Photon drag of superconducting fluctuations in 2D systems
aa r X i v : . [ c ond - m a t . s up r- c on ] A p r Photon drag of superconducting fluctuations in 2D systems
M. V. Boev
Novosibirsk State Technical University, Novosibirsk 630073, Russia (Dated: April 30, 2020)The theory of photon drag of superconducting fluctuations in the two-dimensional electron gasis developed. It is shown that the frequency dependence of the induced current is qualitativelysimilar to the case of photon drag of conventional two-dimensional degenerate electron gas. Withthe decreasing temperature the magnitude of the effect increases dramatically and the current ofsuperconducting fluctuations carries an additional power of reduced temperature in comparison withthe Aslamazov-Larkin contribution. The magnitude of the developed effect is expected sufficient tobe visible against the conventional photocurrent background.
I. INTRODUCTION
Light absorption by a condensed matter is accompa-nied by the transfer of the momentum of photons, besidestheir energy, to charged excitations. Such transfer resultsin the occurrence of the electric current in the systemand this effect was called photon drag. The magnitudeand direction of photocurrent depend on many factors,such as light polarization, incident angle and frequency.Moreover, the transport features of carriers in a con-densed matter and the microscopic mechanism of light-carriers interaction drastically influence on the photon-drag. That is the reason why this effect is widely usedin the study of numerous systems: semiconductors ,two-dimensional electron gas , graphene , topologicalinsulators , metal-semiconductor nanocomposite , two-dimensional exciton gas and metal films .At the same time, the investigation of superconduc-tivity phenomenon in two-dimensional systems takes agreat part in condensed matter physics. Starting fromthin metallic films, the samples fabrication technologiesand experimental tools become suitable for the study ofhighly crystalline superconductors possessing extremelysmall thicknesses down to a monolayer . Among atom-ically thin superconductors, the systems based on tran-sition metal dichalcogenides (TMD), e.g. MoS , havearoused interest in recent years . The remarkablefeature of such systems is the use of ionic liquid gate forcreating a large density of electrons, reaching the valuesup to 3 · cm − .To date, the transition of TMD and main-groupmetal dichalcogenide flakes from the normal (resis-tive) to superconductive phase have been studied inexperiments . However, in the range of temperatureclose to the phase transition, T & T c , the behaviour ofTMD flakes in the electromagnetic (EM) field has notbeen completely studied experimentally, as well as the-oretically. In this direction, it was observed that thesuperconductive fluctuations in the normal phase makethe effect of magnetochiral anisotropy be noticeably moredistinct .By now, the investigations of fluctuation phenom-ena in superconductors have opened comprehensive fa-cilities in identification of fundamental properties of j x j y x z y E K k x (cid:570) E FIG. 1. System sketch. The EM-wave illuminate the 2Dsuperconductor, which lies in ( x, y )-plane, at some angle Θand produces the electric current in x and y directions. Here K is the wave-vector of the EM-wave, E is the electric fieldamplitude, while E is its projection to the ( x, y )-plane. superconductors . In the present work, we suggest thephoton drag effect as an additional approach to the in-vestigations of transport features of 2D superconductorsin the fluctuating regime. We explore the classical limitof this effect. It means that no transition between sub-bands happens. In other words, it is supposed that theincident EM-wave frequency is much less than any energygap in the systems. In this case, the physical mechanismof photon drag just consists in the momentum transferfrom the EM-wave to fluctuations.To develop the theory, the Ginsburg-Landau (GL) ap-proach is used . Although the microscopic treatment ismore exact, it is simultaneously more difficult and cum-bersome than the GL one. Since our aim is to achieve aqualitative picture, the GL theory seems to be appropri-ate as a good approximation. II. MODEL
Let us consider the 2D-superconductor being in thenormal phase and irradiated by an electromagnetic wavewith an electric field amplitude E , wave vector K andfrequency Ω (Fig.1). In present work we consider apurely 2D system. Thus, the electron motion in the z-direction is neglected and the superconducting fluctua-tions respond to the projection of EM-wave amplitude E on a superconductor surface only. For later computa-tions, it is convenient to define E in the complex form: E ( R , t ) = E e i ( kR − Ω t ) + E ∗ e − i ( kR − Ω t ) , (1)where E and E ∗ are complex amplitudes of electromag-netic wave and k is a projection of K to the superconduc-tor plane. We focus on the stationary and homogenouspart of the electric current, which does not vanish afterthe averaging in space and time. Thus, in the lowestorder of the wave amplitude, the photon drag currentcorresponds to the second-order response: j α = σ αβγ ( k , Ω) E β E ∗ γ , (2)because any odd term of expansion will give zero contri-bution after averaging. In Eq.(2) σ is the second-orderconductivity and the subscripts denote components inthe Cartesian axes. For convenience, let k to be ori-ented along the x-axis (Fig.1). Then the system is sym-metric under reflection y → − y and, therefore, only σ xxx , σ xyy , σ yxy , σ yyx are nonzero. After separating σ tosymmetric and antisymmetric parts in accordance with σ αβγ = σ sαγβ − σ aαγβ , the current reads (cid:18) j x j y (cid:19) = σ sxxx | E x | + σ sxyy | E y | σ syxy ( E x E ∗ y + E ∗ x E y ) − iσ ayxy ( E x E ∗ y − E ∗ x E y ) ! , (3)where the imaginary unit in the second line is introducedto make σ ayxy real.Eqs. (3) are just a general form of the second-orderresponse. However, the explicit expressions for conduc-tivity are the goal. To succeed in it, we start from thedefinition of an electric current in the form of the varia-tional derivative (further we will omit variables ( R , t ) forbrevity): j = − δ F [Ψ] δ A , (4)where the GL free energy has the form F [Ψ] = αT c Z dV (cid:8) ǫ | Ψ | + ξ | (ˆ p − e A )Ψ | (cid:9) , (5)Ψ is the order parameter, A is the EM-wave vector poten-tial, α = (4 mT c ξ ) − is the GL expansion coefficient, m is the electron mass, T c is the temperature of transition to the superconductive state, ξ is the coherent length,ˆ p = − i ∇ , ǫ = ln( T /T c ) ≈ ( T − T c ) /T c is the reducedtemperature. In writing (5) it is supposed that the EM-field does not change the coefficients in the GL free energyexpansion and is just included via the minimal coupling − i ∇ → − i ∇ − e A . Combining (4) and (5) we can seethat, as usual, the current proves to be a sum of dia- andparamagnetic terms: j D = − e αT c ξ A | Ψ | , (6a) j P = 4 eαT c ξ Re[Ψ ∗ ˆ p Ψ] . (6b)In (6) the order parameter is still undefined. To proceed,let us note that including the vector potential to the GLfree energy makes the order parameter dependent on it,Ψ = Ψ( A ). To obtain the explicit expression of thisdependence, we explore the Time-Dependent Ginzburg-Landau (TDGL) equation : (cid:26) γ ∂∂t + αT c h ǫ + ξ (ˆ p − e A ( r, t )) i(cid:27) Ψ( r, t ) = f ( r, t ) , (7)where parameter γ has both real and imaginary parts, γ = γ ′ + iγ ′′ . The real part is proportional to the life-time of fluctuating Cooper pair, γ ′ = αT c ǫτ GL . Thelifetime τ GL goes to infinity near the critical point and,in the BCS theory, it has the form τ GL = π/ T − T c ).Thus, γ ′ = πα/
8. The appearance of the imaginary term, γ ′′ = − αT c ∂ ln( T c ) /∂E F , in the TDGL is shown to bea consequence of the gauge invariance of GL-theory .From the microscopic point of view, the origin of γ ′′ canarise from either the electron-hole asymmetry or thetopological structure of the Fermi surface . The quan-tity γ ′′ plays the crucial role in some effect, for instance,in the fluctuation Hall conductivity . Further, it is as-sumed that γ ′′ /γ ′ ≪
1. In eq. (7), f is a Langevin ran-dom force, which defines the white noise in the systemand is completely uncorrelated: h f ∗ ( r, t ) f ( r ′ , t ′ ) i = 2 T γ ′ δ ( r − r ′ ) δ ( t − t ′ ) . (8)Here the angle brackets designation h ... i means fluctu-ations averaging. In writing the TDGL equation, wechoose the gauge of EM-wave with zero scalar potentialthat means the connection E = − ∂ t A . Assuming thevector potential to be a perturbation, let us utilize themethod of progressive approximation, i.e. we will find thesolution of (7) in the form of expansion in the powers of A : Ψ = Ψ + Ψ + Ψ ... (9)where Ψ i ∼ A i . Since the second order response isneeded, we should keep the terms ∼ A after the sub-stitution of expansion (9) to (6) yielding: h j D i ≈ − e αT c ξ A ( h Ψ ∗ Ψ i + h Ψ ∗ Ψ i ) , (10a) h j P i ≈ eαT c ξ Re[ h Ψ ∗ ˆ p Ψ i + h Ψ ∗ ˆ p Ψ i ++ h Ψ ∗ ˆ p Ψ i ] . (10b)For the next step the explicit form of approximate so-lution is required. To derive it, we rewrite (7) in termsof operators: n ˆ L − − ˆ M − ˆ M o Ψ( R , t ) = f ( R , t ) , (11)where ˆ L − = γ ∂∂t + αT c [ ǫ + ξ p ] , (12a)ˆ M = αT c ξ e (ˆ pA + A ˆ p ) , (12b)ˆ M = − αT c ξ (2 e ) A . (12c)Thus, the formal solution of (11) can be obtained withmultiplying (11) by ˆ L from the left. So, we find the following expressions for the terms in expansion (9):Ψ ( R , t ) = ˆ Lf ( R , t ) , (13a)Ψ ( R , t ) = ˆ L ˆ M ˆ Lf ( R , t ) , (13b)Ψ ( R , t ) = ( ˆ L ˆ M ˆ L ˆ M + ˆ L ˆ M ) ˆ Lf ( R , t ) . (13c)Returning to Eq.(12) we can see that operator (12a) isdiagonal in the plane wave basis and has the eigenvalue: L q ω = 1 ε q − iγ , (14)where ε q = αT c [ ǫ + ξ q ] . (15)So, it is convenient to deal with Fourier transformedfunctions, Ψ( R , t ) = P q ω Ψ q ω e i ( qR − ωt ) and f ( R , t ) = P q ω f q ω e i ( qR − ωt ) . Substituting (13) to (10), perform-ing Fourier transformation and assuming γ ′′ ≪ γ ′ , aftersome computations, we arrive at expressions: h j D i = − e T ( αT c ξ ) X p ε − ( Re[ A ( pA ∗ )] γ ′′ Ω( ε − + ε + ) + γ ′ Ω (cid:20) ε − − ε + )( ε − + ε + )( ε − + ε + ) + γ ′ Ω (cid:21) ++ 2Im[ A ( pA ∗ )] γ ′ γ ′′ Ω ( ε − − ε + )[( ε − + ε + ) + γ ′ Ω ] ) (16a) h j P i = 8 e T ( αT c ξ ) X p ( ( p − k ) | pA | ε − γ ′′ Ω( ε − + ε + ) + γ ′ Ω (cid:20) ε − − ε + )( ε − + ε + )( ε − + ε + ) + γ ′ Ω (cid:21) ++ ( p + k ) | pA | ε − ε + γ ′′ Ω( ε − − ε + )( ε − + ε + )[( ε − + ε + ) + γ ′ Ω ] ) (16b)where ε ± = ε ( p ± k ) / and A is a complex amplitude ofvector potential A ( R , t ) = A e i ( kR − Ω t ) + c.c. . The fullintegration of expressions (16) is quite difficult but thepolar angle integration can be performed. To make thetext be not overloaded, we set the cumbersome integralsto the appendix section and produce the second-orderconductivity in the following form: σ sαβγ ( ˜ T , ˜Ω , Θ) = γ ′′ γ ′ e ˜ T ξ I sαβγ ( ˜ T , ˜Ω , Θ) ~ c ˜ T c ˜Ω cos (Θ) , (17a) σ ayxy ( ˜ T , ˜Ω , Θ) = − πγ ′′ γ ′ e ˜ T ξ I ayxy ( ˜ T , ˜Ω , Θ) ~ c ˜ T c ˜Ω cos (Θ) , (17b)where dimensionless factors I sαβγ and I ayxy are given inthe appendix, ˜Ω = Ω ξ/c , ˜ T ( c ) = k B T ( c ) ξ/ ~ c and the re-lation | k | = cos(Θ)Ω /c has been used. III. RESULTS AND DISCUSSION
The qualitative dependence of (17a) on dimensionlessfrequency ˜Ω is shown in Fig.2. It is proved that the ab-solute value of each component of the symmetrical partof the second-order conductivity monotonically increases,while the frequency decreases and, furthermore, it con-verges to the constant value at Ω = 0. The numericalcomputation shows that, in fact, no all components areindependent and the following equality is obeyed: σ sxxx − σ sxyy = 2 σ syxy , (18)where we omit arguments ( ˜ T , ˜Ω , Θ). This relation is notaccidental and is a result of the system symmetry withregard to the rotation around the z-axis. Usually, thewave-vector of EM-wave is the smallest in comparisonwith a wave-vector of any excitation in a solid. Withthis assumption, we would expand the second-order con-ductivity in powers of k : σ αβγ ( k ) ≃ σ αβγ (0) + D αδβγ k δ , s [ a r b i t r a r y un i t s ] /c [10 -3 ] xxx xyy yxy FIG. 2. The frequency dependence of symmetric part of thesecond-order conductivity calculated by Eq.(17a). where D αδβγ is a forth-rank tensor. Owing to the pres-ence of the inversion symmetry in the system, the firstterm of expansion vanishes, σ αβγ (0) = 0. Then therequirement of invariance under the rotation at an ar-bitrary angle around the z-axis produces the relation D xxxx − D xxyy = 2 D yxxy that gives the formula (18).But in deriving (16), the smallness of k is not used ex-plicitly and that is clear from the dependence of ε ± on k . However, we can still represent the second-order con-ductivity in the form σ αβγ ( k ) = D αδβγ k δ , which willbe rotational-invariant. For example, let us consider thefirst term in (16b) and rewrite it in the form: j P, α = X p f ( ε ± )( p − k ) | pA | = X p f ( ε ± ) M αδ p β p γ k δ A β A ∗ γ = D ( | k | ) αδβγ k δ A β A ∗ γ , (19)where M αδ is the matrix transforming k to p − k . Fur-ther it is not difficult to check that (19) does not changeunder the rotation in the (x,y)-plane. In practice, thelight polarization is often defined by Stokes parameters.So, it is convenient to rewrite the first line of (3) in thecorresponding form: j x = σ xxx + σ xyy | E x | + | E y | ) + σ yxy ( | E x | − | E y | ) . (20)Further, the dependence of drag current magnitudeon temperature arouses great interest. But, to beginwith, it is necessary to confine the temperature rangeof applicability of the theory. First, the inequality ǫ ≈ ( T − T c ) /T c ≪ s yxy [ a r b i t r a r y un i t s ] /c [10 -3 ] T/T c FIG. 3. The frequency dependence of σ syxy at different tem-peratures: T /T c = 1 .
06, 1 .
08, 1 . · − , 2 . · − , 0 . · − for orange, blue and olive curves, respectively. omit the term ∼ | Ψ | in the GL free energy. At an essen-tially small ǫ the fluctuations become strong and this con-tribution cannot be neglected. The analysis producesthe so-called Ginsburg-Levanyuk parameter Gi ≈ T c /E F which characterizes the temperature range of strong fluc-tuations. Our theory is correct for the case of weak fluc-tuations only, i.e. under the condition ǫ ≫ Gi .The temperature dependence of symmetrical part ofthe second-order conductivity is similar for each com-ponent. So, it is enough to show the qualitative re-sults for one component only, for instance, for the σ syxy -component (Fig.3). It is proved that the current sub-stantially increases when the temperature is close to itscritical value. To obtain the obvious temperature de-pendence, let us consider the range of small frequency.For that purpose, the frequency should be turned to zeroand that allows us to perform the integration in (A.6) ex-plicitly. After some computations we obtain the simpleexpression: σ syxy (Ω →
0) = γ ′′ γ ′ e T ξ cos(Θ)48 ~ cT c ǫ . (21)We can see that the reduced temperature dependence atzero frequency, σ s ∼ /ǫ , is rather dramatic and in-cludes an additional power of ǫ in comparison with theAslamazov-Larkin conductivity.The frequency dependence of (17b) is non-monotonicand possesses its extremum at a small value of ˜Ω (Fig.4).With the decreasing temperature the extremum depthincreases and moves towards zero frequency. We want toremind here that the current, defined by the asymmetriccomponent of conductivity, is nonzero in response to thecircular-polarized EM-wave only, which is characterizedby the direction of vector E rotation. Thus, the switching T/T c = 1.1 1.08 1.06 a yxy [ a r b i t r a r y un i t s ] /c [10 -3 ] FIG. 4. The frequency dependence of antisymmetric part ofthe second-order conductivity calculated by Eq.(17b) from the clockwise polarization to the reverse one changesthe sign of E x E ∗ y − E ∗ x E y as well as the direction of current y -projection.The important feature of the obtained frequency de-pendence of the second-order conductivity consists inthat its qualitative behaviour is the same as for the caseof photon drag effect in conventional systems, for exam-ple, based on graphene . Apparently, the reason ofsuch similarity lies in a certain affinity of the TDGL-equation and the Boltzmann one, which is widely usedfor analyzing the nonlinear response of 2D electron gas.At the end of this subsection, we discuss the magnitudeof the examined effect. For this purpose, let us comparethe contribution of superconducting fluctuations with theone of normal electron gas. For the estimation, it isenough to use the simplest classical expression for thephoton drag current in 2D systems, which has the fol-lowing form : j n = 2 e n Ω m τ τ ) | E | k , (22)where n is the electron gas density and τ is the momen-tum relaxation time. With utilizing Eqs.(21) and (22),the ratio of two contributions in the zero-frequency limitcan be easily composed: j s j n = 83 γ ′′ γ ′ T nξ T c (cid:18) σ AL σ n (cid:19) , (23)where σ n = ( e /h ) k F l is the Drude conductivity and σ AL = e / ~ ǫ is the Aslamazov-Larkin conductivity forthe 2D system. Eq.(23) is convenient to be consideredpiecemeal. First, the ratio T /T c ≈ σ AL /σ n ) ,we take the typical values of normal conductivity σ n =10 − ÷ − Ω − , that gives ( σ AL /σ n ) ≃ − ÷ − at ǫ = 0 .
1. Further, the quantity | γ ′′ /γ ′ | was supposed to be much less then unity. For instance, in tantalumnitride thin films, it takes the value ∼ − . For TMDsuperconductors, the measurements of | γ ′′ /γ ′ | have notbeen performed yet. However, we can estimate it as-suming the power dependence of T c on E F , that gives | γ ′′ /γ ′ | ∼ | ∂T c /∂E F | ∼ T c /E F ∼ − at T c = 9 Kand n = 10 cm − for MoS . The above argumentsgive an idea that the ratio (23) possesses a small mag-nitude. However, the rest dimensionless factor nξ isoften very large. In TaN films nξ ≃ · , while, inMoS , nξ ≈ nξ l ≃ · (where ξ is the BCS coher-ent length at T = 0 K and l is mean free path). Finally,the estimation becomes j s /j n ∼ − ÷ IV. CONCLUSION
In the presented work we developed the theory of pho-ton drag of the superconducting fluctuations based on us-ing the TDGL-equation. The calculation showed that themagnitude of the photo-drag current strongly grows whenthe temperature comes down to the critical point. In thelow-frequency domain the drag-current is proportional tothe squared Aslamazov-Larkin conductivity, that was notevident from the beginning. We would like to emphasizethat the induced current is proportional to the imaginarypart of the γ -parameter as it has a place in the Hall-effectand, consequently, the presented effect can be treated asan additional approach in the fluctuation spectroscopy. Itis interesting to note that in thin superconducting filmswith three-dimensional electrons and a simple electronspectrum, parameter γ ′′ is negative. Thus, the photondrag of fluctuations will compensate the photocurrent ofnormal electrons. Taking into account the commensura-bility of these currents, as it was shown by the estimationabove, the reduction of the full photocurrent near T c canbe considerable. ACKNOWLEDGMENTS
We thank A.G. Semenov for his useful discussion. Thiswork was supported by the Foundation for the Advance-ment of Theoretical Physics and Mathematics ”BASIS”,RFBR (Grant No. 18-29-20033) and by the Ministry ofScience and Higher Education of the Russian Federation(project ”Nonlinear electrodynamics of electron systemsin micro- and nanostructures”).
Appendix: Explicit expressions for integrals I αβγ Let us introduce for brevity the following notations: a = a ( T, ˜Ω , Θ) = 4 ǫ ˜Ω cos (Θ) (A.1) b = b ( ˜Ω , Θ) = (cid:18) π T c ˜Ω cos (Θ) (cid:19) (A.2) Then the dimensionless factors from the resulting expres-sions (17) have the form: I ayxy ( T, ˜Ω , Θ) = ∞ Z a dy y + b ) y (cid:16) y − p ( y − + 4 a (cid:17)p ( y − + 4 a , (A.3) I sxxx ( T, ˜Ω , Θ) = 2 ∞ Z a dy y + b ) ((cid:2) y + b (2 y − (cid:3) " − y p ( y − + 4 a + 4 by ( y − y − − a )[( y − + 4 a ] / ) , (A.4) I sxyy ( T, ˜Ω , Θ) = − ∞ Z a dy y + b ) ((cid:2) y + b ( y − (cid:3) " − y p ( y − + 4 a + 2( y − − a )( b + 2 y ) p ( y − + 4 a ) , (A.5) I syxy ( T, ˜Ω , Θ) = − ∞ Z a dy y + b ) ((cid:2) y + b (4 y − (cid:3) " − y p ( y − + 4 a + 8 b ( y − − a ) p ( y − + 4 a ) , (A.6) A. M. Danishevskii, A. A. Kastalskii, S. M. Ryvkin, andI. D. Yaroshetskii, Sov. Phys. JETP , 292 (1970). A. F. Gibson, M. F. Kimmit, and A. C. Walker, Appl.Phys. Lett. , 75 (1970). S. Graf, H. Sigg, K. Kohler, and W. Bachtold, Phys. Rev.B , 10301 (2000). A. D. Wieck, H. Sigg, and K. Ploog, Phys. Rev. Lett. ,463 (1990). V. A. Shalygin, H. Diehl, C. Hoffmann, S. N. Danilov,T. Herrle, S. A. Tarasenko, D. Schuh, C. Gerl,W. Wegscheider, W. Prettl, and S. D. Ganichev, JEPTLetters , 570 (2006). M. M. Glazov and S. D. Ganichev, Physics Reports ,101 (2014). H. Plank, L. E. Golub, S. Bauer, V. V. Belkov,T. Herrmann, P. Olbrich, M. Eschbach, L. Plucinski,C. M. Schneider, J. Kampmeier, M. Lanius, G. Mussler,D. Grutzmacher, and S. D. Ganichev, Phys. Rev. B ,125434 (2016). G. M. Mikheev, A. S. Saushin, V. M. Styapshin, and Y. P.Svirko, Scientific Reports , 8644 (2018). V. M. Kovalev, M. V. Boev, and I. G. Savenko, Phys. Rev.B , 041304(R) (2018). V. M. Kovalev, A. E. Miroshnichenko, and I. G. Savenko,Phys. Rev. B , 165405 (2018). M. V. Boev, V. M. Kovalev, and I. G. Savenko, JETPLett. , 737 (2018). A. S. Vengurlekar and T. Ishihara, Appl. Phys. Lett. ,091118 (2005). J. H. Strait, G. Holland, W. Zhu, C. Zhang, B. R. Ilic,A. Agrawal, D. Pacifici, and H. J. Lezec, Phys. Rev. Lett. , 053903 (2019). Y. Saito, T. Nojima, and Y. Iwasa, Nat. Rev. Mater. ,16094 (2016). J. M. Lu, O. Zheliuk, I. Leermakers, N. F. Q. Yuan,U. Zeitler, K. T. Law, and J. T. Yewasa, Science ,1353 (2015). D. Costanzo, S. Jo, H. Berger, and A. F. Morpurgo, Na-ture Nanotechnology , 339 (2016). Y. Saito, Y. Nakamura, M. S. Bahramy, Y. Kohama,J. Ye, Y. Kasahara, Y. Nakagawa, M. Onga, M. Tokunaga,T. Nojima, Y. Yanase, and Y. Iwasa, Nature Physics ,144 (2016). E. Piatti, D. D. Fazio, D. Daghero, S. R. Tamalampudi,D. Yoon, A. C. Ferrari, , and R. S. Gonnelli, Nano Lett. , 4821 (2018). C. H. Sharma, A. P. Surendran, S. S. Varma, and M. Tha-lakulam, Communications Physics , 90 (2018). J. T. Ye, Y. J. Zhang, R. Akashi, M. S. Bahramy, R. Arita,and Y. Iwasa, Science , 1193 (2012). S. Jo, D. Costanzo, H. Berger, and A. F. Morpurgo, NanoLett. , 1197 (2015). W. Shi, J. Ye, Y. Zhang, R. Suzuki, M. Yoshida,J. Miyazaki, N. Inoue, Y. Saito, and Y. Iwasa, Scientific
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