Generation and Detection of Spin Currents in Semiconductor Nanostructures with Strong Spin-Orbit Interaction
Fabrizio Nichele, Szymon Hennel, Patrick Pietsch, Werner Wegscheider, Peter Stano, Philippe Jacquod, Thomas Ihn, Klaus Ensslin
GGeneration and Detection of Spin Currents in Semiconductor Nanostructures withStrong Spin-Orbit Interaction
Fabrizio Nichele, ∗ Szymon Hennel, Patrick Pietsch, Werner Wegscheider, Peter Stano,
2, 3
Philippe Jacquod, Thomas Ihn, and Klaus Ensslin Solid State Physics Laboratory, ETH Z¨urich, 8093 Z¨urich, Switzerland RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan Institute of Physics, Slovak Academy of Sciences, Dubravska cesta 9, 84511 Bratislava, Slovakia HES-SO, Haute Ecole d’Ing´enierie, 1950 Sion, Switzerland (Dated: September 30, 2018)Storing, transmitting, and manipulating information using the electron spin resides at the heartof spintronics. Fundamental for future spintronics applications is the ability to control spin currentsin solid state systems. Among the different platforms proposed so far, semiconductors with strongspin-orbit interaction are especially attractive as they promise fast and scalable spin control withall-electrical protocols. Here we demonstrate both the generation and measurement of pure spincurrents in semiconductor nanostructures. Generation is purely electrical and mediated by thespin dynamics in materials with a strong spin-orbit field. Measurement is accomplished using aspin-to-charge conversion technique, based on the magnetic field symmetry of easily measurableelectrical quantities. Calibrating the spin-to-charge conversion via the conductance of a quantumpoint contact, we quantitatively measure the mesoscopic spin Hall effect in a multiterminal GaAsdot. We report spin currents of 174 pA, corresponding to a spin Hall angle of 34%.
The generation and detection of spin currents in nanos-tructures is the central challenge of semiconductor spin-tronics. On the one hand, spin injection cannot be easilyachieved by coupling semiconductors to ferromagnets [1]because of the lack of control over material interfaces [2].On the other hand, magnetoelectric alternatives exploit-ing the celebrated spin Hall effect (SHE) [3, 4], have deliv-ered only qualitative measurement protocols in transportexperiments [5]. Alternatively to all-electrical setups,spin polarizing the current through a quantum point con-tact (QPC) with a magnetic field allows a quantitativecontrol over spin current generation and detection at thenanoscale [6–8]. The latter approach typically requiressuch high magnetic fields (6 − electrically generate and quanti-tatively measure spin currents in a two-dimensional semi-conductor nanostructure.It is predicted that charge currents flowing throughspin-orbit interaction (SOI)-coupled nanostructures aregenerically accompanied by spin currents, if the spin-orbit time is shorter than the electron dwell time [9–12].This spin current generation mechanism is purely elec-trical and based on the mesoscopic SHE (MSHE) [9, 10],where the electronic orbital dynamics in chaotic nanos-tructures cooperates with the SOI to make transportspin dependent. We will consider an open three-terminalquantum dot as represented in Fig. 1(a), where each lead i is a QPC carrying N i spin degenerate modes. Runninga charge current I between terminals 1 and 2, a spin cur-rent in all terminals, including 3, is expected due to the MSHE.For a weak SOI, the spin currents’ amplitude fluctu-ates from sample to sample with zero average. For cav-ities with a strong SOI, geometric correlations betweenthe spin and the orbital electronic dynamics lead to spincurrents with large, predictable nonzero average values[13]. In the latter case, the spin currents’ amplitude isdetermined by the nanostructure geometry and the SOIstrength. This particular mechanism renders spin cur-rents robust against decoherence and allows us to differ-entiate them from mesoscopic fluctuations. This is es-sential for spintronics applications, where spin currentsmust be reproducible regardless of the microscopic detailsof the sample.To detect and measure the spin currents describedabove, we employ the scheme of Ref. [14], based onthe magnetic field parity of the voltage behind a QPC.With reference to Fig. 1(a), we are interested in the spincurrent I ( S )3 emitted from QPC . The energy depen-dent transmission probability of QPC , T ( s )33 , is shownin Fig. 1(b). At zero field, QPC is tuned to a con-ductance of e /h by a suitable gate voltage, correspond-ing to a spin-independent transmission probability of onehalf. A weak in-plane magnetic field B modifies the elec-trons’ kinetic energy via Zeeman coupling, selectively in-creasing or decreasing the transmission probability ac-cording to the spin eigenstate and magnetic field sign.For simplicity, we assume I ( S )3 to be a pure spin cur-rent at B = 0, arising as two counterpropagating andinversely spin-polarized charge currents of equal magni-tude, as schematically shown in Fig. 1(c), where arrowsindicate current amplitude and colors spin polarization.A magnetic field affects the QPC spin dependent trans-mission probability, enhancing one of the two charge cur- a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y I =0 B=0 I <0 B>0I >0 B<0 B<0 B=0B>0EnergyE (s) I (a) (b)(c) ( S p i n ) T E Z E =g μ B Z B
FIG. 1. (a) Schematic of the system used to generate spincurrents. Charge currents are depicted as black lines, spincurrents are depicted as red and blue lines. (b) Energy de-pendent, spin sensitive, transmission probability of QPC . Atzero field the transmission is tuned to 1 /
2. A positive (nega-tive) field suppresses (enhances) the transmission probabilityof one spin eigenstate. (c) Representation of spin and chargecurrents in QPC as a function of magnetic field. The netcharge current in QPC varies with the magnetic field. rents and suppressing the other. The result is the flowof a net charge current I in the QPC. The sign of I re-verses by reversing the magnetic field sign. Note that thenet spin current remains constant in the three situationsdepicted in Fig. 1(c).By operating QPC as a floating probe, the chargecurrent I is constantly fixed to zero. In this case, thepresence of a spin current I ( S )3 in the QPC reflects itselfin an antisymmetric component of the voltage behindit: V ( B ) (cid:54) = V ( − B ). Remarkably, theory predicts thatthe zero-field derivative of the measured voltage, ∂ B V ,is proportional to the spin current I ( S )3 polarized alongthe applied magnetic field. The proportionality coeffi-cient between the spin-to-charge signal ∂ B V and I ( S )3 isgiven by the QPC g factor and its energy sensitivity ¯ hω measured at N = 0 . I ( S )3 = e h hωπgµ B ∂ B V . (1)More generally, for N ≤ ∂ B V ∝ ∂ V G /G . For N = 0 . ∂ V G /G = − π/ ¯ hω . Details aboutthe derivation of this proportionality and Eq. (1) are re-ported in the Supplemental Material [15]. Equation (1)not only allows us to detect the presence of a spin cur-rent flowing in QPC , but also to quantitatively measureand express it in units of ampere, giving the difference incurrents carried by electrons with opposite polarization.Given the large SOI of our system, the measurement pro-cess requires only weak magnetic fields that do not affect V g4,g5 (V) B ( T ) −0.8 −0.6 −0.4 −0.2 0−3−1.501.53 6.26.36.4 −3 −1.5 0 1.5 33.33.43.53.6 B (T) V /I ( k Ω ) −3 −1.5 0 1.5 366.16.26.36.46.5 B (T) V C /I ( k Ω ) V g4,g5 (V) B ( T ) −0.8 −0.6 −0.4 −0.2 0−3−1.501.53 3.33.43.53.6 −0.8 −0.6 −0.4 −0.2 00123 V g4,g5 (V) G ( / h ) (a)(c) (b)(d)(f)(e)
12 T 0 T109 mK 585 mK109 mK 585 mKV /I (kΩ)V C /I (kΩ) g g g g g QPC1 QPC2 I V QPC3 V C FIG. 2. (a) Atomic force micrograph of our sample, wheredark lines indicate insulating trenches. The frame size is 5 × µ m . (b) QPC conductance as a function of the voltageapplied to g and g , for different values of magnetic field. (c)Cavity four-terminal resistance as a function of the voltageapplied to g and g and magnetic field. (d) Linecut of (c)along the dashed line for different temperatures. (e) V /I asa function of the voltage applied to g and g and magneticfield. (f) Linecut of (e) along the dashed line for differenttemperatures. the generated spin currents. We note that our approachis restricted to the linear response regime, and to termi-nal 3 being a weak probe, i.e., N ≤ (cid:28) N , N . Themeasurement protocol described here is independent ofthe spin current generation mechanism. In particular,the detection method can be used to measure spin cur-rents of other origin.Motivated by the theory above, we study a three-terminal chaotic cavity embedded in a p -type GaAs two-dimensional hole gas (2DHG) with a strong Rashba SOI.Our sample, shown in Fig. 2(a), lacks any spatial sym-metry and consists of three leads and five in-plane gates,named QPC i and g j respectively. The gates g and g ,colored in blue, tune the conductance of QPC with littleinfluence on the dot average conductance. The gates g ,g and g , depicted in red, tune the conductance of QPC and QPC , and also affect the dot shape. The lateral ex-tent of the cavity is about 2 µ m, the hole mean free path l e = 4 . µ m and the spin-orbit length l SO = 36 nm [15].Spin rotational symmetry is then completely broken and, −0.4 −0.2 0−20020406080 V g4,g5 (V) ∂ B V /I ( Ω T − ) ∂ V G / G ( a . u . ) −0.6 −0.4 −0.2−20020406080 V g4,g5 (V) ∂ B V /I ( Ω T − ) ∂ V G / G ( a . u . ) −0.4 −0.2 0−20020406080 V g4,g5 (V) ∂ B V /I ( Ω T − ) ∂ V G / G ( a . u . ) −0.4 −0.2 0−20020406080 V g4,g5 (V) ∂ B V /I ( Ω T − ) ∂ V G / G ( a . u . ) (a)(c) (b)(d) N =4 N =4 N >10 N >10N =5 N =4 N =4 N =5 FIG. 3. (a)-(d) Comparison between ∂ B V /I (markers) and ∂ V G /G (solid line) as a function of side gate voltage fordifferent number of modes in QPC and QPC , as indicated ineach subfigure. Dots and squares in (a) represent two identicalmeasurements performed in different cooldowns [16]. with such a strong SOI, our cavity is in the so-called spinchaos regime [13]. Unless differently stated, a charge cur-rent I flows from terminal 1 to terminal 2, while terminal3 is connected to a high impedance voltage amplifier andis used to measure spin currents.To measure the spin current in terminal 3, we firstextract the detector electric and magnetic sensitivity viaa standard QPC characterization. Figure 2(b) shows thedetector conductance G as a function of side gate voltagefor different values of a magnetic field aligned with theQPC axis. The three well-developed plateaus visible atzero field split at finite field. The zero field slope andthe finite field splitting give, respectively, ∂ V G and the g factor [15].After the detector calibration, QPC is operated asa voltage probe. The spin current measurement is per-formed running an AC current ( I = 4 nA unless statedotherwise) between terminals 1 and 2, and measuring themagnetic dependence of the voltages V C and V as definedin Fig. 2(a). The magnetic field is applied in-plane, tominimize its orbital effects, and aligned with the detec-tor axis (unless stated otherwise). The finite zero fieldderivative ∂ B V , and its correlation with ∂ V G /G , indi-cates the presence of the spin current I ( S )3 .Figures 2(c) and (e) show the resistances R C = V C /I and V /I as a function of B and the voltage applied tog and g . Panels (d) and (f) show line cuts along thedashed lines of Fig. 2(c) and (e), respectively, for differ-ent temperatures. These line cuts are taken for N = 0 . R C is symmetric upon magnetic field reversal both in the slowly varying background andin the CFs, as expected from a two-terminal measure-ment in the linear regime [17]. V is, on the contrary,strongly asymmetric. We first address the slowly varyingbackground of V ( B ). We will discuss the nature of theCFs below.Large CFs do not allow us to measure meaningful volt-age asymmetries at small magnetic fields. We thereforeaverage them out by a linear regression of V ( B ) in amagnetic field range between − I and show the detectortransconductance in arbitrary units. Panels (a)-(d) rep-resent different cavity configurations, with different N and N as indicated in every subfigure. Despite the largeerror bars introduced by the CFs, in Figs. 3(a), 3(b) and3(c) we observe what theory anticipates: a correlation ofthe two quantities below the last detector conductancestep (right-hand side of each subfigure) and disappear-ance of this correlation beyond the first plateau (left-hand side of each subfigure). This is the key observationfrom which we conclude that we measure a spin current.Where the detector is most energy sensitive, we observeuseful signals with ∂ B V /I ≈
70 ΩT − , with a typicalbackground of 20 ΩT − . The latter value was also typ-ically observed for N , N >
6, as shown in Fig. 3(d),where the voltage asymmetry is uncorrelated with thetransconductance. This is expected due to suppressionof spin currents by energy averaging when many modescontribute to transport. In the following we give fur-ther evidence supporting the spin current origin of theobserved voltage asymmetry.To confirm the spin related nature of our signal, weexploit a key ingredient for the spin-to-charge conver-sion: the magnetic field sensitivity of the detector QPC.A detector with zero g factor should result in a vanish-ing voltage asymmetry, regardless of the spin current in-tensity. We confirmed this prediction in the two waysshown in Fig. 4. In Fig. 4(a) we modified the measure-ment scheme such that the current flows from terminal1 to 3, while terminal 2 is used as the detector. The lat-ter is characterized in Ref. [18], and the first mode shows g = 0. As expected, we observe a vanishing voltage asym-metry for N → ◦ in the 2DHG plane, to have themagnetic field perpendicular to QPC . Along this direc- V g3 (V) ∂ B V /I ( Ω T − ) −0.500.51 ∂ V G / G ( a . u . ) −0.6 −0.4 −0.2−40−20020406080 V g4,g5 (V) ∂ B V /I ( Ω T − ) −0.500.51 ∂ V G / G ( a . u . ) −0.6 −0.4 −0.2−40−20020406080 V g4,g5 (V) ∂ B V /I ( Ω T − ) −0.500.51 ∂ V G / G ( a . u . ) −0.6 −0.4 −0.2−40−20020406080 V g4,g5 (V) ∂ B V /I ( Ω T − ) −0.500.51 ∂ V G / G ( a . u . ) (a)(c) (b)(d) N =4 N =5 N =4 N =4N =4 N =4 N =5 N =4 FIG. 4. (a) Comparison between ∂ B V /I acquired with twodifferent cavity shapes (dots and squares) both having N = N = 4 and ∂ V G /G (solid line) as a function of the voltageapplied to g . The electrical setup was modified to use QPC as detector. (b), (c) and (d) as in Fig. 3 but with the magneticfield aligned perpendicularly to QPC . tion, the g factor vanishes for all modes [15], which isa well-known anisotropy of p -type QPCs [18, 19]. Thelatter is a particularly powerful approach as it leaves thespin current unaltered and only suppresses the spin-to-charge conversion efficiency. In Figs. 4(b), 4(c) and 4(d)we show three of such measurements for the same modeconfigurations as in Figs. 3(a), . 3(b) and . 3(c). In allthe three cases, the voltage asymmetry is comparable tothe background level of Fig. 3 and uncorrelated to thedetector transconductance, proving the importance of amagnetic field sensitive measurement lead for observingan asymmetric voltage signal.The spin-to-charge signal shown in Fig. 3 reflects a ro-bust property of the system and is not linked to CFs. TheCFs are phase coherent effects originating from electro-static cavity shape distortion and magnetic flux penetra-tion [20], or from a purely in-plane field as a consequenceof the asymmetric and finite-width confinement poten-tial [21]. We carefully checked that our results do notdepend on the CFs’ pattern first by changing g , g andg while keeping N and N constant, second by apply-ing a strong voltage asymmetry on g and g . In bothcases, the CFs are completely changed without a signifi-cant modification of the voltage asymmetry extracted byaveraging. Additionally, Fig. 3(a) includes a measure-ment performed during a different cooldown (squares)[16]. Despite the completely different CFs’ fingerprint,an identical voltage asymmetry is obtained, proving therobustness of the measured effect. As visible in Fig. 2(f),coherent contributions are almost entirely suppressed at T = 530 mK, which is a standard temperature scale for the disappearance of coherent effects in quantum dotswith few open channels [22]. The average signal is, onthe other hand, more resistant to temperature increasesbecause of its diffusionlike origin [13]. We performedadditional analysis of the temperature and charge cur-rent amplitude dependence of the spin-to-charge signal.These measurements, reported in the Supplemental Ma-terial [15], confirm the distinct nature of CFs and thespin-to-charge signal, as well as the fact that the spin-to-charge signal is a linear effect.So far, we discussed the presence of a robust spincurrent in QPC visible from the slowing varying back-ground of ∂ B V /I . As discussed in the context of theMSHE, CFs might also reflect the presence of mesoscopicspin CFs [9–12]. Although the CFs occasionally show afinite zero field slope [see Fig. 2(f)], it was not possible tounivocally assign them to spin related or orbital effects,not considered in Ref. [14]. In particular, we could nottest the QPC transmission dependence of ∂ B V /I forsingle CFs due to the influence of g and g on the cavityshape.We now evaluate the spin current amplitude for N =0 .
5. With the measured detector sensitivity ¯ hω =0 .
46 meV, its g factor g = 0 .
27 and the typical volt-age asymmetry of ∂ B V /I = 60ΩT − , Eq. (1) gives I ( S )3 = 174 pA. We compare this value with theoreti-cal predictions on geometrical correlation induced spincurrents. The spin transmission of QPC , calculated fora three-terminal cavity in the spin chaos regime, is [13]: (cid:104) T ( S )13 (cid:105) = C ξ l SO k F N N N + N + N ≈ . × C . (2)To evaluate this expression we used ξ = 1, appropriatefor a ballistic dot, N = N = 4, N = 0 . C is a systemspecific prefactor, of order unity. Neglecting spin flipscaused by QPC itself, the expected spin current is I ( S )3 ≈ e h (cid:104) T ( S )13 (cid:105) ( V − V ) = C ×
134 pA (3)for a charge current of 4 nA. The very good agreementwith our measurement further supports the interpreta-tion that our signal goes beyond a mere spin current de-tection, but provides a quantitatively reliable magnitude.As shown by Eq. 2, the spin conductance depends on N ,allowing larger spin currents to be generated by openingQPC further. However, since the detection scheme re-quires N = 0 .
5, we could not probe this scenario.Similarly to bulk materials, the spin-to-charge con-version efficiency of the cavity can be expressed viathe spin Hall angle Θ, defined as the ratio betweenspin and charge current densities. In our case we getΘ = ( I ( S )3 /N ) / ( I/N ) ≈ N for N (cid:28) N , N . This efficiency is substantially higher thanany reported for semiconductors [4], making this systeminteresting for future semiconductor spintronics applica-tions. SUPPLEMENTARY MATERIAL
In this Supplemental Material section we provide ad-ditional information on the wafer structure used for theexperiment and on the techniques adopted to measurethe sample and characterize the detector QPC. We fur-thermore show the current and temperature dependenceof the spin-to-charge signal discussed in the main text.We provide an analytical treatment of the spin-to-chargeconversion effect directly applicable to our experimentand we discuss its dependence on the detector QPC con-ductance.
Materials and Methods
The two-dimensional hole gas (2DHG) in use consistsof a 15 nm GaAs quantum well placed 45 nm below thesurface. The sample is grown along the [001] directionand remotely doped with carbon. The wafer has beenextensively characterized by magnetotransport measure-ments in Ref. [23]. The total hole density is n = 3 . × m − and the strong Rashba spin-orbit interaction(SOI) results in a splitting of the dispersion relation inthe two subbands with different total angular momentumprojection along the growth direction. The densities ofthe two spin-orbit split subbands have been derived fromthe periodicity of the Shubnikov-de Haas oscillations as n = 1 . × m − and n = 1 . × m − result-ing in a cubic Rashba parameter β = 1 . × − eVm [24]. The spin-orbit length, defined as the length scaleover which the electron spin rotates by 2 π is calculated,as in Ref. [13], as l SO = v F τ SO = (¯ hk F /m ∗ )(2¯ h/ ∆ SO ).∆ SO = 2 β R k F is the spin-orbit energy splitting, m ∗ thehole effective mass and k F the Fermi wavevector. Be-cause of the large difference in m ∗ and k F between thetwo spin-orbit split subbands [23], we use their density-averaged values m ∗ = 0 . m e and k F = 1 . × m − .Two nominally identical cavities were defined by elec-tron beam lithography and wet etching (see Fig. 1(a)of the main text). The samples were measured in a di-lution refrigerator with a base temperature of 110 mKusing standard low-frequency lock-in techniques. Thetilt angle between 2DHG and magnetic field could betuned in-situ with an accuracy of less than 0 . ◦ . Theasymmetry in V ( B ) and the conductance fluctuations(CFs) visible in Fig. 2(f) of the main text are genuine ef-fects due to the in-plane magnetic field only. Tilting the2DHG angle with respect to the external magnetic fieldbetween − . ◦ and 0 . ◦ leaves V ( B ) unaffected, prov-ing that an eventual residual out-of-plane component ofthe magnetic field is negligible for the effects under dis-cussion. The voltage measurements are performed in alongitudinal configuration, so that an out-of-plane mag-netic field would not result in the appearance of a Hallslope. Changing the orientation of the device with re- g g g g g QPC1 QPC2QPC3 IV V /2 in -I-V /2 in FIG. S.1. False color atomic force micrograph of the sampleunder study, together with a schematic of the electrical setupused to characterize QPC . spect to the in-plane component of the magnetic fieldrequired warming up the sample and manually changingits bonding configuration. The electronic properties ofthe sample and the characteristics of the QPC remainedlargely unchanged by this carefully done procedure. Thetwo devices showed quantitatively comparable behavior,reproducible over multiple cool-downs. In the main textwe show data from a single sample, characterized by alarger tunability of the three QPCs used as leads. Characterization of the detector QPC
In the following we describe in more detail the char-acterization measurements performed on QPC . Thedata presented allows the extraction of the QPC g -factorand energy sensitivity, used to quantify the spin currentintensity from the spin-to-charge signal. Furthermore,we show the existence of a pronounced Zeeman splittinganisotropy that allowed us to obtain the data presented inFig. 4 (b), (c) and (d) of the main text. A similar proce-dure has been discussed in greater detail in Ref. [18, 19].Fig. S.1 shows the electrical setup used for the char-acterization of QPC . A low-frequency AC bias V in =15 µ V was symmetrically applied to the cavity leads 2and 3 and the current flowing in the structure was mea-sured as a function of the voltage applied to g and g .At the same time, the voltage drop between terminal 1and 3 was recorded, allowing to extract a four terminalQPC conductance that does not depend on QPC andQPC . During the characterization of QPC , QPC andQPC were set to a very high conductance by negativelybiasing g , g and g . We carefully checked that the pre-cise voltages applied to g , g and g have no effects onthe presented results.Via finite bias measurements, we extracted the voltagedependent lever arm of g and g . The lever arm al-lows one to convert the gate voltage axis into energy and FIG. S.2.
QP C transconductance (numerical derivative of the linear conductance with respect to the gate voltage axis) as afunction of gate voltage and magnetic field for different field orientations. Dark lines indicate high transconductance, brightregions indicate low transconductance. (a) Magnetic field applied perpendicular to the plane of the 2DHG. (b) Magneticfield applied in the plane of the 2DHG and along the QPC axis. (c) Magnetic field applied in the plane of the 2DHG andperpendicular to the QPC axis. extract quantities such as the g -factor and the energysensitivity ¯ hω from a conductance measurement. For theexperiment under consideration, the determination of thelever arm is irrelevant if both ¯ hω and the g -factor are ex-tracted from the same conductance plot and in a narrowgate voltage range, as we do here. In fact, as shown byEq. (1) of the main text, the spin-to-charge conversionamplitude is only given by the ratio ¯ hω/g that does notdepend on the gate lever arm.Close to a conductance G = e /h , the relevant regimefor the effects presented in the main text, the lever armwas measured to be 4 meVV − . We determined ¯ hω , i.e.the curvature of the harmonic potential, by fitting a sad-dle point model [25] to the QPC conductance [26]. Sincewe do know the lever arm in this case, we can convertthe curvature into energy units by fitting the equation: G ( E ) = 2 e h N (cid:18)
11 + exp (2 π ( E − E ) / (¯ hω )) (cid:19) , (S.1)with the fitting parameters being a constant energy shift E and the potential curvature ¯ hω . N is the modenumber of the plateau under consideration, in this case N = 1.The QPC is characterized by a strong g -factoranisotropy, typical for QPCs embedded in 2DHGs grownalong the [001] crystallographic direction [18, 19]. TheQPC transconductance (numerical derivative with re-spect to gate voltage) is shown in Fig. S.2 for three dif-ferent magnetic field orientations. A finite spin splittingis present when the magnetic field is oriented perpen-dicularly to the plane of the 2DHG (Fig. S.2(a)) or inthe plane of the 2DHG and aligned along the QPC axis(Fig. S.2(c)). No spin splitting up to 12 T is visible whenthe magnetic field is aligned in the plane of the 2DHGand perpendicular to the QPC axis (Fig. S.2(b)). In orderto use the QPC to detect a spin current via the spin-to-charge conversion scheme, it is necessary to have a finite g -factor (see Eq. (1) of the main text). In the presentcase, this is possible by performing the experiment with the magnetic field oriented as in Fig. S.2(b). The g -factoris obtained from Fig. S.2(b) by tracking the separation ingate voltage between spin split plateaus as a function ofan in-plane magnetic field. The gate voltage separationis then converted into Zeeman energy using the gate de-pendent lever arm. For the first mode we find g = 0 . ◦ in theplane of the 2DHG, obtained in the situation shown inFig. S.2(c). In this case all the modes are characterizedby g = 0.As visible in Fig. S.2(b), the levels splitting as a func-tion of an in-plane magnetic field is not symmetric ingate voltage. For each pair of spin-split levels, the leftbranch rises in energy (more negative values of gatevoltage) faster then the right branch. The anomalousmagnetic field dependence of the QPC levels in an in-plane magnetic field was recently studied in Ref. [18] andfound to be caused by a quadratic SOI peculiar for holegases. This anomalous spin-splitting makes the points ofstrongest magnetic field dependence of G not to corre-spond to the points of strongest gate-voltage dependence,as assumed in Ref. [14], but to be slightly shifted to morenegative gate voltage. This effect might explain the smallhorizontal offset between ∂ B V /I and ∂ V G /G seen inFig. 3(a), (b) and (c) of the main text. Current and temperature dependence of thespin-to-charge signal
The spin current detection method we employ is lim-ited to the linear transport regime [27], estimated tobreak-down in our samples for currents of a few nA. In thelinear regime, ∂ B V /I should be independent of I . Weplot this current dependence in Fig. S.3(a) for the sameconfiguration as Fig. 2(a) of the main text and N = 0 . ∂ B V /I is independent of I for I < −0.5 −0.4 −0.3 −0.2 −0.1 0−40−20020406080 V g3,g4 (V) ∂ B V /I ( Ω T − ) T = 109 mKT = 250 mKT = 530 mK0.5 5 5010204080
I (nA) ∂ B V /I ( Ω T − ) (a) (b) I = 4 nAT = 109 mKV g3,g4 = −0.115 V
FIG. S.3. (a) ∂ B V /I as in Fig. 3(a) of the main text and N = 0 . ∂ B V /I as inFig. 3(a) of the main text for different temperatures. V ( B ) is a linear effect. For currents higher than 5 nA,in addition to sample heating, we enter the non-lineartransport regime where the detector voltage asymmetryand transconductance are not necessarily linked.Fig S.3(b) shows the temperature dependence of theextracted ∂ B V /I as in Fig. 3(a) of the main text, forthree different temperatures. At T = 530 mK we stillrecover the same gate dependence of ∂ B V as shown inFig. 3(a)of the main text, but with a reduction in peak-to-peak height by a factor of 15. The signal eventuallydisappears entirely at higher temperature because of theensuing energy-averaging, affecting also the geometriccorrelation corrections. Furthermore, for T >
600 mK, G loses the step-like behavior (hence its energy sensitiv-ity), becoming a smooth function of the gate voltage. Spin-to-charge conversion
We review here the theoretical basis for the spin-to-charge conversion effect in a three-terminal mesoscopiccavity as the one depicted in Fig. 1(a) of the main text.We extend the theory presented in Ref. [14] to prove theproportionality between ∂ B V /I and ∂ V G /G .The cavity under consideration has three contacts (la-beled 1,2 and 3), and each of them is at a voltage V i and carries N i spin degenerate modes. We consider thesituation where N ≤ N , N (cid:29)
1. By applyinga voltage bias V − V between leads 1 and 2, a chargecurrent I will flow in the cavity. If contact 3 is grounded,a charge current I (0)3 will flow in it. In the following weare interested in the situation in which no charge currentflows in contact 3 at zero magnetic field. This situationcan be achieved either by applying a voltage V that sets I (0)3 to zero at zero magnetic field, or by leaving it floatingand connected to a volt meter that measures V . In thefirst case, the application of a magnetic field will makethe current vary from zero, in the second case the cur-rent will always remain zero and the measured V willvary. As we will show in the following, the zero fieldspin current I ( S )3 in contact 3 is directly proportional tothe zero-field derivative of I , or V , with respect to the in-plane magnetic field: I ( S )3 = ¯ hωπµ ∂ B I (0)3 | B =0 , (S.2) I ( S )3 = ¯ hωπµ e h (cid:16) − T (0)33 (cid:17) ∂ B V | B =0 , (S.3)where ¯ hω is the magnetic field energy sensitivity, µ = gµ B / − T (0)33 )is the charge transmission coefficient of contact 3. TheseQPC parameters are easily measured experimentally.In the following theoretical treatment we do not in-clude the presence of any orbital effect. The latter as-sumption is well justified if no out-of-plane magnetic fieldis applied. An in-plane field can give rise to other or-bital effects [21] which will not be accounted for in thissection, assuming that these are small enough. We willalways consider that the voltages applied to the systemare within the linear response regime (i.e. small com-pared to other energy scales). The generic current I ( α ) i at a contact i can be calculated using Landauer-B¨uttikerformalism, resulting in: I αi = (cid:88) j (cid:16) N j δ ij δ α − T ( α ) ij (cid:17) V j , (S.4)where i = 1 , , α = 0 , x, y, z denotes the spin polarization of the current. The generictransmission coefficient is: T ( α ) ij = (cid:88) m ∈ i,n ∈ j T r (cid:16) t † mn σ ( α ) t mn (cid:17) , (S.5)where σ ( α ) are spin matrices, with σ (0) the identity ma-trix. The 2 × t mn indicate the probability of anelectron entering the cavity from the n -th mode of QPC j to exit the cavity from the m -th mode of QPC i , theirelements t σσ (cid:48) m,n take into account spin flipping. It can beshown that the transmission coefficients for charge andspin are: τ (0) ij = (cid:88) σσ (cid:48) τ σσ (cid:48) ij (S.6) , τ ( S ) ij = (cid:88) σσ (cid:48) στ σσ (cid:48) ij . (S.7)The transmission probabilities from contact 1 or 2 to con-tact 3 are assumed to take the form: T σσ (cid:48) i ( B ) = τ σσ (cid:48) i ( B )Γ ( E F − σµB ) , (S.8)hence they can be separated into a spin dependent partand an energy dependent part. The spin affects the sec-ond term only via the Zeeman energy. In Eq. (S.8) itwas assumed that the QPC has high energy sensitiv-ity, hence Γ ( E F − σµB ) varies faster than τ σσ (cid:48) i ( B ) with B . τ σσ (cid:48) i ( B ) are phenomenological parameters, describingthe spin transmission of the QPC when it is fully open.Eq. (S.8) is valid only in the limit when an electron re-flected back in the cavity from contact 3 has a negligibleprobability to come back to contact 3 again. This limitis achieved when N , N (cid:29) N .Using Landauer-B¨uttiker expressions for charge andspin current through contact 3 one has: I (0)3 = e ¯ h (cid:16) T (0)31 ( V − V ) + T (0)32 ( V − V ) (cid:17) , (S.9) I ( S )3 = e ¯ h (cid:16) T ( S )31 ( V − V ) + T ( S )32 ( V − V ) (cid:17) . (S.10)Imposing I (0) = 0 allows to find an analytical form forthe spin current: V = (cid:16) T (0)31 V T (0)32 V (cid:17) / (cid:16) T (0)31 + T (0)32 (cid:17) , (S.11) I ( S )3 = e ¯ h (cid:16) T ( S )31 ( V − V ) + T ( S )32 ( V − V ) (cid:17) . (S.12)Equations (S.2) and (S.3) are obtained by evaluating ∂ B I | B =0 with constant V and ∂ B V | B =0 for I (0)3 = 0,respectively. For obtaining a simpler analytical form ofthese expressions, we will make an assumption of the spe-cific form of the QPC transmission Γ, though the resultsof this section do not qualitatively depend on this choice.We choose the saddle point potential model, which givesthe QPC energy dependent transmission probability as[25]:Γ( E F , V g , B ) = 11 + exp ( − π ( E F − αV g − σµB ) / ¯ hω ) . (S.13) The partial derivative of the QPC transmission with re-spect to magnetic field is: ∂ B Γ( E F , V g , B ) = σµ∂ V g Γ( E F , V g , B ) . (S.14)This allows us to write the magnetic field derivative ofthe charge transmission coefficients T (0)3 i in terms of spintransmission coefficients T ( S )3 i : ∂ B T (0)3 i = ∂ B (cid:88) σσ (cid:48) T σσ (cid:48) ij | B =0 (S.15)= (cid:88) σσ (cid:48) τ σσ (cid:48) ij ∂ B Γ( E F , V g , | B =0 (S.16)= (cid:88) σσ (cid:48) τ σσ (cid:48) ij σµ∂ V g Γ( E F , V g , | B =0 (S.17)= (cid:88) σσ (cid:48) τ σσ (cid:48) ij σµ Γ( E F , V g , ∂ V g Γ( E F , V g , | B =0 Γ( E F , V g , µT ( S )3 i ∂ V g Γ( E F , V g , | B =0 Γ( E F , V g , . (S.19)In the middle of the first slope in the QPC conductance,the energy sensitivity is maximal and it gives ( V g = E F ): ∂ V g Γ( E F , V g , | B =0 Γ( E F , V g ,
0) = − π ¯ hω . (S.20)With this, the zero-field derivative of the charge currentis ( V , V , V are constant): ∂ B I (0)3 | B =0 = − e ¯ h (cid:16) ∂ B T (0)31 ( V − V ) + ∂ B T (0)32 ( V − V ) (cid:17) (S.21)= − e ¯ h (cid:18) µ ∂ V g Γ( E F , V g , | B =0 Γ( E F , V g , (cid:19) (cid:16) T ( S )31 ( V − V ) + T ( S )32 ( V − V ) (cid:17) (S.22)= − µ ∂ V g Γ( E F , V g , | B =0 Γ( E F , V g , I ( S )3 (S.23)= πµ ¯ hω I ( S )3 . (S.24)Finally, Eq (S.2) is obtained solving Eq. (S.24) for I ( S )3 . Similarly, the zero field derivative of V is: ∂ B V | B =0 = − µ ∂ V g Γ( E F , V g , | B =0 Γ( E F , V g ,
0) 12 − T (0)33 he I ( S )3 . (S.25)In the point of highest energy sensitivity we have: ∂ B V | B =0 = he πµ ¯ hω − T (0)33 I ( S )3 , (S.26) which results in Eq. (S.3) upon solving for I ( S )3 .It is interesting, in the light of our experimental re-sults, to investigate the behavior of ∂ B V | B =0 around thepoint of highest energy sensitivity. 2 − T (0)33 is the exact expression for the charge current transmission coefficientof contact 3. It can be approximated as (see Eq. (S.8)):2 − T (0)33 = T (0)31 + T (0)32 = (cid:88) σσ (cid:48) (cid:16) τ σσ (cid:48) + τ σσ (cid:48) (cid:17) Γ( E F , V g, B ) . (S.27)We can apply the approximation from Eq. (S.8) to Eq. (S.10), finding a relation between I ( S )3 and Γ: I ( S )3 = − e h (cid:32)(cid:88) σσ (cid:48) τ σσ (cid:48) ( V − V ) + (cid:88) σσ (cid:48) τ σσ (cid:48) ( V − V ) (cid:33) Γ( E F , V g , B ) . (S.28)Combining the last three equations we get the expression: ∂ B V | B =0 = µC ∂ V g Γ( E F , V g , | B =0 Γ( E F , V g , , (S.29)where C is a prefactor containing the voltages V i andthe coefficients τ σσ (cid:48) i . It is assumed to be constant withrespect to gate voltage. The results obtained here can besummarized with the following proportionality relation: ∂ B V | B =0 ∝ ∂ V g Γ( E F , V g , | B =0 Γ( E F , V g , . (S.30)It is worth reminding that the last proportionality is validin the limit N , N (cid:29) N and N ≤ τ σσ (cid:48) i are supposed to weakly depend on magnetic field.In the absence of SOI, reversing the magnetic field di-rection reverses the sign of the spin polarization S . Inthis case τ (0)3 i ( B ) = τ (0)3 i ( − B ) and τ ( S )3 i ( B ) = − τ ( S )3 i ( − B )(see Eq. (S.6) and (S.7)). Since Γ( E F , V g , B ) is an evenfunction of B , it results that T σσ (cid:48) i ( B ) = − T σσ (cid:48) i ( − B ), and I S (0) = 0 (see Eq. (S.12)): in the absence of SOI, both ∂ B I (0)3 and ∂ B V vanish.0We acknowledge Christian Gerl for growing the waferstructure. The authors wish to thank the Swiss NationalScience Foundation via NCCR QSIT “Quantum Scienceand Technology” for financial support. ∗ , R4790 (2000).[2] S. Sharma, A. Spiesser, S. P. Dash, S. Iba, S. Watanabe,B. J. van Wees, H. Saito, S. Yuasa, and R. Jansen, Phys.Rev. B , 075301 (2014).[3] M. I. D’yakonov and V. I. Perel’, JETP Lett. , 467(1971).[4] T. Jungwirth, J. Wunderlich, and K. Olejnik, Nat Mater , 382 (2012).[5] C. Br¨une, A. Roth, E. G. Novik, M. Konig, H. Buh-mann, E. M. Hankiewicz, W. Hanke, J. Sinova, andL. W. Molenkamp, Nat Phys , 448 (2010).[6] R. M. Potok, J. A. Folk, C. M. Marcus, and V. Umansky,Phys. Rev. Lett. , 266602 (2002).[7] S. K. Watson, R. M. Potok, C. M. Marcus, and V. Uman-sky, Phys. Rev. Lett. , 258301 (2003).[8] S. M. Frolov, A. Venkatesan, W. Yu, J. A. Folk, andW. Wegscheider, Phys. Rev. Lett. , 116802 (2009).[9] W. Ren, Z. Qiao, J. Wang, Q. Sun, and H. Guo, Phys.Rev. Lett. , 066603 (2006).[10] J. H. Bardarson, I. Adagideli, and P. Jacquod, Phys.Rev. Lett. , 196601 (2007).[11] Y. V. Nazarov, New Journal of Physics , 352 (2007).[12] J. J. Krich and B. I. Halperin, Phys. Rev. B , 035338(2008).[13] I. Adagideli, P. Jacquod, M. Scheid, M. Duckheim,D. Loss, and K. Richter, Phys. Rev. Lett. , 246807(2010). [14] P. Stano and P. Jacquod, Phys. Rev. Lett. , 206602(2011).[15] See Supplemental Material for material and methods,temperature and current dependence studies and a theo-retical treatment of the spin-to-charge conversion effect.[16] For a direct comparison, we horizontally shifted the redcurve to account for the different pinch-off voltage ofQPC in this cool-down.[17] H. B. G. Casimir, Rev. Mod. Phys. , 343 (1945).[18] F. Nichele, S. Chesi, S. Hennel, A. Wittmann, C. Gerl,W. Wegscheider, D. Loss, T. Ihn, and K. Ensslin, Phys.Rev. Lett. , 046801 (2014).[19] A. Srinivasan, L. A. Yeoh, O. Klochan, T. P. Martin,J. C. H. Chen, A. P. Micolich, A. R. Hamilton, D. Reuter,and A. D. Wieck, Nano Letters , Nano Lett. , 148(2013).[20] C. Beenakker and H. van Houten, in Semiconductor Het-erostructures and Nanostructures , Vol. Volume 44, editedby H. Ehrenreich and D. Turnbull (Academic Press,1991) pp. 1–228.[21] D. M. Zumb¨uhl, J. B. Miller, C. M. Marcus, V. I. Fal’ko,T. Jungwirth, and J. S. Harris, Phys. Rev. B , 121305(2004).[22] A. G. Huibers, J. A. Folk, S. R. Patel, C. M. Marcus,C. I. Duru¨oz, and J. S. Harris, Phys. Rev. Lett. ,5090 (1999).[23] F. Nichele, A. N. Pal, R. Winkler, C. Gerl, W. Wegschei-der, T. Ihn, and K. Ensslin, Phys. Rev. B , 081306(2014).[24] R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems , Springer Tractsin Modern Physics, Vol. 191 (Springer-Verlag, Berlin,2003).[25] M. B¨uttiker, Phys. Rev. B , 7906 (1990).[26] C. R¨ossler, S. Baer, E. de Wiljes, P.-L. Ardelt, T. Ihn,K. Ensslin, C. Reichl, and W. Wegscheider, New Journalof Physics , 113006 (2011).[27] P. Stano, J. Fabian, and P. Jacquod, Phys. Rev. B85