Nonlinear analog spintronics with van der Waals heterostructures
S. Omar, M. Gurram, K. Watanabe, T. Taniguchi, M.H.D. Guimarães, B.J. van Wees
NNonlinear analog spintronics with van der Waals heterostructures
S. Omar, ∗ M. Gurram, K. Watanabe, T. Taniguchi, M.H.D. Guimar˜aes, and B.J. van Wees The Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG, Groningen, The Netherlands National Institute for Material Science, 1-1 Namiki, Tsukuba, 305-044, Japan (Dated: April 23, 2020)
The current generation of spintronic devices,which use electron-spin relies on linear operationsfor spin-injection, transport and detection pro-cesses. The existence of nonlinearity in a spin-tronic device is indispensable for spin-based com-plex signal processing operations. Here we for thefirst time demonstrate the presence of electron-spin dependent nonlinearity in a spintronic de-vice, and measure up to th harmonic spin-signalsvia nonlocal spin-valve and Hanle spin-precessionmeasurements. We demonstrate its applicationfor analog signal processing over pure spin-signalssuch as amplitude modulation and heterodynedetection operations which require nonlinearityas an essential element. Furthermore, we showthat the presence of nonlinearity in the spin-signal has an amplifying effect on the energy-dependent conductivity induced nonlinear spin-to-charge conversion effect. The interaction ofthe two spin-dependent nonlinear effects in thespin transport channel leads to a highly efficientdetection of the spin-signal without using ferro-magnets. These effects are measured both at 4Kand room temperature, and are suitable for theirapplications as nonlinear circuit elements in thefields of advanced-spintronics and spin-based neu-romorphic computing. Nonlinear elements, such as transistors and diodes, ledShockley and coworkers [1] to lay the foundation for elec-tronics revolution, and underlie the modern-day electron-ics. However, such elements lack in the field of spintron-ics. Major ideas in the field of spintronics thus far havesuggested the possibility of achieving a gate operationemploying, for example, a Datta-Das transistor [2, 3].The possibility of spin-signal amplification and process-ing has not been explored experimentally and forms amore fundamental building block to replace conventionalelectronics with the spin-based analogues [4–8].The current generation of state-of-the-art spintronicdevices can only execute linear operations. In such de-vices, the output differential spin-signal v s [4, 9] scaleswith the applied input ac charge current i , i.e. v s = p inj R s p det i (1)Here, p inj(det) , the differential spin injection(detection)efficiency and R s , the effective spin resistance of the spintransport channel [11] are constant, and thus the relation v s ∝ i is established. Therefore, a nontrivial operationrequiring nonlinearity can not be executed.Interestingly, the differential spin-injection efficiency p inj of ferromagnetic (FM) tunnel contacts with atom-ically flat and pinhole-free thin hBN flakes as a tunnelbarrier [12, 13] depends on the input dc bias current I [1, 15, 16], and renders them as a viable platform todemonstrate spin-dependent nonlinear effects. Nonlinear spin-injection
We perform nonlinear spin-transport experiments ona van der Waals heterostructure of Graphene (Gr), en-capsulated between a thick boron nitride (hBN) sub-strate and a trilayer hBN tunnel barrier with ferro-magnetic cobalt contacts as shown in Figs. 1(a,b). Westart by characterizing the tunnelling behaviour of thecontacts. Contacts with(out) the tunnel-barrier shownonlinear(linear) current-voltage characteristics [inset ofFig. 1(c)] for an applied dc charge current I and themeasured voltage V c across the contact in a three-probemeasurement geometry. Next, we probe the presence ofnonlinear behaviour in the nonlocal signal v nl in a four-probe measurement geometry. For an input ac current i at frequency f =6 Hz, v nl is measured using the scheme inFig. 1(a), and its Fourier transform is shown in Fig. 1(c).For a linear device, an applied current at a certain fre-quency should yield a voltage at the same frequencyalone. The appearance of voltage at integral multiplesof the input-current frequency, the so called higher har-monics, is a smoking gun signature of nonlinearity. In ourmeasurements, higher harmonics at 2 f, f, ... appear in v nl only when the tunnel contact C1 is used as an injector(blue spectrum in Fig. 1(c)), and thus underline the cru-cial role of tunnel contacts for introducing nonlinearityin v nl .The concept of nonlinear spintronic measurements isschematically demonstrated in Fig. 1(d). For an inputcharge current i + I at a ferromagnetic contact, higherharmonics in the spin-signal v s are measured at the out-put, due to the nonlinearity in the spin-injection pro-cess, present in a spintronic device. To probe the spin-dependent origin of nonlinearity in v nl , we perform biasdependent nonlocal spin-valve (SV) measurements [1].Here, we apply an ac+dc charge current i + I and measurethe 1 st harmonic response of v nl via the lock-in detectionmethod. An in-plane magnetic field B || is swept to switch a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r b B || B ┴ a L= µm Cobalt3L-hBNGraphenebottom-hBN Si/SiO I + i C1 v nl C2 = ++ C1 C2 +- C1 C2 d Ii p inj f ( I ) p det R s v nl - 1 0 1- 1010 6 1 2 1 8 2 4 3 01 0 - 7 - 6 - 5 - 4 - 3 - 2 - 1 I( m A) V c ( V ) c - 2 0 002 0 0 t u n n e l ( C 1 ) t r a n s p a r e n t ( C 2 ) FFT vnl(V2/Hz) f ( H z ) f FIG. 1.
Device geometry and tunnel characteristics. a.
Graphene encapsulated between a thick hBN at the bottomand a 3L-hBN tunnel barrier on the top. The cobalt (inner) injector electrode C1 is located on top of the tunnel barrier andthe (inner) detector electrode C2 is directly in contact with the graphene flake. The outer injector and detector electrodes(transparent orange) are far enough to be spin-sensitive, and serve as reference electrodes. A charge current i + I is appliedacross C1 for spin-injection and a nonlocal ac voltage v nl is measured at C2 via the lock-in detection method. b. An opticalimage of the stack with the actual positions of electrodes drawn schematically. The hBN-tunnel barrier is highlighted withfalse blue colour. c. An ac charge current i =50nA(20 µ A) at f =6 Hz is applied at C1 (C2) and the Fourier transform of thenonlocal signal measured at C2(C1) is plotted in blue(red). In the inset, I-V characteristics of the tunnel contact C1 (blue)and the transparent contact C2 (red). d. The concept of nonlinearity is presented schematically via a circuit diagram. Asinusoidal charge current i along with a dc current I is applied at the input of a nonlinear element (inside the triangle) and adistorted non-sinusoidal spin-signal is measured at the output. The harmonic components which construct the output signalare also shown. The equivalent circuit representing the bias dependent spin-injection ( p inj = f ( I )) and spin transport ( R s ) ishighlighted in pink. the magnetization-orientation of C1 and C2 from paral-lel to anti-parallel and vice-versa using the connectionscheme in Fig. 1(a). Via SV measurements, we obtainbackground free pure spin-signal v s = v pnl − v apnl2 , where v apnl ( v pnl ) is the nonlocal signal v nl measured at the (anti-)parallel magnetization-direction alignment of the elec-trodes C1 and C2, as labeled in Fig. 2(a). In order toobtain the bias dependence of the spin-signal, we mea-sure v p(ap)nl as a function of I , as shown in Fig. 2(d), andobtain v s . At I =0, there is a very small spin-signal v s ∼ I acrossthe injector electrode, in line with the previous studieson Gr-hBN tunnel barrier systems [1, 15], v s increases inmagnitude and changes its sign on reversing the polarityof I (Fig. 2 (d)). Similarly, we also measure the 2 nd and3 rd harmonic spin-signals via SV measurements and itsbias dependence, as shown in Figs. 2(b,c,e,f). The unam-biguous measurement of the higher harmonic spin-signalsclearly suggests a presence of nonlinear processes in the spin-signal.To confirm the spin-dependent origin of the nonlinear-ity in the spin-signal, we perform Hanle spin-precessionmeasurements on 1 st , nd and 3 rd harmonic spin-signals.Here, for a fixed in-plane magnetization configuration ofthe injector-detector electrodes (parallel or anti-parallel),as labeled in Figs. 2(a,b,c), a magnetic-field B ⊥ is ap-plied perpendicular to the plane of the device, as shownin Fig. 1(a). The injected in-plane spins diffuse towardsthe detector and precess around B ⊥ with the Larmor fre-quency ω L ∝ B ⊥ . The whole dynamics is given by theBloch equation, D s (cid:53) −→ µ s − −→ µ s τ s + −→ ω L × −→ µ s = 0, with thespin diffusion constant D s , spin relaxation time τ s , spin-accumulation −→ µ s = v s /p det in the transport channel, andthe spin diffusion length λ s = √ D s τ s . The measured 1 st ,2 nd and 3 rd harmonic Hanle curves are fitted with thesolution to the Bloch equation. From the fitting, we con-sistently obtain D s ∼ s − and τ s ∼ λ s ∼ µ m for the 1 st and higher harmonic - 4 0 - 3 0 - 2 0 - 1 0 0- 2 0 0- 1 0 00 - 4 0 - 3 0 - 2 0 - 1 0 0- 3 0- 2 0- 1 00 - 4 0 - 3 0 - 2 0 - 1 0 0- 1 0- 50 r d h a r m .2 n d h a r m .1 s t h a r m .v pn l vnl(nV) B || ( m T ) I = 1 3 0 n Av a pn l s v pn l vnl(nV) B || ( m T ) I = 0 n Av a pn l cb v pn l vnl(nV) B || ( m T ) I = 3 0 n Av a pn l a - 0 . 2 - 0 . 1 0 . 0 0 . 1 0 . 2- 1 0 0- 5 005 01 0 01 5 0 - 0 . 2 - 0 . 1 0 . 0 0 . 1 0 . 2- 5051 01 52 02 5 - 0 . 2 - 0 . 1 0 . 0 0 . 1 0 . 2- 1 0- 5051 0 fed vs(nV) I ( m A ) 9 01 0 01 1 01 2 0 v pn l v a pn l vnl( m V) vs(nV) I ( m A ) - 1 5- 1 0- 5051 0 v pn l v a pn l vnl( m V) vs(nV)
I ( m A ) - 6- 3036 v pn l v a pn l vnl( m V) - 1 0 0 - 5 0 0 5 0 1 0 00123 05 01 0 01 5 0 - 1 0 0 - 5 0 0 5 0 1 0 00246 02 04 06 0 - 1 0 0 - 5 0 0 5 0 1 0 0013467 051 01 52 02 5 ih vs(nV) Rs( W ) B ^ ( m T )D s ~ 0 . 0 2 4 m s - 1 t s ~ 6 5 2 p s l s ~ 4 m m g vs(nV) Rs( W / m A) B ^ ( m T )D s ~ 0 . 0 2 5 m s - 1 t s ~ 6 4 3 p s l s ~ 4 m m vs(nV) Rs( W / m A2) B ^ ( m T )D s ~ 0 . 0 2 m s - 1 t s ~ 7 3 3 p s l s ~ 3 . 8 m m FIG. 2.
Higher harmonic spin-signals. a. st , b. nd and c. rd harmonic spin-valve measurements. d-f. spin-signal v s (orange) as a function of I applied at C1, using the measurement geometry in Fig. 1(a). The ac injection current i is keptfixed at 50nA. v pnl (red) and v apnl (blue) are the nonlocal signal v nl measured at the parallel and anti-parallel magnetizationconfigurations of the injector-detector electrodes, respectively. g. st , h. nd and i. rd harmonic Hanle spin-signal v s as afunction of out-of-plane magnetic field B ⊥ . Hanle data is symmetrized in order to remove the linear background and is offsetto zero. SV measurements in (a)-(f) are performed at RT, and Hanle curves in (g)-(h) are measured at 4K. measurements in Figs 2(g-i). Since the spin transportparameters are the same for all harmonics, we concludethat the higher harmonic spin-signals do not have its ori-gin in the spin-transport process, and pinpoint the originof the spin-dependent nonlinearity to the spin-injectionprocess.To understand the concept of nonlinearity during spin-injection, we now develop an analytical framework. As the differential spin-injection polarization depends on theinput dc bias current I , the expression for p inj using theTailor expansion around I = 0 with a small ac chargecurrent i can be written as: p inj ( i ) | I =0 = p (1 + C i + C i + ... ) (2)where p inj = p in the absence of nonlinear processes,which are enabled via the nonzero constants C , C , ... .Now, using Eq. 1, we obtain: v s ∝ p i + p C i + .... (3)which enables us to measure the presence of higher har-monic spin-signals ∝ i , i , ... due to the nonlinearity in-troduced by the spin-injection process in Eq. 2.As shown in the spin-transport measurements in Fig. 2,the nonlinearity can be experimentally probed by usingthe mixed signal (ac+dc) measurements. When an inputcurrent i + I is applied to such nonlinear system, the ex-pression for the 1 st harmonic spin-signal v s , obtained byreplacing i with i + I in Eq. 3, acquires a different func-tion form (see Supplementary Material for derivation)and contains a bias I dependent term : v s ∼ { p (1 + 2 C I ) } R s p det i. (4)As a consequence of the nonlinearity present in the spin-signal (Eq. 3), additional terms with the mixing of i and I appear, and now p inj ∼ p (1+2 C I ) is obtained insteadof p (at I = 0). For such case, one would expect a gainin p inj ∝ I . Indeed, corroborating with the hypothesisin Eq. 4, v s increases in magnitude with the applied dcbias I and reverses its sign with the dc current polarity(Fig. 2(d)).Similarly, the expressions for n th ( n ≥
2) harmoniccomponents of v s ∝ ( C n − + C n I ) i n due to nonzero C j sare obtained using the mixed signal analysis (see Sup-plementary Material for detailed expressions). Here, thecontribution from C n − , i.e. the n th order term in Eq. 3would appear even if only the ac current is applied. Inpresence of a nonzero I , the higher order term C n wouldalso contribute to the n th order spin-signal and introducethe dc bias dependence on the spin-signal.For SV measurements in our device, we can only mea-sure the even harmonic spin-signal , i.e. 2 nd (Fig. 2(b,e))and 4 th (Supplementary Material) harmonic using thepure ac current injection ( I = 0). However, similar to the1 st harmonic spin-signal, higher odd (3 rd ) harmonic spin-signal (Fig. 2(c,f)) can be measured unambiguously onlywith the application of the dc bias. When a nonzero I isapplied, the contribution of even harmonic signals cou-ples to the odd harmonic spin-signals, and now the oddharmonic responses can also be measured. The domi-nance of only even harmonic components in the spin-signal is peculiar, and is not clear at the moment. Also,the bias-dependent behaviour of higher harmonic spin-signals can be explained via the expressions obtainedfrom the mixed-signal analysis only near the zero-bias,where higher ( ≥ th ) harmonic components do not playa major role. A complete understanding of this behaviourwarrants the inclusion of higher order terms in the expres-sion for contact polarization as well as higher harmonicSV measurements for the estimation of the proportional-ity constant C j ( ≥ th harmonic). Analog signal-processing of spin-signal due tononlinear effects
The presence of nonlinearity which gives rise to signal-amplification , is fundamental to analog signal-processingoperations [17]. In our spintronic device, we exploit thespin-dependent nolinearity and demonstrate its applica-tions straightaway by performing the spin analogues ofwell established analog electronic operations.
Amplitude modulation
For amplitude modulation (AM) signal-processing [18],a modulating input i m along with a reference input i ref ,both having the same frequency f , are applied to a non-linear element (Fig. 3(a)). As a result of signal mixing,for our nonlinear spintronic device, the output spin-signal v s ∝ ( i ref + i m ) is detected at frequency 2 f . For a con-stant i ref , if i m << i ref , the measured spin-signal will be ∝ i ref i m , implying the effect will be linear in i m at thedetection frequency 2 f ( 2 nd harmonic response), and wecan realize an analog spin-signal multiplier.In order to measure this effect, we inject i ref = 200nA and modulate i m in the range of 0-120 nA (both at f =7 Hz) at the injector (Fig. 1(a)). We measure the 2 nd harmonic v s via SV measurements. The measured spin-signal is linear in i m (Fig. 3(b)) and thus the device actsas a spin-signal multiplier. For the other situation, i.e.when i m >> i ref , v s ∝ ( i m ) . In this case, we fix i ref =30nA and modulate i m in the range 30-60 nA. The mea-sured response of v s (Fig. 3(c)) clearly deviates from theearlier measured linear response in Fig. 3(c). However,due to the contribution of higher-order terms to the 2 nd harmonic signal, the measurement in Fig. 3(c) is betterexplained by the 4 th order polynomial fit instead of aparabolic fit. Heterodyne detection
As another demonstration of signal-processing, in aheterodyne detection method the input signal frequen-cies are not equal, i.e. f m (cid:54) = f ref , and one obtains thesignal at the heterodyne frequencies f ref ± f m at theoutput of the nonlinear element [18, 19]. In order torealize this operation, i ref at the frequency f ref = f and i m at f m = 2 f are applied at the injector input(Fig. 3(d)). The nonlinear component of the spin-signal v s is ∝ ( i ref sin(2 πf t ) + i m sin(2 π f t )) . If i m =0, onewould expect the spin-signal v s at 2 f . Interestingly,for i m (cid:54) = 0, v s ∝ i ref i m can also be detected at the1 st ( f = 2 f − f ) and 3 rd (3 f = 2 f + f ) harmonic com-ponents.In our measurements, for i m =0 and i ref =200 nA ( f =7Hz), only the 2 nd harmonic spin-signal is measured ( AM ( ) f m =f f ref =f 2ff0 filterOscillatorSignal f m =2f f ref =f f ref +f m
0 filterOscillatorSignal mixer f ref -f m f ref v s v s Amplitude modulationHeterodyne detection filter ad fe cb i r e f = 3 0 n A2 n d h a r m . vs(nV) i m ( n A )2 n d h a r m .i r e f = 2 0 0 n A vs(nV) i m ( n A ) f vnl(nV) B || ( m T ) f vnl(nV) B || ( m T ) FIG. 3.
Analog spin-signal processing a.
Amplitude modulation (AM) scheme. b. AM measurement of the spin-signal v s for i m << i ref and the linear fit (black) c. i m ≥ i ref and the nonlinear fit (4th order polynomial). d. Heterodyne detectionscheme e. for i m = 0 no spin-signal is present at the frequency f and 3 f . f. For i m (cid:54) = 0, due to the frequency-shifting of thesignal present at 2 f to f and 3 f , equal strength signal appears at both frequencies. Both measurements were performed atRT. SV measurements in (e) and (f) are offset to zero for clear representation. Fig. 3(e)). When we also apply i m =150 nA at the in-put frequency 2 f , spin-valve signals of similar magni-tudes are detected at frequencies both at f and 3 f (Fig. 3(f)), which is a clear demonstration of heterodyne detection of spin-signals. Note that earlier there was nomeasurable odd (1 st and 3 rd ) harmonic spin-signal at I = 0 (Figs. 2(d,f)) due to low injection-polarization/high-noise present in the signal. Now, using the hetero-dyne detection method we can clearly measure v s in the1 st harmonic even without applying I . In fact, this effectis equivalent to applying a dc current, as both hetero-dyne and ac+dc measurements couple the higher har-monic spin-signals to the 1 st harmonic spin-signal. Thismethod can be used to detect spin-signals at low frequen-cies where the spin-dependent noise would dominate inspintronic circuits [20, 21]. Furthermore, the method canalso be used as an electrical analog of the heterodynedetection in the field on optical spin-noise spectroscopy[22, 23]. Nonlinear spin-to-charge conversion
So far we have demonstrated that the nonlinearitypresent in the spin-signal in a Gr/hBN heterostructurehas its origin in the spin-injection process, not in thespin transport parameters. However, the nonlinearity in the spin-injection has an important consequence, and canamplify another nonlinear effect present in a small mag-nitude, i.e. spin-to-charge conversion [2, 3] in the spin-transport channel. The effect requires the energy depen-dent conductivity of the transport channel and the pres-ence of spin-accumulation as prerequisites. A nonlocalcharge-signal v c due to energy-dependent spin-to-chargeconversion is given by: v c = C µ = C ( p inj R s e ) i , (5)where C is a proportinality constant (see SupplementaryMaterial for details), and we have used the relation µ s = v s e/p det and Eq. 1 to obtain the v c - i dependence.Now, to probe the spin-to-charge conversion effect andthe spin-dependent origin of the nonlocal charge volt-age, we perform Hanle measurements. Since the spin-to-charge conversion is a 2 nd harmonic effect for the appliedcharge current i , we inject a pure ac current i in the range100-400 nA and measure the 2 nd harmonic response of v p(ap)nl as a function of B ⊥ using the measurement ge-ometry in Fig. 1(a). In our measurements, we observean asymmetry between the magnitudes of v pnl and v apnl inFigs. 4(a-c), which is present in a small magnitude for i =100 nA and grows rapidly for i =400 nA to such extentthat the Hanle-dephasing of v nl is measured properly onlyin the parallel configuration. - 6 0 - 3 0 0 3 0 6 0- 6 0- 4 0- 2 002 04 06 0 - 9 0 - 6 0 - 3 0 0 3 0 6 0 9 0- 1 0 0- 5 005 01 0 01 5 02 0 0 - 3 0 0 3 0- 1 0 001 0 02 0 03 0 04 0 0 0 1 0 0 2 0 0 3 0 0 4 0 001 0 02 0 03 0 0- 5 0 0 5 002 04 06 0 - 5 0 0 5 00369 vnl(nV) B ^ ( m T )i = 1 0 0 n A b vnl(nV) B ^ ( m T )i = 3 0 0 n A e fc vnl(nV) B ^ ( m T )i = 4 0 0 n A a d a t a f i t vc(nV) i ( n A )i = 1 0 0 n A vs(nV) B ^ ( m T ) l s ~ 4 m mv s = ( v pn l - v a pn l ) / 2 v c = ( v pn l + v a pn l ) / 2i = 1 0 0 n A d l s ~ 1 . 8 m m vc(nV) B ^ ( m T ) FIG. 4.
Nonlocal nd harmonic spin-to-charge conversion. nd harmonic Hanle measurements in the parallel (green)and anti-parallel (red) configuration for i ac = a.
100 nA, b.
300 nA and c.
400 nA. The enhancement in the spin-accumulationinduced charge signal v c with i is visible in the asymmetry between v pnl and v apnl with respect to the spin-independent background(dashed black line). The measured data is symmetrized in order to remove the spin-independent linear component presentin the data and is offset to zero. All measurements are performed at 4K. 2 nd harmonic Hanle spin-precession measurementsof d. the spin-signal v s and e. the spin-accumulation induced charge-signal v c . Both data are obtained from (a) for i =100nA and fitted with the solution to the Bloch equation. From the fit in (d) and (e), we obtain λ s ∼ µ m and 1.8 µ m for thespin and charge signal, respectively. f. v c − i dependence for the 2 nd harmonic data is fitted (blue curve) with a 4 th orderpolynomial function. The dark(light)-gray dashed line is the calculated magnitude of 2 nd harmonic component of v c due to thenonlinear(linear) spin-injection. To understand the origin of this asymmetry, we plotthe nonlocal charge voltage v c = ( v pnl + v apnl ) / v s = ( v pnl − v apnl ) / v c ( B ⊥ ) in Fig. 4(e) immediately con-firms that indeed the nonlocally measured charge volt-age v c has the spin-dependent origin, and is reduced tozero in the absence of spin-accumulation (at B ⊥ ∼ µ s in Eq. 5, v c should decay with the characteristic spin-relaxationlength λ s / λ s [2]. In order to verify this hy-pothesis (Eq. 5), we fit v c - B ⊥ dependence in Fig. 4(e)with the solution to the Bloch equation, and obtain λ s ∼ µ m, which is about half the spin relaxation length ob-tained via the Hanle spin-precession measurements onthe spin-signal v s (Fig. 4(d)). The same effect appearsin the 1 st harmonic v c due to its coupling with the 2 nd harmonic effect in presence of a nonzero I (see Supple-mentary Material). In this way, we unambiguously estab-lish the spin-dependent origin and the square dependenceof the nonlocal charge voltage on spin-accumulation viaHanle measurements.Lastly, the v c - i dependence is plotted in Fig. 4(f). Thedark (light) grey dashed line is the calculated magnitudeof v c while considering the contribution from nonlinear(linear) spin injection with i ( i ) dependence on the in-jected current (see Supplementary Material for details).The measured data is in close agreement with the calcu-lated v c due to the nonlinear spin-injection, and is bet-ter fitted with a 4 th order polynomial than a parabolicfunction. Clearly, such efficient spin-to-charge conversioncannot be explained only via the linear spin-injection pro-cess, and the contribution from the nonlinear processeshas to be taken into account. In conclusion, Gr/hBNheterostructures due to the presence of nonlinear spin-injection offer a highly efficient platform to probe non-linear spin-to-charge conversion effect. The interactionof the two nonlinear effects produces a measurable effectwithout needing any additional effect such as spin-orbitcoupling [26, 27].To summarize, we, for the first time demonstrate thepresence of spin-dependent nonlinearity in a spintronicdevice via all electrical measurements. This effect is thekey ingredient in signal-processing, and opens up the por-tal for the development of the field of analog spintronics,following the pathway of the electronic revolution. Ourresults suggest that nonlinearity can be exploited in mul-tiple ways to manipulate spin-information such as viacomplex signal-processing and spin-to-charge conversion,and develop advanced multi-functional spintronic devices[2–5, 28] and spin-based neuromorphic computing [29]. METHODSa. Sample fabrication
We prepare a fully hBN encapsulated graphene stackvia a dry pick-up transfer method. The hBN (thickness ∼ /Sisubstrate and identified via optical contrast analysis us-ing an optical microscope. The thickness of hBN layeris measured via the atomic force microscopy and is ∼ substrate is brought in contact with the Gr/3L-hBN onthe PC film, the whole assembly is heated up to 150 ◦ Cand the PC film with the Gr/3L-hBN is released onto the bottom hBN substrate. Afterward, the bottom-hBN/Gr/3L-hBN stack is put in chloroform solution atroom temperature to dissolve the PC film. In order toremove the remaining polymer residues on top of the top3L-hBN layer, the stack is annealed at 250 ◦ C in Ar-H environment for 7 hours. b. Device Fabrication The electrodes are patterned via the electron-beamlithography on the PMMA (poly-methyl methacrylate)spincoated sample. Then the sample is developed in aMIBK:IPA solution for 60 seconds in order to remove thepolymer from the electron-beam exposed area. Next, toobtain the spin-sensitive electrodes, 65 nm thick cobalt isdeposited on the sample via electron-beam evaporation.In order to prevent the oxidation of cobalt, a 3 nm thicklayer of aluminium is deposited on top. The residualmetal on top of the polymer is removed by performingthe lift-off in hot acetone at 40 ◦ C. c. Measurements Measurements were performed both at 4K (Heliumtemperature) and room temperature in vacuum in a flowcryostat. Differential ac signal measurements were per-formed using low frequency lock-in detection method.For mixed signal (ac+dc) measurements and back-gateapplication, Keithley 2410 dc source was used. ∗ corresponding author; [email protected][1] Shockley, W., Sparks, M. & Teal, G. K. p-n JunctionTransistors. Phys. Rev. , 151–162 (1951).[2] Datta, S. & Das, B. Electronic analog of the electroopticmodulator. Appl. Phys. Lett. , 665–667 (1990).[3] Gmitra, M. & Fabian, J. Proximity Effects in BilayerGraphene on Monolayer WSe : Field-Effect Spin ValleyLocking, Spin-Orbit Valve, and Spin Transistor. Phys.Rev. Lett. , 146401 (2017).[4] Zeng, M. et al.
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ACKNOWLEDGEMENTS
We acknowledge J. G. Holstein, H.H. de Vries, T.Schouten, H. Adema and A. Joshua for their technicalassistance. We thank B.N. Madhushankar, RUG for thehelp in sample preparation, and A. Kamra, NTNU forcritically reading the manuscript. Growth of hexago-nal boron nitride crystals was supported by the Elemen-tal Strategy Initiative conducted by the MEXT, Japanand the CREST(JPMJCR15F3), JST. MHDG acknowl-edges financial support from the Dutch Research Council(NWO VENI 15093). This research work was fundedby the the Graphene flagship core 1 and core 2 pro-gram (grant no. 696656 and 785219), Spinoza Prize (forB.J.v.W.) by the Netherlands Organization for ScientificResearch (NWO) and supported by the Zernike Institutefor Advanced Materials.
AUTHOR CONTRIBUTIONS
S.O., M.G. and B.J.v.W. conceived the experiment.S.O., K.W. and T.T. carried out the sample fabrication.S.O. and M.H.D.G. carried out the experiment. S.O.,M.G.,M.H.D.G. and B.J.v.W. carried out the analysisand wrote the manuscript. All authors discussed the re-sults and the manuscript.
Supplementary Information
I. MODEL FOR NONLINEAR SPIN-INJECTION
Current/voltage bias dependence of the spin-injectionefficiency across the cobalt/hBN/Gr contact introducesnonlinearity in the spin-injection process. As discussedin the main text, the differential polarization depends onthe input bias current I , the expression for p inj using theTailor expansion around I = 0 can be written as: p inj ( i ) = p (1 + C i + C i + ... ) (S1)where p is the unbiased differential contact polarization.The spin-signal in the nonlocal geometry is: v s = { p inj } × R s × p det i. (S2)Here, p det is the unbiased detector polarization, R s is theeffective graphene spin resistance. By substituting Eq.S1into Eq.S2, we obtain the following expression for v s : v s = { p (1 + C i + C i + ... ) i } × R s × p det = { p × R s × p det } i + { p C × R s × p det } i + { p C × R s × p det } i + ... (S3)Due to the presence of nonlinearity in the spin-injectionprocess, even the 2 nd , 3 rd and 4 th harmonic spin-signalscan be measured. II. MIXED-SIGNAL ANALYSIS
The nonlinearity in the spin-signal can be measured viamixed-signal analysis which is a well established frame-work in the field of analog electronic-circuit design. Weuse this tool to measure and analyze the nonlinearitypresent in the spin-injection process.When two independent inputs are supplied to a linearsystem, its response will be a linear combination of theinput signals. However, this is not the case when the sys-tem possesses nonlinearity, and additional contributionswould appear due to the mixing of the independent in-put signals. In order to measure the response of the acand dc input signals and their mixing, we apply a chargecurrent I + i sin ωt . We apply a dc current using a homebuilt dc current source and the ac current using a lock-in source across the injector electrodes. Since we knowfrom the measurements presented in the main text that R s and p det remain constant, for sake of simplicity weomit the constants in Eq.S3, and to explain the analysis,we assume p inj = p (1 + C I ) and omit the higher order terms. Now, v s is: v s (cid:119) { p (1 + C ( I + i sin ωt )) } × ( I + i sin ωt ) (cid:119) { p ( I + i sin ωt ) } + p C × { ( I + i sin ωt ) } (cid:119) { p ( I + C I ) } + { p (1 + 2 C I ) i sin ωt } + { p C i sin ωt } (S4) v s in Eq. S4 has three distinct contributions, a dc, a 1 st harmonic (i.e. ∝ sin ωt ) and a 2 nd harmonic ( ∝ sin ωt )contribution, separated in curly brackets. In a lock-inmeasurement, if an ac current i is applied to the sample ata lockin reference frequency f = ω π , only the componentsof v s appearing at frequency f or at higher harmonics2 f, f, ... would be measured via the lock-in detectionmethod. Other contributions are filtered out and arenot measured via the lock-in amplifier. In order to doso, the smaple output v s in Eq. S4 at the lock-in inputis multiplied with the reference signal ∝ sin nωt , where n = 1 , , ... in order to measure the 1 st , 2 nd , ... harmoniccontributions. Then the output is low-pass filtered toobtain a dc output.In order to filter out the 1 st harmonic contribution, v s in Eq. S4 is multiplied with the reference signal sin ωt ,and v s ∝ sin ωt is only filtered out and measured in the1 st harmonic response: v s = { p (1 + 2 C I ) } × R s × p det i (S5)Here, we would like to emphasize that due to the nonlin-ear term present in contact polarization, i.e. because of anonzero C , contact polarization is not equal to p inj = p anymore and is modified to p inj = p (1 + 2 C I ). There-fore, in presence of the nonlinearity the dc and differentialcontact polarization will not be equal and the differentialone may exceed the dc polarization [1].Now, in the same way, the 2 nd harmonic component of v s can be filtered out by multiplying v s in Eq. S4 withsin 2 ωt : v s = { p C } × R s × p det i (S6)It is evident from the expression in Eq. S6 that if C (cid:54) = 0, the spin-valve effect would be observed in the2 nd harmonic measurements as well.Using the analysis presented above, higher order non-linearity can be included in the contact polarization p inj with nonzero C , ... as in Eq. S1 in the same way. With-out the loss of generality, following the arguments pre-sented above, one would expect the spin-valve effect toappear in the 3 rd harmonic measurements for C (cid:54) = 0and the C I dependent terms in the 1 st and 2 nd har-monic spin-signals, as can be seen in Fig.2 of the maintext. In this way, for highly nonlinear spin-injection pro-cesses, there will be contributions of higher order spin-injection processes appearing in the low-order terms dueto the coupling of higher order processes with the chargecurrent.0 III. NONLINEAR SPIN-TO-CHARGECONVERSION
Ferromagnets (FM) have a nonzero spin-dependentconductivity σ s , i.e. spin-up and spin-down electronsin FM materials have different conductivity σ ↑ and σ ↓ where σ s = σ ↑ − σ ↓ . In presence of a nonzero spin-current,i.e gradient of the spin-accumulation µ s = ( µ ↑ − µ ↓ ) / v c ∝ σ s ∇ µ s in the FM.However, nonmagnets (NM) have σ s = 0.Because of energy-dependent density of states ingraphene, in presence of a large spin-accumulation µ s ,spin-up and spin-down electrons experience different con-ductivity [2, 3]. Therefore, in spite of being a nonmag-net, graphene develops a nonzero spin-polarization P d away from the Fermi level, and behaves as a pseudo -ferromagnet. As a consequence, a nonlocal charge voltage v c is developed along the spin-transport channel length: v c = − P d ( µ s /e ) (S7)where P d is the spin-to-charge conversion efficiency andcan be represented as: P d = − µ s × σ × δσδE (S8)where σ is the energy-dependent conductivity ofgraphene. Now, v c can be written as : v c = 1 σ × δσδE × ( µ s ) e = C µ s ∝ i (S9)The proportionality constant C can be derived fromthe carrier-density dependent conductivity measure-ments of graphene and the density of states in graphene(bilayer graphene in our case). The procedure is as fol-lows:The total number of carriers n can be calculated usingthe relation: n = (cid:90) E F ν ( E ) dE (S10)where the density of states ν ( E ) of the BLG is: ν ( E ) = g s g v π ¯ h v F (2 E + γ ) (S11)Here g s and g v are electron spin and valley degener-acy(=2), ¯ h is the reduced Planck coefficient, v F =10 m/sis the electron Fermi velocity, and γ =0.37 eV is the inter-layer coupling coefficient. We extract the carrier densityin graphene from the Dirac measurements, and use it cal-culate δσδE as a function of n using Eq. S10 and Eq. S11.Then, we can easily estimate C using Eq. S9 for oursample. Using this procedure, we obtain C as a func-tion of the back-gate voltage V bg (Fig. S1). Since, weperform all measurements at V bg =0 V, we use C ∼ − for further calculations. - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 01 0 0 02 0 0 03 0 0 04 0 0 05 0 0 06 0 0 0 Rsq( W ) V b g ( V ) - 4 5- 3 0- 1 501 53 04 5 C0(V-1)
FIG. S1. R sq (left y-axis)- V bg dependence of graphene (red).On the right y-axis the spin-to-charge conversion efficiency C − V bg dependence (in black) is plotted. C at V bg =0V ismarked with a solid black dot. III. MODEL FOR SPIN TO CHARGECONVERSION
We know, from Eq. S9, the spin-accumulation inducedcharge voltage v c = C µ . If an ac charge current i isapplied across the injector electrode, it creates a spinaccumulation µ s = p inj iR s underneath the injector elec-trode, which is measured at a distance L away from theinjector electrode. v c = C µ = C ( p inj iR s e ) = C ( p inj i ) (S12)where C = C ( R s e ) . While deriving the expressionfor v c , we also consider the role of the nonlinear spin-injection and assume p inj = p (1 + C i ), and substitutethis expression into Eq. S15: v c = Cp (1 + C i ) i = Cp ( i + 2 C i + C i ) (S13)Now, we can perform the mixed signal analysis on theexpression in Eq. S14 by replacing i with I + i sin ωt inthe same way as described in supplementary section IIand obtain expressions for the 2 nd harmonic componentsof v c :1 v = Cp { i + 6 C Ii + 6 C I i + C i } cos 2 ωt (S14)Here, we would like to remark that for I = 0, we wouldexpect v ∝ i for small i and v ∝ i for the large i values. The consequences of this dependence are signifi-cant as because of a nonzero C , at large i values, v c ∝ i while v s ∝ i (Fig. 4 in the main text) . Therefore, atlarge i the spin-accumulation induced charge-voltage v c would be comparable to, and can even surpass the spin-signal v s . SPIN TO CHARGE CONVERSION IN 1 st HARMONIC RESPONSE
As explained in Sec.III, spin-to-charge conversion is anonlinear effect, and is measured as higher harmonic ofthe nonlocal charge signal. However, similar to the non-linear spin transport measurements, when a dc chargecurrent I is injected along with i , due to the mixing be-tween i and I , we can also measure v c in the 1 st harmonicresponse. The expression for 1 st harmonic component of v c , obtained by using the mixed signal analysis is: v = Cp { Ii + 6 C I i + (3 / C i + 3 C Ii + 4 C I i } sin ωt (cid:39) Cp Ii { C I + 2( C I ) } ( i << I ) . (S15)Here, the terms with C in the expression in Eq. S15appear due to the nonlinear spin-injection process. If C I >
1, the term ∝ ( C I ) will dominate and set am-plification factor for the spin-to-charge conversion effectin the 1 st harmonic response, which is absent for lin-ear spin-injection. Now, v c (cid:39) Cp Ii ( C I ) , and due tothe nonlinearity present in the spin-injection process, wewould expect an amplification in the spin-to-charge con-version effect in the 1 st harmonic signal, proportional to( C I ) .In order to verify this hypothesis, we revisit the 1 st harmonic Hanle measurements, and obtain v c as an av-erage of v pnl and v apnl . We first present the case whenno bias is applied at the injector. For I=0, we expect v c = 0 and symmetric parallel and anti-parallel Hanlecurves. Since we cannot measure any spin-signal clearlywhile using the tunnel contact C I to enhance its spin-detection efficiency p det and the spin-signal can be measured. Since I =0 at the injector, thereshould be no coupling between the 1 st and 2 nd harmonic v c and the parallel and anti-parallel Hanle signals shouldbe symmetric. In our measurement while using C2 asan injector and C1 as a detector we measure roughlysymmetric Hanle curves with respect to the background(black dashed line) for the parallel and anti-parallel con-figurations (Fig. 4(d)). However, there is no asymmetryexpetected for this case. The reason for such behavior isnot clear to us at the moment.Now, when we use the tunnel contact C1 as an injec-tor and C2 as a detector, and inject a charge current I + i , as expected we see a huge asymmetry betweenthe magnitudes of parallel and the anti-parallel Hanlesignals in Fig. S2(b,c) for I = ±
250 nA due to the pres-ence of spin-to-charge conversion effect. Simar to the 2 nd harmonic measurements in Fig. 4(c) in the main text,the spin-accumulation induced charge signal which is theaverage of the parallel and anti-parallel spin-signals iscomparable to the spin-signal. Due to the dominant 2 nd order spin-injection and its contribution to the 1 st har-monic spin-to-charge conversion, the higher order termsin Eq. S15 contribute significantly, and we also see thesign-reversal of v c with the polarity of I . In order to ap-preciate the role of the nonlinear spin-injection to mea-sure such effect, we also present a similar case for spin-injection in a different sample using a 2L-hBN, in ref. [1]where, where the nonlinear constant C ∼ A − is notdominant enough compared to C ∼ A − for 3L-hBNin our sample. Here, even for high enough I = ± µA , v c is significantly small compared to the spin-signal v s (Fig. S3), where v c and v s are of similar order magnitude.Therefore, we also do not measure a strong modulationin Hanle shapes in Fig. S3 as in Fig. S2 for nonzero I .However, the asymmetry between the magnitudes of par-allel and anti-parallel Hanle curves, along with the signreversal in v c with the polarity of I is consistently presentin both measurements.We would also like to remark that the observed be-havior is not due to the contribution of outer injec-tor/detector FM electrodes to the measured spin-signal.For nonlocal SV measurements, we consistently observeonly two distinct levels in the spin-valve effect corre-sponding to parallel and anti-parallel configuration of theinjector-detector pair. It confirms the contribution ofonly one FM injector and one detector in contrast withthe recently reported two-probe spin-transport measure-ments in ref. [1, 4 ? ] where both FM electrodes act asspin-injector and detector contacts and result in asym-metric Hanle curves for parallel and anti-parallel config-urations.Lastly, similar to the 2 nd harmonic spin-to-charge con-version effect, shown in Fig. 4(e) of the main text, wealso plot the 1 st harmonic v c - B ⊥ dependence, and ob-tain the same information. The Hanle-like magnetic-field dependence of v c in Fig. S2(e) confirms its spin-accumulation induced origin. The fitting of the Hanlecurves in Fig. S2(d) and Fig. S2(e) results in λ s ∼ - 1 0 0 - 5 0 0 5 0 1 0 0- 0 . 4- 0 . 20 . 00 . 20 . 40 . 6 - 1 0 0 - 5 0 0 5 0 1 0 0- 10123 - 1 0 0 - 5 0 0 5 0 1 0 0- 4- 3- 2- 101- 1 0 0 - 5 0 0 5 0 1 0 00123 05 01 0 01 5 0 - 1 0 0 - 5 0 0 5 0 1 0 002 04 06 0 5 0 1 0 0 1 5 003 06 09 0I d e t = - 1 2 0 n A Rnl( W ) B ^ ( m T )I in j = 0 n A I d e t = 0 n A Rnl( W ) B ^ ( m T )I in j = 2 5 0 n A I d e t = 0 n A Rnl( W ) B ^ ( m T )I in j = - 2 5 0 n A vs(nV) Rs( W ) B ^ ( m T ) l s ~ 4 m mI = 1 3 0 n Ai = 5 0 n A I = 1 3 0 n Ai = 5 0 n A vc(nV) l s ~ 2 m mB ^ ( m T )( e ) ( f )( d ) ( c )( b ) vc(nV) v s ( n V )( a ) FIG. S2. 1 st harmonic measurements (a) R p(ap)nl = v p(ap)nl i are roughly symmetric with respect to the background (black dashedline) for i =20 µ A and I = 0 at the injector. Here, the transparent contact C2 is used as an injector and the tunnel contact C1is used as a detector. When the tunnel contact is used as an injector, a strong asymmetry is measured between R pnl (green)and R apnl (pink) with respect to the background for (b) I = +250 nA and (c) I = -250 nA due to the coupling of the 2 nd harmonic spin-to-charge conversion effect to the applied dc bias I . R pnl (green) is negative in (a) and (c) because of the negativebias current I . 1 st harmonic (d) spin-signal v s . (e) Nonlocal charge-signal v c - B ⊥ dependence. The data is symmetrized and aconstant background is subtracted from the raw data. (f) Summary of v c - v s dependence. The measurements are performed at4K. µ m and ∼ µ m, respectively. By fitting the v c − B ⊥ dependence in Fig. S2(e), according to the expectation v c ∝ v , λ s ∼ µ m is obtained which is half of the spin-relaxation length obtained from the spin-signal v s , andagain corroborates the square dependence of the nonlo-cal charge-signal on spin-accumulation, as obtained inthe 2 nd harmonic measurements in the main text. th HARMONIC SPIN SIGNAL
We also measure the 4 th harmonic component of thespin-signal for the input ac current i = 100 nA. Thespin-valve effect is shown in Fig. S4(b). Similar to the2 nd harmonic spin-signal, the 4 th harmonic spin-signalcan be measured unambiguously without applying any dc current. As soon as a finite dc bias I is applied alongwith the ac charge current i , again in line with the bias-dependence of the 2 nd harmonic spin-signal (Fig. 2(e) inthe main text), the magnitude of the 4 th harmonic signalis also reduced as shown in Fig. S4(a). ∗ corresponding author; [email protected][1] Gurram, M., Omar, S. & Wees, B. J. v. Bias inducedup to 100% spin-injection and detection polarizationsin ferromagnet/bilayer-hBN/graphene/hBN heterostruc-tures. Nature Communications , 248 (2017).[2] Vera-Marun, I. J., Ranjan, V. & van Wees, B. J. Nonlinearinteraction of spin and charge currents in graphene. Phys.Rev. B , 241408 (2011). - 0 . 1 0 . 0 0 . 1- 0 . 50 . 00 . 5 - 0 . 1 0 0 . 0 0 0 . 1 0- 0 . 2- 0 . 10 . 00 . 10 . 2 - 0 . 1 0 . 0 0 . 1- 1 . 0- 0 . 50 . 00 . 51 . 0- 0 . 1 0 . 0 0 . 1- 0 . 0 20 . 0 00 . 0 20 . 0 40 . 0 60 . 0 8 - 0 . 1 0 . 0 0 . 1- 0 . 0 20 . 0 00 . 0 20 . 0 4 - 0 . 1 0 . 0 0 . 1- 0 . 2 0- 0 . 1 5- 0 . 1 0- 0 . 0 50 . 0 00 . 0 5 vnl( m V) B ^ ( T ) I = 2 5 m A vnl( m V) B ^ ( T ) I = 5 m A vnl( m V) B ^ ( T ) I = - 2 5 m A vc( m V) B ^ ( T ) I = 2 5 m A vc( m V) B ^ ( T ) I = 5 m A b c vc( m V) B ^ ( T ) I = - 2 5 m A ad e f FIG. S3. 1 st harmonic measurements for parallel (green) and anti-parallel (pink) configurations, using a bilayer hBN tunnelbarrier as a spin-injector at i =3 µ A and I =(a) 25 µ A (b) 5 µ A and (c) -25 µ A, and the corresponding v c at (d), (e) and (f)respectively. The data is symmetrized and a constant background is subtracted from the raw data. The measurements areperformed at RT.[3] Vera-Marun, I. J., Ranjan, V. & van Wees, B. J. Nonlineardetection of spin currents in graphene with non-magneticelectrodes. Nature Physics , 313–316 (2012).[4] Gurram, M., Omar, S. & Wees, B. J. v. Electrical spininjection, transport, and detection in graphene-hexagonal boron nitride van der Waals heterostructures: progressand perspectives.
2D Mater. , 032004 (2018). - 0 . 1 0 . 0 0 . 1- 1 5- 1 0- 5051 01 52 0 1 0 2 0 3 0 4 0 5 0- 2 0- 1 001 0( a ) vs(nV) I ( m A ) ( b )- 303 v pn l v a pn l vnl( m V) v pn l vnl(nV) B || ( m T )i = 1 0 0 n A v a pn l FIG. S4. (a) 4 th harmonic spin signal (orange) and its biasdependence. The (anti) parallel, data in (blue)red is plottedagainst the right-y axis. (b) 4 thth