Observation of Large Unidirectional Rashba Magnetoresistance in Ge(111)

TL;DR

This study reports a significant unidirectional magnetoresistance in Ge(111) surfaces, linked to spin textures and Rashba effects, with potential for semiconductor device applications.

Contribution

First observation of large unidirectional Rashba magnetoresistance in Ge(111), highlighting the role of spin-split surface states and external magnetic fields.

Findings

Magnetoresistance is linear in current density and magnetic field.

At 15 K, the effect reaches 0.5% of zero field resistance.

The effect diminishes with temperature and gate voltage changes.

Abstract

Relating magnetotransport properties to specific spin textures at surfaces or interfaces is an intense field of research nowadays. Here, we investigate the variation of the electrical resistance of Ge(111) grown epitaxially on semi-insulating Si(111) under the application of an external magnetic field. We find a magnetoresistance term which is linear in current density j and magnetic field B, hence odd in j and B, corresponding to a unidirectional magnetoresistance. At 15 K, for I = 10 $\mu$A (or j = 0.33 A/m) and B = 1 T, it represents 0.5 % of the zero field resistance, a much higher value compared to previous reports on unidirectional magnetoresistance. We ascribe the origin of this magnetoresistance to the interplay between the externally applied magnetic field and the current-induced pseudo-magnetic field in the spin-splitted subsurface states of Ge(111). This unidirectional magnetoresistance is independent of the current direction with respect to the Ge crystal axes. It progressively vanishes, either using a negative gate voltage due to carrier activation into the bulk (without spin-splitted bands), or by increasing the temperature due to the Rashba energy splitting of the subsurface states lower than $\sim$58 k$_B$. The highly developed technologies on semiconductor platforms would allow the rapid optimization of devices based on this phenomenon.

Equations

$$\mathcal{H}=-\frac{{\hslash }^2{k}^2}{2m^{\ast }}+\alpha (\text{k}\times \text{upsigma})\cdot \hat{\text{z}}+g{\mu }_{\text{B}}\text{upsigma}\cdot \text{B},$$

$${\left(\Delta R/R\right)}_{\text{AMR}}=-A{B}^2{\cos }^2(\phi )=A{B}_y^2-A{B}^2$$

$$\Delta R/R=-A{B}^2{\text{cos}}^2(\phi )+2A(\alpha \beta /g{\mu }_{\text{B}})jB\text{sin}(\phi )$$

Full text

After decades of studies, spintronics has driven its most successful industrial revolutions in the read-out head of magnetic hard disk and in magnetic random access memory Akerman2005 . In both cases, a long-range magnetic order is the ultimate ingredient, since these applications rely on the giant magnetoresistance (GMR) effect Baibich1988 ; Binasch1989 . Due to the seek of magnetic ordering, the investigation of GMR has been wide and successful in ferromagnetic-based layers but still rare in semiconductors. Since a connection with magnetism in semiconductors would be desirable, lots of interest has been devoted to doping semiconductors with magnetic impurities Ohno1998 ; Dietl2000 . The low solubility of magnetic ions Akerman2005 and Curie temperatures below ${200\leavevmode\nobreak\ \textup{K}}$ Chen2011 still limit the applicability of this technology. An alternative to long-range magnetic order in semiconductors comes from spin-orbit coupling (SOC), the main core of the so-called spin-orbitronics field in semiconducting films and in topological insulators Chappert2007 ; Manchon2015 .

Within this field, the investigation of magnetoresistance has recently moved over the standard ferromagnet-related effects Locatelli2014 ; Kim2016 ; Latella2017 ; Qiu2018 ; He2017 ; Zhang2018 ; He2018 , and a promising new type of magnetoresistance has been observed in the topological insulator Bi2Se3 He2017 , and in the two-dimensional electron gas at the SrTiO3(111) surface He2018 . Since, in both cases, no magnetic order is present, the effect has been related to the characteristic spin-momentum locking He2017 ; Zhang2018 ; He2018 . The detected magnetoresistance exhibits two characteristic features: it is unidirectional (i.e. odd) and linear with the applied magnetic field and electrical current, therefore it has been classified among the unidirectional magnetoresistances (UMRs)He2017 ; Zhang2018 ; He2018 . Despite the same angular dependence, this SOC-related UMR has a different origin compared to another type of UMR recently investigated and involving a ferromagnetic layer Olejnik2015 ; Avci2015 ; Yasuda2016 ; Lv2018 .

Here, we report the observation of UMR in Ge(111). We ascribe its origin to the Rashba SOC, which generates spin-momentum locking inside the subsurface states of Ge(111). Their presence and spin-texture have already been demonstrated exploiting angle and spin-resolved photoemission spectroscopy Ohtsubo2010 ; Ohtsubo2013 ; Aruga2015 ; Yaji2015 . Experimentally, we find that the UMR in the Ge(111) subsurface states is drastically larger compared to previous reports He2017 ; He2018 . We detect a maximum UMR value equivalent to $0.5\%$ of the zero field resistance, when a magnetic field of ${1\leavevmode\nobreak\ \text{T}}$ and a current of ${10\leavevmode\nobreak\ \text{\textmu A}}$ are applied at ${15\leavevmode\nobreak\ \textup{K}}$. The effect progressively vanishes when increasing the temperature or applying a negative gate voltage due to carrier activation in the bulk valence bands of Ge and to the low value of the Rashba spin-orbit coupling ($\sim 58\,k_{B}$)Ohtsubo2010 .

We perform magnetotransport measurements on a $2\leavevmode\nobreak\ \text{\textmu m}$-thick Ge(111) using lithographically defined Hall bars (length ${\ell=120\leavevmode\nobreak\ \text{\textmu m}}$, width ${w=30\leavevmode\nobreak\ \text{\textmu m}}$ and aspect ratio $Z=\ell/w=4$) as shown in Fig. 1(a) (further details in the Supplementary Material). We apply a DC charge current and measure the longitudinal ($R_{l}=U_{l}/I$) and transverse ($R_{t}=U_{t}/I$) resistances under the application of an external magnetic field B. The direction of B is determined by its polar ($\theta$) and azimuth ($\varphi$) angles as shown in Fig. 1(a). In DC measurements, the UMR term is odd with respect to the applied current and thus defined as: ${R_{l}^{\,\textup{odd}}=\,[R_{l}(I)-R_{l}(-I)]/2}$. We also measure ${R_{t}^{\,\textup{odd}}=\,[R_{t}(I)-R_{t}(-I)]/2}$ and the longitudinal resistance which is even with respect to the applied current ${R_{l}^{\,\textup{even}}=\,[R_{l}(I)+R_{l}(-I)]/2}$. All the measurements are carried out as a function of the temperature from ${15\leavevmode\nobreak\ \text{K}}$ to ${295\leavevmode\nobreak\ \text{K}}$. The conductivity is $p$-type in the whole temperature range, at ${15\leavevmode\nobreak\ \textup{K}}$ the carrier density reaches ${p\approx 6\times 10^{15}\leavevmode\nobreak\ \text{cm}^{-3}}$.

We report in Fig. 1(b) the 4-probe temperature dependence of the zero magnetic field resistance $R_{l,0}$. The resistance saturation at low temperature is a fingerprint of a conduction channel in parallel with the bulk (black dashed line) which we attribute to the presence of subsurface states. The angular dependence of $R_{l}^{\,\textup{even}}$ at ${15\leavevmode\nobreak\ \textup{K}}$ in the $(xy)$ plane is shown in Fig. 1(c) for ${I=10\leavevmode\nobreak\ \text{\textmu A}}$. This MR signal exhibits maxima (resp. minima) for ${\textbf{B}\parallel\hat{\textbf{y}}}$, $\varphi=90^{\circ}$ (resp. ${\textbf{B}\parallel\hat{\textbf{x}}}$, $\varphi=0^{\circ}$). Since the sign is not reversed when reversing the magnetic field direction, we call this term anisotropic magnetoresistance (AMR) by analogy with ferromagnets. At ${15\leavevmode\nobreak\ \textup{K}}$, we find an AMR of ${0.4\%}$ under a magnetic field of 1 T. The same behaviors are obtained for angular dependencies within $(zy)$ and $(zx)$ planes.

In Fig. 2, we report the angular dependence of $R_{l}^{\,\textup{odd}}$ and $R_{t}^{\,\textup{odd}}$ in the ($xy$), ($xz$) and ($yz$) planes for ${B=1\leavevmode\nobreak\ \text{T}}$, ${I=10\leavevmode\nobreak\ \text{\textmu A}}$ at ${15\leavevmode\nobreak\ \textup{K}}$. We observe a unidirectional behavior for both longitudinal and transverse resistances: the maximum (minimum) of $R_{l}^{\,\textup{odd}}$ is observed for {\textbf{B}\parallel\mathbin{\vbox{\hbox{ \oalign{\hfil\scriptscriptstyle-$\hfil\cr\kern-1.29167pt\cr$\scriptstyle({+})\cr} }}}\hat{\textbf{y}}}, and the maximum (minimum) of $R_{t}^{\,\textup{odd}}$ is observed for {\textbf{B}\parallel\mathbin{\vbox{\hbox{ \oalign{\hfil\scriptscriptstyle-$\hfil\cr\kern-1.29167pt\cr$\scriptstyle({+})\cr} }}}\hat{\textbf{x}}}. Thus, experimentally, ${R_{l}^{\,\textup{odd}}=-R_{l,\textup{max}}^{\,\textup{odd}}\,\sin{(\varphi)}\sin{(\theta)}}$ and ${R_{t}^{\,\textup{odd}}=-R_{t,\textup{max}}^{\,\textup{odd}}\,\cos{(\varphi)}\sin{(\theta)}}$. These functions are shown as solid lines in Fig. 2. The angular dependence of the transverse resistance reveals the presence of the Nernst effect due to a current-induced vertical temperature gradient (along $\hat{\textbf{z}}$) in the Ge(111) film. This effect generates spurious thermal UMR signal in the longitudinal resistance. The Nernst effect contribution to $R_{l}^{\,\textup{odd}}$ can be written as: ${R_{l,\textup{max}}^{\,\textup{odd,Nernst}}=Z\,R_{t,\textup{max}}^{\,\textup{odd}}}$, with $Z$ being the aspect ratio of the channel (${Z=4}$ in our case) Avci2015 . Hence, to remove the Nernst effect contribution from the longitudinal signal, we study ${R_{\textup{UMR}}=R_{l,\textup{max}}^{\,\textup{odd}}-Z\,R_{t,\textup{max}}^{\,\textup{odd}}}$. We find that the Nernst effect contribution is negligible at ${15\leavevmode\nobreak\ \textup{K}}$ for low currents while it dominates when approaching room temperature and/or applying large currents (more details are given in the Supplementary Material).

In Fig. 3 we investigate the dependencies of $R_{\textup{UMR}}$ on the applied current [Fig. 3(a)], magnetic field [Fig. 3(b)], temperature [Fig. 3(c)] and gate voltage [Fig. 3(d)]. The signal is normalized with respect to the zero field resistance $R_{{l},0}$ at the corresponding current. In agreement with previous reports on UMR generated by spin-momentum locking Olejnik2015 ; Avci2015 we observe a signal proportional to the current and the magnetic field. ${R_{\textup{UMR}}/R_{l,0}}$ is maximum and almost constant at low temperature ($T<{20\leavevmode\nobreak\ \text{K}}$) and sharply decreases when the temperature becomes comparable to the Rashba spin-splitting energy ($\approx 60\leavevmode\nobreak\ \text{K}$). As shown in Fig. 3(d), the application of a top gate voltage modulates the channel resistance $R_{l,0}$. In Fig. 3(d), we also plot both the longitudinal and transverse odd resistance components as a function of the gate voltage. The transverse component we attribute to the Nernst effect stays constant with the gate voltage. This observation is consistent with the fact that this effect is due to vertical temperature gradient in the Ge(111) film and is almost unaffected by the top gate voltage. By contrast, the longitudinal component attributed to the UMR effect is much affected by the gate voltage: it increases from $V_{g}=-10\leavevmode\nobreak\ \text{V}$ to $V_{g}=+10\leavevmode\nobreak\ \text{V}$ by a factor $\approx 3$. $R_{\textup{UMR}}$ cancels out at $V_{g}=-10\leavevmode\nobreak\ \text{V}$ and increases from $-10\leavevmode\nobreak\ \text{V}$ to $+10\leavevmode\nobreak\ \text{V}$.

To make a comparison with previous results on different systems, we can define a figure of merit $\eta$. Since the UMR signal is proportional to the current and magnetic field, a natural definition is: ${\eta=R_{\textup{UMR}}/(R_{l,0}\,j\,B)}$. At ${15\leavevmode\nobreak\ \text{K}}$, in Ge(111), we obtain ${\eta=2.5\times 10^{-4}\leavevmode\nobreak\ \textup{cm}^{2}/(\textup{A\,T})}$ when considering the charge current flowing in the whole Ge(111) film (${2\leavevmode\nobreak\ \text{\textmu m}}$) and ${\eta=4.2\times 10^{-7}\leavevmode\nobreak\ \textup{cm}^{2}/(\textup{A\,T})}$ if we consider that the current completely flows within the spatial extension of the subsurface states (10 atomic layers from Ref. Aruga2015, ). In the worst case scenario, the value of $\eta$ obtained in Ge(111) is orders of magnitude larger than the one of SrTiO3 at 7 K [${\eta=2\times 10^{-9}\leavevmode\nobreak\ \textup{cm}^{2}/(\textup{A\,T})}$ from Ref. He2018, ] and the one of Bi2Se3 at ${60\leavevmode\nobreak\ \text{K}}$ [${\eta=2\times 10^{-11}\leavevmode\nobreak\ \textup{cm}^{2}/(\textup{A\,T})}$ from Ref. He2017, ]. In this second case, if we compare the results extrapolated at ${58\leavevmode\nobreak\ \text{K}}$ for Ge(111) we still obtain a larger $\eta$ value [${3\times 10^{-8}\leavevmode\nobreak\ \textup{cm}^{2}/(\textup{A\,T})}$].

At variance with previously reported systems He2017 ; He2018 the UMR is isotropic with respect to the direction of the current flow in the surface Brillouin zone (SBZ). In fact, in the data shown in Figs. 1$-$3, the current flows along the $\Gamma\text{M}$ direction of the Ge(111) SBZ, but no difference, within the experimental error, is detected with the current flowing along other reciprocal lattice directions (see Supplementary Material). In Refs. He2017, ; He2018, , the magnetoresistance is affected by the direction of the current flow in the SBZ, indicating that, in such a case, the UMR originates from the out-of-plane spin texture. In the case of Ge, this contribution appears to be negligible. We thus propose an alternative mechanism, in which the UMR in Ge(111), results from a combination of the applied magnetic field and the current-induced pseudo-magnetic field in the spin-splitted subsurface states of Ge(111) shown in Fig. 4(a). Ge(111) subsurface states are located close to the top of the valence bands and can only contribute to transport in $p$-type Ge(111) Ohtsubo2013 . This interpretation is supported by the fact that we do not observe this effect for $n$-type Ge(111) (see Supplementary Material). It also explains the gate voltage dependence of $R_{\textup{UMR}}$ in Fig. 3(d). Applying negative gate voltage shifts the Fermi level down into the valence band which leads to the activation of bulk conduction and $R_{\textup{UMR}}\approx 0\leavevmode\nobreak\ \Omega$ for $V_{g}=-10\leavevmode\nobreak\ \text{V}$. At variance, by ramping the gate voltage from ${-10\leavevmode\nobreak\ \text{V}}$ to ${+10\leavevmode\nobreak\ \text{V}}$, the Fermi level shifts into the subsurface states thus increasing $R_{\textup{UMR}}$. Finally, this interpretation also explains the temperature dependence of the UMR. By increasing the temperature, bulk conduction in the valence band is activated and shorts the subsurface states. Moreover, the Rashba spin-orbit coupling of $\sim 58\,k_{B}$ in Ge subsurface states Ohtsubo2013 becomes negligible with respect to $k_{\textup{B}}T$ suppressing spin-momentum locking.

For the Fermi level crossing the subsurface states as shown in Fig. 4(a), the Fermi contour is made of two concentric rings [C and D in Fig. 4(b)] with opposite spin helicities. To describe the magnetotransport inside the subsurface states, we consider the following model Hamiltonian :

[TABLE]

with $\hbar$ being the reduced Planck constant, $m^{*}$ the effective mass of holes in the subsurface states, $\alpha$ the Rashba spin-orbit interaction, $\bm{\upsigma}$ the vector of Pauli matrices, $g$ the Landé factor and $\mu_{\textup{B}}$ the Bohr magneton. When a 2D charge current density j flows in the subsurface states, in the Boltzmann approach, the momentum acquires an extra component ${\Delta\textbf{k}=\beta\textbf{j}}$ with ${\beta=4\pi/(ev_{\textup{F}}k_{\textup{F}})}$, $v_{\textup{F}}$ and $k_{\textup{F}}$ the Fermi velocity and wavevector we consider ($e=\lvert e\rvert$). A well-known consequence of such shifts of Rashba Fermi contours is the Rashba-Edelstein spin polarization Chappert2007 due to the unbalance between the opposite spin polarizations induced by the shifts in the same direction of the Rashba-splitted Fermi contours of opposite helicity. In parallel with the Rashba-Edelstein effect, the shift $\Delta\textbf{k}$ introduces a current-induced out-of-equilibrium energy term which, from Eq. 1, is equal to $\alpha(\Delta\textbf{k}\times\bm{\upsigma})\cdot\hat{\textbf{z}}=\alpha\beta(\textbf{z}\times\textbf{j})\cdot\bm{\upsigma}$ and acts on the spins as a pseudo-magnetic field ${\textbf{B}_{\textup{E}}=(\alpha\beta/g\mu_{\textup{B}})\,\hat{\textbf{z}}\times\textbf{j}}$. As illustrated in Fig. 4(c), for a current along $\pm\leavevmode\nobreak\ \hat{\textbf{x}}$ with $\alpha>0$ , this field is directed along $\pm\leavevmode\nobreak\ \hat{\textbf{y}}$ and proportional to the current density. In the presence of an applied magnetic field B, the spin of the subsurface states is submitted to $\textbf{B}+\textbf{B}_{\textup{E}}$, $\textbf{B}_{\textup{E}}$ increasing or decreasing the effect of the $y$ component of B for currents either along + or - $\hat{\textbf{x}}$. In the same way, still for $\alpha>0$ for j along $+\hat{\textbf{x}}$ and $\textbf{B}_{\textup{E}}$ along $\hat{\textbf{y}}$ , there is addition or subtraction of the effects of B and $\textbf{B}_{\textup{E}}$ for opposite orientations of B along $\hat{\textbf{y}}$. The physics of the UMR thus comes from the pseudo-field $\textbf{B}_{\textup{E}}$ induced by the out-of-equilibrium situation of a current flow and acting on the spins. We can go a little further by assuming that the AMR term shown in Fig. 1(c) (the only MR in the limit $j\rightarrow 0$) is also due to the effect of B on the spins. We thus follow Taskin et al. Taskin2017 who explain the AMR of Rashba systems by the re-introduction of some backscatterings by a partial re-alignement of the spins by B and we neglect contributions such as the effect of the Lorentz force on the trajectories. Then, in the situation of finite $j$, we add $\textbf{B}_{\textup{E}}$ to B in the $B^{2}$ term of the AMR to derive the expression of UMR. The AMR term can be written as:

[TABLE]

Where ${A\approx 0.004}$. Adding $B_{\textup{E}y}=\alpha\beta j/g\mu_{\textup{B}}$ to $B_{\textup{y}}$, and keeping only the terms of first order in $j$ gives :

[TABLE]

Where the second term, proportional to $j\,B$, is the UMR. Our experimental results with an UMR proportional to $j\,B\,\textup{sin}(\varphi)$, see [Fig. 2], correspond to a negative value of the Rashba coefficient $\alpha$, that is to the clockwise chirality of the spin orientation in the outer Fermi contour. This chirality is in agreement with the chirality derived from spin-resolved ARPES measurements for the subsurface states inside Ge at Ge/Bi interfaces, as shown in Fig. 3a of [Ohtsubo2010, ]. Quantatively, taking reasonable values for the parameters in the expression of the UMR amplitude. By setting ${B=1\leavevmode\nobreak\ \text{T}}$, ${j=0.33\leavevmode\nobreak\ \textup{A}\,\textup{m}^{-1}}$ in the subsurface states, ${\alpha=-0.2\leavevmode\nobreak\ \textup{eV}\cdot\textup{\AA}}$ (in Ohtsubo2010 , this value corresponds to Bi covered subsurface states, in our case it is probably an upper bound), ${k_{\textup{F}}=0.025\leavevmode\nobreak\ \textup{\AA}^{-1}}$ (Rashba splitting $\lvert\alpha k_{\textup{F}}\rvert=5$ meV$\sim 58\,k_{B}$), $m^{*}=0.4\,m_{e}$ ioffe , $m_{e}$ being the electron mass, ${v_{\textup{F}}=\hbar k_{\textup{F}}/m^{*}}$ and ${g=2}$, we find a UMR amplitude of ${\approx 0.2\%}$. This value is in good agreement with our low temperature experimental data. We indeed find a maximum value of $0.5\%$ at ${15\leavevmode\nobreak\ \textup{K}}$. Therefore, by using simple arguments, we capture the physics of UMR in the Ge Rashba-splitted subsurface states.

In conclusion, we performed magnetoresistance measurements on Ge(111) and detected a unidirectional magnetoresistance (UMR) which scales linearly with both the current and the applied magnetic field. We ascribe the UMR to the spin-momentum locking generated by the Rashba effect in the subsurface states of Ge(111) and interpret our results in a simple model relating the UMR to the Rashba coefficient and the characteristic parameters of the subsurface states. Such unidirectional effects can be expected in any Rashba 2DEG and can be used to obtain information about the electronic structure details. The amplitude of the detected UMR signal is much larger than the ones previously reported. We also showed that this UMR is tunable by turning on and off the Rashba coupling in the conduction channel by applying a gate voltage. Ultimately, these findings lead towards the development of a semiconductor-based spin transistor where the spin information can be manipulated by a gate-tunable Rashba field.

The authors acknowledge the financial support from the ANR project ANR-16-CE24-0017 TOPRISE. One of us (AF) acknowledges fruitful discussions with A .Dyrdal and J. Barnas (Poznan University), as well as with S. Zhang (University of Arizona).