 𝑔𝑔 𝐼𝐼𝐼𝐼∗ of the first quantum level (level I ) can be extracted from the energy difference ΔE ( B X ) between the two spin states according to ΔE ( B X ) = E C + | 𝑔𝑔 𝐼𝐼𝐼𝐼∗ μ B B X |, where μ B is the Bohr magneton. Figure 2(b) shows the measured energy separations of the two spin states as a function of B X . By a line fit to the measurements [red solid line in figure 2(b)], we extract a value of 𝑔𝑔 𝐼𝐼𝐼𝐼∗ ~ I in the QD. Similar measurements are performed with magnetic fields B Y and B Z applied along the Y and Z axes, and the corresponding g-factors, 𝑔𝑔 𝐼𝐼𝐼𝐼∗ ~ 𝑔𝑔 𝐼𝐼𝐼𝐼∗ ~ inset of figure 2(b) summarizes the three extracted g-factor values of the quantum level at magnetic fields applied along the X, Y and Z axes. Clearly, the three g-factor values are different, implying that the g-factor of the quantum level in the QD is anisotropic. Figure 2(c) shows the differential conductance dI ds /dV ds measured along the line cut A in figure 1(c) as a function of source-drain bias voltage V ds and magnetic field B X . Here, the three high conductance stripes are observed. The upper one involves a transition of quantum states from the spin- ↑ doublet ground state | 𝐷𝐷 ↑ ⟩ to the two-electron singlet ground state | 𝑆𝑆⟩ at a finite magnetic field and can be associated with the tunneling process of a spin- ↓ electron through the QD. As a result, this high conductance stripe moves to higher energy with increasing B X . Note that negative values of applied bias voltage V ds are given in figure 2(c). Since both the doublet and singlet ground states, | 𝐷𝐷 ↑ ⟩ and | 𝑆𝑆⟩ , involve only the lowest quantum level in the QD, we can extract the electron g-factor of the quantum level in the QD from the slope of the high conductance stripe. The result is 𝑔𝑔 𝐼𝐼𝐼𝐼∗ ~ 𝑔𝑔 𝐼𝐼𝐼𝐼∗ ~ 𝑔𝑔 𝐼𝐼𝐼𝐼∗ ~ ↑ doublet ground state | 𝐷𝐷 ↑ ⟩ to three two-electron excited states | 𝑆𝑆 ∗ ⟩ , | 𝑇𝑇 ⟩ , and | 𝑇𝑇 +∗ ⟩ involving both the first and the second quantum level in the QD [36]. Among the two, the low-energy high conductance stripe involves the transition from | 𝐷𝐷 ↑ ⟩ to | 𝑇𝑇 +∗ ⟩ and thus tunneling of spin- ↑ electrons through the QD, while the high-energy high conductance stripe involves the transitions of | 𝐷𝐷 ↑ ⟩ → | 𝑆𝑆 ∗ ⟩ and | 𝐷𝐷 ↑ ⟩ → | 𝑇𝑇 ⟩ and tunneling of spin- ↓ electrons through the QD. Since the three two- electron excited states consist of an electron of the same spin (the ↑ spin in this case) in the first quantum level and an electron of different spins in the second quantum level in the QD, the energy difference between the two high conductance stripes at a given magnetic field B X is a measure of the Zeeman splitting ΔE ( B X ) = | 𝑔𝑔 𝐼𝐼𝐼𝐼𝐼𝐼∗ μ B B X | of the second quantum level (level II ) in the QD, where 𝑔𝑔 𝐼𝐼𝐼𝐼𝐼𝐼∗ is the effective electron g-factor of a spin-1/2 electron at the second quantum level. Figure B X . Through a line fit (red solid line) to the data in figure 2(d), an effective electron g-factor of 𝑔𝑔 𝐼𝐼𝐼𝐼𝐼𝐼∗ ~ with the magnetic field applied along the Y and Z directions and the corresponding g-factors of 𝑔𝑔 𝐼𝐼𝐼𝐼𝐼𝐼∗ ~ 𝑔𝑔 𝐼𝐼𝐼𝐼𝐼𝐼∗ ~ The above measurements demonstrate that the extracted g-factor of a spin-1/2 electron in the InSb nanowire QD is sensitively dependent on the energy level and on the direction of the applied magnetic field. Thus, in general, the g-factor of a spin-1/2 electron in a quantum level m of the InSb nanowire QD should be described by a tensor 𝑔𝑔 𝑚𝑚∗ and its value determined under an applied magnetic field B of an arbitrary direction can be expressed as [8,9] | 𝑔𝑔 𝑚𝑚∗ ( 𝐁𝐁 )| = �𝑔𝑔 𝐵𝐵 +𝑔𝑔 𝐵𝐵 +𝑔𝑔 𝐵𝐵 | 𝐁𝐁 | , (1) where 𝑔𝑔 , 𝑔𝑔 and 𝑔𝑔 are the three principal values of the g-factor tensor defined with respect to the three orthogonal principal axis directions, and 𝐵𝐵 , 𝐵𝐵 and 𝐵𝐵 are the magnetic field components along these directions. We emphasize that these principal axis directions are generally different from the axis directions in the coordinate system we have set above in the measurements. However, the principal axis coordinate system of the g-factor tensor can be transformed from our measurement coordinate system by means of the three successive rotations defined through the Euler angles of rotation (α, β, γ). To determine the principal values and the orientations of the principal axes of the g-factor tensor 𝑔𝑔 𝑚𝑚∗ in our InSb nanowire QD, we perform the measurements for the values of 𝑔𝑔 𝑚𝑚∗ with the rotations of the magnetic field B in three orthogonal planes and fit the measured values to Eq. (1). These three orthogonal planes are defined as the X-Y, Z-X, and Z-Y planes as described by the three schematics shown in figure 3. Figures 3(a)-3(c) show the results of such measurements for the g-factor tensor 𝑔𝑔 𝐼𝐼∗ of a spin-1/2 electron in the first quantum level of the InSb nanowire QD in device A. Here, the two current peaks, corresponding to the resonant transport through the Zeeman splitting states of the first quantum level, are found in the measurements of current I ds as a function of V bg at V ds = 0.2 mV under rotations of applied magnetic field B in the three orthogonal X-Y, Z-X, and Z-Y planes. The strength of the magnetic field is set at | B | = 0.4 T, which is within a range of magnetic fields in which the Zeeman splitting is expected to show a good linear dependence on | B | in all the magnetic field directions (see experimental observations as shown in figure 2(b) for example). The extracted energy difference between the two Zeeman splitting states from the measurements is given by ΔE ( B )= E C + 𝑔𝑔 𝐼𝐼∗ μ B | B |. Figure 3(a) displays the experimentally extracted values of 𝑔𝑔 𝐼𝐼∗ with B applied in the X-Y plane. It is found that 𝑔𝑔 𝐼𝐼∗ oscillates with the rotation of B with its maxima appearing approximately at rotation angles of ϕ ~
0º and 180º and its minima appearing approximately at ϕ ~
90º and 270º. Oscillations of 𝑔𝑔 𝐼𝐼∗ with the rotation of B are also found in figures 3(b) and 3(c), where B is applied in the Z-X and Z-Y planes. However, in difference from the case shown in figure 3(a), the maximum (minimum) values of 𝑔𝑔 𝐼𝐼∗ do not appear at about 0º and 180º (90º and 270º) in figures 3(b) and 3(c). The red solid curves in figures 3(a)-3(c) are the fit of the measured 𝑔𝑔 𝐼𝐼∗ values to Eq. (1) with the principal values ( 𝑔𝑔 , 𝑔𝑔 , 𝑔𝑔 ) and the Euler angles (α, β, γ) as free fitting parameters. The fit yields ( 𝑔𝑔 , 𝑔𝑔 , 𝑔𝑔 ) = (60.5, 50.1, 56.5) and ( α , β , γ ) = (96.4°, 71.7°, 155.6°). Clearly, it is seen that the effective g-factor tensor in the QD is spatially anisotropic and is level dependent. In the absence of SOI, the g-factor in a symmetric QD should be isotropic. The anisotropy of the 𝑔𝑔 𝐼𝐼∗ tensor observed here can thus be attributed to the complex profile of the quantum confinement in the InSb nanowire QD and to anisotropic orbital contributions [17,25] due to the presence of strong SOI in the InSb quantum structure [7] as we will discuss below. For the principal axis directions of the 𝑔𝑔 𝐼𝐼∗ tensor, an intuitive view is to present them in terms of the polar angle 𝜗𝜗 𝑖𝑖 with respect to the Z axis and the azimuth angle 𝜑𝜑 𝑖𝑖 defined in the X-Y plane with respect to the X direction of the measurement coordinate system. Here, subscript i =1, 2 or 3 is the index for a principal axis. For the 𝑔𝑔 𝐼𝐼∗ tensor of the first quantum level in the QD of device A, the orientations of the three principal axes can be mapped out from the obtained Euler angles of rotation of ( α , β , γ ) = (96.4°, 71.7°, 155.6°) as ( 𝜗𝜗 , 𝜑𝜑 )=(66.9°, 1.7°), ( 𝜗𝜗 , 𝜑𝜑 )=(108.3°, 83.6°) and ( 𝜗𝜗 , 𝜑𝜑 )=(30.1°, 138.9°), respectively. These results show that the first principal axis of the 𝑔𝑔 𝐼𝐼∗ tensor with the largest principal value of 𝑔𝑔 ~ 𝑔𝑔 𝐼𝐼𝐼𝐼∗ i.e., the g-factor values for the second quantum level of the QD with the magnetic field applied in the X-Y, Z-X, and Z-Y planes, respectively. Here, the Zeeman splitting energies of the quantum level of the QD are extracted from the excitation spectra measured along the line cut A in figure 1(c) at a fixed magnetic field magnitude of | B | = 0.8 T, which is within a range of 0 to 0.9 T in which the magnetic field evolution of the Zeeman energy is, to a good approximation, linear in | B | for all magnetic field directions [see, for example, a result shown in figures 2(c) and 2(d)]. The red solid curves in figures 4(a)-4(c) are the results of the fit of the measurements to Eq. (1). The 𝑔𝑔 𝐼𝐼𝐼𝐼∗ tensor extracted from the fit has the three principal values ( 𝑔𝑔 , 𝑔𝑔 , 𝑔𝑔 ) = (50.7, 46.0, 47.8) and the Euler angles of rotation ( α , β , γ ) = (96.1°, 97.4°, 2.1°) of the principal axes with respect to the measurement coordinate system. The directions of the three principal axes viewed with respect to the measurement coordinate system are found to be ( 𝜗𝜗 , 𝜑𝜑 )=(87.9°, 8.1°), ( 𝜗𝜗 , 𝜑𝜑 )=(96.1°, 97.8°) and ( 𝜗𝜗 , 𝜑𝜑 )=(6.5°, 117.2°), respectively. Again, here we see that 𝑔𝑔 𝐼𝐼𝐼𝐼∗ is anisotropic and its first principal axis with a principal value of 𝑔𝑔 ∼ 𝑔𝑔 𝐼𝐼∗ and 𝑔𝑔 𝐼𝐼𝐼𝐼∗ tensors, we find that both the principal values and the principal axes are level dependent, similar to the results reported previously for the g-factors in other quantum structures [9,14].
For convenience, we list the obtained orientations of the principal axes of the 𝑔𝑔 𝐼𝐼∗ and 𝑔𝑔 𝐼𝐼𝐼𝐼∗ tensors together with their corresponding principal values in Table I. 3.4.
Measurements of g-factors and SOI for the QD in device B
We have also studied device B made from a nanowire with a slightly smaller diameter and obtained similar results for the g-factor as in device A. Figure 5(a) shows the differential conductance dI ds /dV ds of the InSb nanowire QD in device B measured as a function of source-drain bias voltage V ds and back-gate voltage V bg . Here, several well-defined Coulomb blockade diamond structures are observed. Since no more Coulomb diamond structure is observable at back-gate voltages V bg lower than −1.35 V, the QD is empty of electrons at V bg below − n . At n = 2, 4, and 6, the first, second and third quantum levels (labelled as quantum levels I , II , and III ) are successively fully filled with electrons in the QD. Figure 5(b) shows the current I ds of the device measured at V ds = 0.2 mV as a function of V bg and magnetic field B Z applied along the Z direction. Three pairs of high current stripes are seen in the figure and each pair corresponds to electron occupations of the two spin states of a quantum level in the QD. Here each spin state is again marked by an arrow. Figure 5(c) shows the extracted spin splitting energies of the three quantum levels as a function of B Z . By line fits to the measurement data, the values of the g-factor 𝑔𝑔 𝐼𝐼𝐼𝐼∗ ~ 𝑔𝑔 𝐼𝐼𝐼𝐼𝐼𝐼∗ ~ 𝑔𝑔 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∗ ~ I , II and III , respectively. Similar measurements are performed with magnetic fields B X and B Y applied along the X and Y axes and the corresponding g-factor values of 𝑔𝑔 𝐼𝐼𝐼𝐼∗ ~ 𝑔𝑔 𝐼𝐼𝐼𝐼𝐼𝐼∗ ~ 𝑔𝑔 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∗ ~ 𝑔𝑔 𝐼𝐼𝐼𝐼∗ ~ 𝑔𝑔 𝐼𝐼𝐼𝐼𝐼𝐼∗ ~ 𝑔𝑔 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∗ ~ Finally, we briefly demonstrate, using device B, the presence of strong SOI in our InSb nanowire QDs. Figure 5(d) displays the differential conductance dI ds /dV ds measured along the line cut B in figure 5(a) as a function of V ds and B Z . The three high conductance stripes are found in the figure. On the low field side, these three high conductance stripes, from top to bottom, involve the transitions from the spin- ↑ doublet ground state | 𝐷𝐷 ↑ ⟩ to the two-electron singlet ground state | 𝑆𝑆⟩ , to the two-electron triplet excited state | 𝑇𝑇 +∗ ⟩ , and to the two-electron singlet and triplet excited states | 𝑆𝑆 ∗ ⟩ and | 𝑇𝑇 ⟩ in the QD, as marked in figure 5(d). With increasing B Z , the top high conductance stripe involving the transition from state | 𝐷𝐷 ↑ ⟩ to state | 𝑆𝑆⟩ shifts towards higher | V ds |. The other two high conductance stripes move apart and the one involving the transition from state | 𝐷𝐷 ↑ ⟩ to state | 𝑇𝑇 +∗ ⟩ is found to shift towards lower | V ds | with increasing B Z , leading to an avoided crossing with the top high conductance stripe at B Z ~ In order to extract the strength of SOI, ∆ 𝑠𝑠𝑠𝑠 , seen at this experiment, we fit the energy positions of the quantum states extracted from the two anti-crossed high conductance stripes to a simple two-level perturbation model [36] 𝐸𝐸 ± = 𝐸𝐸 S +𝐸𝐸 𝑇𝑇+∗ ± � ( 𝐸𝐸 S −𝐸𝐸 𝑇𝑇+∗ ) + ∆ so2 , (2) where 𝐸𝐸 S and 𝐸𝐸 𝑇𝑇 +∗ are the energy levels of the singlet state | 𝑆𝑆⟩ and the triplet state | 𝑇𝑇 +∗ ⟩ without taking SOI into account. Figure 5(e) displays the energy positions (data points) extracted from the two anti-crossed high conductance stripes shown in figure 5(d), while the solid lines in figure 5(e) are the fit. The fit yields a large value of ∆ 𝑠𝑠𝑠𝑠 ∼ μ eV, consistent with the results reported previously [7]. The values of the g-factors, 𝑔𝑔 𝐼𝐼𝐼𝐼∗ ~ 𝑔𝑔 𝐼𝐼𝐼𝐼𝐼𝐼∗ ~
4. Conclusions
In summary, we have measured the Zeeman splitting energies of electrons in single InSb nanowire QDs under different, three-dimensionally oriented magnetic fields. The measurements show that the extracted g-factor in the QDs remains large but varies from level to level. The measurements also show that the g-factor in the QDs is spatially anisotropic and should be described by a tensor for completeness. We have demonstrated the experimental determination of the g-factor tensors in an InSb nanowire QD by the Zeeman energy measurements of a spin-1/2 electron at quantum levels of the QD under rotations of a magnetic field in three orthogonal planes. The anisotropy and the level dependence of the g-factor can be attributed to the presence of strong SOI and asymmetric confining potentials in the QDs. In this work, a SOI strength of ∆ 𝑠𝑠𝑠𝑠 ∼ μ eV is observed in the measurements of the magnetic field evolution of the excitation spectra of an InSb nanowire QD. Our work provides a first solid demonstration for the complex nature of the g-factor in InSb nanowire quantum structures. Such information and its determination play a crucial role in successful realization and accurate manipulation of spin qubits and topological quantum states made from such semiconductor nanowire structures. Acknowledgements
We thank Philippe Caroff for the growth of the InSb nanowires employed in this work. This work is supported by the Ministry of Science and Technology of China through the National Key Research and Development Program of China (Grant Nos. 2017YFA0303304, 2016YFA0300601, 2017YFA0204901, and 2016YFA0300802), the National Natural Science Foundation of China (Grant Nos. 91221202, 91421303, 11874071 and 11974030), the Beijing Academy of Quantum Information Sciences (No. Y18G22), and the Beijing Natural Science Foundation (Grant No. 1202010). References [1] Nadj-Perge S, Frolov S M, Bakkers E P A M, Kouwenhoven L P 2010 Spin–orbit qubit in a semiconductor nanowire
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E 2008 High‐quality InAs/InSb nanowire heterostructures grown by metal– organic vapor‐phase epitaxy
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Rev. Lett. Phys. Rev. Lett. CAPTIONS Figure 1. (a) SEM image (in false color) of a representative device made on a Si/SiO substrate. A single QD is defined in an InSb nanowire between the two Ti/Au metal contacts via naturally formed Schottky barriers. (b) Cross-sectional schematic view of the device displaying the heavily n-doped Si substrate (grey) capped with a 200-nm-thick layer of SiO (maroon) on top, the nanowire (green), and the two metal contacts (orange), and measurements circuit setup. The measurement coordinate system is shown in the lower-left corner, in which the X axis points along the nanowire axis, the Z axis is set to be perpendicular to the substrate plane, and the Y axis is set to be in-plane, perpendicular to the nanowire axis. (c) Differential conductance dI ds /dV ds measured for device A (see the text for the description of the device) as a function of source-drain bias voltage V ds and back-gate voltage V bg (charge stability diagram). Integer numbers n mark the numbers of electrons in the QD formed in the device. Figure 2. (a) Current I ds measured for device A at V ds = 0.2 mV as a function of back-gate voltage V bg and magnetic field B X applied along the X direction. The two high conductance stripes labeled with ↑ and ↓ result from electron tunneling through the spin- ↑ and spin - ↓ states of the first quantum level in the QD. (b) Energy differences between the two spin states (data points) extracted from the measurements in (a). By a line fit to the data, a value of the g-factor of 𝑔𝑔 𝐼𝐼𝐼𝐼∗ ~ dI ds /dV ds measured along the line cut A in figure 1(c) as a function of V ds and magnetic field B X applied along the X direction. The three high conductance stripes, from top to bottom, involve the quantum state transitions from the spin- ↑ doublet ground state | 𝐷𝐷 ↑ ⟩ to the two-electron singlet ground state | 𝑆𝑆⟩ , to the two-electron excited state | 𝑇𝑇 +∗ ⟩ , and to the two-electron excited states | 𝑆𝑆 ∗ ⟩ and | 𝑇𝑇 ⟩ , respectively. (d) Energy differences extracted from the peak values of the two high conductance stripes in (c). By a line fit to the data, a value of the g-factor 𝑔𝑔 𝐼𝐼𝐼𝐼𝐼𝐼∗ ~ the X, Y and Z axes for the second quantum level. Figure 3. (a)
Values of 𝑔𝑔 𝐼𝐼∗ extracted from the Zeeman splitting of the first quantum level of the QD in device A, measured in a similar way as in figures 2(a) and 2(b), as a function of the rotation angle of a magnetic field B with | B |=0.4 T in the X-Y plane as indicated in the schematics (top panel). (b) and (c) The same as in (a) but for the magnetic field rotated in the Z-X and Z-Y planes. The red solid lines are the results of the fit of the experimental data to Eq. (1). Figure 4. (a)-(c)
Values of 𝑔𝑔 𝐼𝐼𝐼𝐼∗ extracted from the Zeeman splitting of the second quantum level of the QD in device A, measured in a similar way as in figures 2(c) and 2(d), (i.e., from the excitation spectra), as a function of the rotation angle of magnetic field B with | B |=0.8 T in the X-Y, Z-X, and Z-Y planes as indicated in the schematics in figure 3. The red lines are the results of the fit of the experimental data to Eq. (1). Figure 5. (a) Differential conductance dI ds /dV ds measured for device B (see the text for the description of the device) as a function of back-gate voltage V bg and source-drain bias voltage V ds (charge stability diagram). Integer numbers n mark the numbers of electrons in the QD. (b) Current I ds measured for device B at V ds = 0.2 mV as a function of back-gate voltage V bg and magnetic field B Z applied along the Z direction. Three pairs of high conductance stripes correspond to electron transport through the spin states of the first three quantum levels (labelled as I , II and III ) in the QD. Corresponding spin fillings of the quantum levels are indicated with arrows in the plot. (c) Energy differences of the two spin states extracted from the measurements for quantum levels I , II and III as shown in (b) as a function of B Z . By line fits to the data points, the g-factor values of 𝑔𝑔 𝐼𝐼𝐼𝐼∗ ~ 𝑔𝑔 𝐼𝐼𝐼𝐼𝐼𝐼∗ ~ 𝑔𝑔 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∗ ~ I , II and III , respectively. (d) Differential conductance dI ds /dV ds measured along line cut B in (a) as a function of V ds and B Z . The three high conductance stripes, from top to bottom, involve the quantum state transitions from the spin- ↑ doublet ground state | 𝐷𝐷 ↑ ⟩ to the two-electron singlet ground state | 𝑆𝑆⟩ , to the two-electron excited state | 𝑇𝑇 +∗ ⟩ , and to the two-electron excited states | 𝑆𝑆 ∗ ⟩ and | 𝑇𝑇 ⟩ . (e) Converted energy positions at the peak values of the two anti-crossed high conductance stripes in (d). The red solid lines show the fit of the data points to a two-level perturbation model and ∆ 𝑠𝑠𝑠𝑠 stands for the strength of SOI seen in the experiment. Table I . Principal values ( 𝑔𝑔 , 𝑔𝑔 , 𝑔𝑔 ) and orientations [( 𝜗𝜗 , 𝜑𝜑 ), ( 𝜗𝜗 , 𝜑𝜑 ), ( 𝜗𝜗 , 𝜑𝜑 )] of the principal axes of g-factor tensors 𝑔𝑔 𝐼𝐼∗ and 𝑔𝑔 𝐼𝐼𝐼𝐼∗ determined for the first and second quantum levels in the InSb nanowire QD of device A. Quantum Level 𝑔𝑔 𝑔𝑔 𝑔𝑔 𝜗𝜗 𝜑𝜑 𝜗𝜗 𝜑𝜑 𝜗𝜗 𝜑𝜑 Ι Figure 1, by Jingwei Mu, et al