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Nonreciprocal electrical transport in multiferroic semiconductor (Ge,Mn)Te

We have investigated the nonreciprocal electrical transport, that is a nonlinear resistance effect depending on the current direction, in multiferroic Rashba semiconductor (Ge,Mn)Te. Due to coexistence of ferromagnetic and ferroelectric orders, (Ge,Mn)Te provides a unique platform for exploring the nonreciprocal electrical transport in a bulk form. (Ge,Mn)Te thin films shows a large nonreciprocal resistance compared to GeTe, the nonmagnetic counterpart with the same crystal structure. The magnetic-field-angle dependence of the nonreciprocal resistance is maximized when magnetic field is orthogonal to both current and electric polarization, in accord with the symmetry argument. From the analysis of temperature and magnetic field dependence, we deduce that inelastic scatterings of electrons mediated by magnons dominantly contribute to the observed nonreciprocal response. Furthermore, the nonreciprocal resistance is significantly enhanced by lowering hole density. The Fermi level dependence is attributed to the deformation of the Rashba band in which the spin-momentum locked single Fermi surface appears by exchange field from the in-plane magnetization. The present study provides a key insight to the mechanisms of novel transport phenomena caused by the interplay of ferroelectric and ferromagnetic orders in a semiconductor.

In materials systems without spatial inversion symmetry, electronic energy bands exhibit momentum-dependent spin-splitting by spin-orbit coupling. The Rashba spin-split bands in polar materials are one such example [1,2], which appears in various materials systems such as heterointerfaces of semiconductors, surfaces of heavy metals, and bulk polar semiconductors [3][4][5][6][7]. They have provided a materials platform to study novel transport and spintronic properties including the spin Hall effect and the spin galvanic effect [8][9][10].
Nonreciprocal electrical transport, in which the electrical resistance depends on the current direction, is one of the characteristic transport phenomena in inversion broken electronic systems [11,12]. In polar electronic structure with electric polarization P, the electronic resistance that is characterized by magnetic field B and current I is expressed by, where the first and second terms represent the resistance at zero magnetic fields and normal magnetoresistance characterized by R0 and β, respectively. ̂ represents the unit vector pointing to P. The third term which depends on both I and B is the nonreciprocal resistance, also termed as unidirectional magnetoresistance. As is clear from the expression, the nonreciprocal response is a nonlinear effect with respect to current, and allowed by symmetry when P, B (or M) and I are orthogonal to each other. The coefficient γ characterizes the nonreciprocal electrical transport, which is the ratio of nonreciprocal resistance and normal resistance R0. Recently, large nonreciprocal responses have been observed in surface/interface systems with ferromagnetism, such as heavy metal/ferromagnetic-metal bilayer interface and magnetic topological insulator surface [13][14][15][16]. A Rashba semiconductor doped with magnetic elements would also be a unique platform to explore the nonreciprocal electrical transport. The introduction of the ferromagnetism is expected to largely deform the Rashba band structure, giving a knob for controlling the nonreciprocal transport. However, the nonreciprocal transport in bulk magnetic Rashba systems has not been studied so far, possibly due to a limited number of conducting materials that show ferroelectricity and ferromagnetism simultaneously.
Mn-doped GeTe (crystal structure shown in Fig. 1(a)) is a multiferroic Rashba semiconductor in which both ferromagnetic and ferroelectric orders appear in a bulk form. The parent compound GeTe is a polar semiconductor with p-type carriers in which inversion symmetry is broken by the displacement of Ge and Te ions along [111] direction below TC FE ~ 700 K [7,[17][18][19][20][21]. The polar crystal symmetry of GeTe results in the large Rashba-type spin splitting of the bulk band structure, whose Rashba parameter R is as large as 2-4 eVÅ [7,22]. By partially substituting Ge with Mn, a ferromagnetic order shows up while keeping the polar crystal symmetry [22][23][24]. The ferromagnetic transition temperature depends on Mn content and hole density, which can be as high as 180 K [23,24]. The coexistence of ferromagnetism and the polar symmetry is demonstrated by angular-resolved photoemission spectroscopy [22].
Here, we investigate the nonreciprocal electrical transport in multiferroic Rashba semiconductor (Ge,Mn)Te thin films. The nonreciprocal resistance in (Ge,Mn)Te is confirmed to be much larger than that in GeTe, suggesting a crucial role of the exchange field effect powered by ferromagnetism. The nonreciprocal resistance is maximized when magnetization is perpendicular to both magnetic-field and electric-current directions, and its sign is switched upon the reversal of magnetization, in accord with the symmetry argument of this multiferroic conductor.
The observed nonreciprocal resistance under variations of applied magnetic field and chemical potentials suggests a crucial role of asymmetry in magnon-mediated scatterings in the uniquely spin-momentum-locked electron system.
We grew 75-nm thick (Ge,Mn)Te films on InP(111)A substrates by molecular beam epitaxy (MBE). The epi-ready substrates were annealed at 380 o C at a base vacuum pressure of approximately 1 × 10 -7 Pa before the deposition of thin films at 200 o C. We inserted Sb2Te3 (1 nm) and GeTe (1 nm) buffer layers beneath the (Ge,Mn)Te layer to stabilize the rhombohedral distortion of (Ge,Mn)Te [25]. The orientation of the (Ge,Mn)Te film is along [111] direction as confirmed by x-ray diffraction [25], which is parallel to the ferroelectric polarization. It is reported that the epitaxial GeTe film grown via MBE technique is single domain over several hundred μm 2 [7,26]. The equivalent pressures of beam flux for Ge and Mn were PGe = 4.5 × 10 -6 Pa and PMn = 5.0 × 10 -7 Pa, respectively. We evaluated the actual composition of Mn in (Ge,Mn)Te by inductively coupled plasma mass spectroscopy and found it to be 9 %. The growth duration for (Ge,Mn)Te layer was 30 minutes, corresponding to the growth rate of approximately 2.5 nm per minute. After the thin film growth, we capped the sample with AlOx by the atomic layer deposition method to prevent the sample from degradation, and then defined Hall bar devices with 10 μm in width by UV photolithography and Ar ion milling. The current direction of the Hall bar devise is along Γ ̅ − K ̅ direction on the surface brillouin zone [6]. The length between voltage probes in the Hall bar devices was 10 μm. The electrodes were Ti (5 nm) and Au (25 nm) formed by electron beam deposition.
We evaluated the nonreciprocal electrical transport properties as a response to ac electric current. We applied ac electric current ( ) = √2 0 sin( ) and measured the quadrature component of the second harmonic voltage 2 by lock-in technique. Since the electrical polarization of (Ge,Mn)Te is parallel to the out-of-plane direction (P || z), the nonlinear resistance Rxx 2ω is to be maximized under the I || x and B || y configuration ( Fig. 1(b)). We anti-symmetrize  shows the current amplitude dependence of Rxx 2ω at By = 0.8 T. Rxx 2ω linearly increases in a low current region and deviates from the linear dependence in the higher current region; magnetic field dependence of Rxx 2ω with different excitation current is shown in Fig. S2 in Supplementary Materials (SM). We define the coefficient of nonreciprocal electrical transport γ (see Eq. (1)) in the low current region and By < 0.8 T. Here, we use γ' = γS, where S represents the cross-sectional area of the sample that is 75 nm × 10 m, to eliminate the sample size dependence [27]. In the present sample, γ' = 1.3 × 10 -11 m 2 T -1 A -1 , which is five orders of magnitude larger than pristine GeTe (see Section S1 in SM and Ref. [28]).
Figure 2(a) shows the magnetic field dependence of Rxx 2ω for several azimuth angles φ (defined as the inset of Fig. 2(b)). The absolute value of Rxx 2ω is the largest at φ = 0° (B || y) and 180° (B ||y), while the sign is the opposite. In addition, Rxx 2ω is negligibly small at φ = 90° (B || x). We plot the φ dependence of Rxx 2ω at |B| = 0.8 T (Fig. 2(b)), which is well fit with the cosine curve (shown in a black line). A similar angular dependence appears with respect to θ, that is a tilt angle of the magnetic field from the z-axis to the y-axis (Fig. 2(c)). From these magnetic-field direction dependences, it turned out that Rxx 2ω is the largest when B is perpendicular to both electric current and electric polarization, as expected from Eq. (1).
In the following, we consider the band structure of (Ge,Mn)Te to identify a microscopic origin for the observed nonreciprocal electrical transport. We employ the following model Hamiltonian that represents a Rashba semiconductor with magnetization M, where ℏ, = ( , , ), * , , , and ̂ respectively represent Planck's constant, Pauli matrices, the effective mass, the Rashba parameter, the parameter for the exchange interaction, and the unit vector pointing to magnetization. Using the band parameters m* = 0.6 m0, = 2.0 eVÅ, and = 50 meV, where m0 represents the mass of a free electron, we describe the band structure of (Ge,Mn)Te near the band edge obtained by the photoemission spectroscopy [22] (Fig. 3(a)). The Rashba bands energetically shift in the opposite direction by 2 ( Fig. 3(b)). Using the model Hamiltonian, we deduce that the hole density p = 1.8 ×10 19 cm -3 of the present sample corresponds to the position of EF close to the band edge (see a broken line in Fig. 3(b) and section S4 in SM), where an isolated spin-momentum-locked Fermi surface exists.
In such a single spin-momentum-locked Fermi surface, the spin scattering of conduction electron associated with magnon can contribute to resistance. Without magnon excitations, the difference in the spin angular momentum between the initial and final states would prohibit the backward scattering of electrons. However, by transferring the difference of angular momentum to magnons (Fig. 3(c)), the back scatterings are allowed and thus contribute to electrical resistance.
The asymmetry in the probability between magnon absorption and emission processes can produce the nonreciprocity of resistance. In fact, the magnon-mediated nonreciprocal resistance was previously discussed in the case of a magnetic topological insulator bilayer which also has a single spin-momentum-locked Fermi surface in the surface Dirac band [15]. In the present experiment, the nonreciprocal resistance ( Fig. 1(c)) decreases in the high field regime (similar magnetic field dependences are observed in other samples as shown in Fig. S5 in SM). This behavior can be explained by the decrease in the population of the magnons as the magnon excitation gap increases with the magnetic field.
To further verify the magnon-mediated scattering mechanism, we examined the temperature dependence of Rxx 2ω under several magnetic fields ( Fig. 4(a)). For By ≤ 6 T, Rxx 2ω monotonically decreases with temperature and vanishes at around 30 K, which is close to the ferromagnetic transition temperature of this sample. On the other hand, the temperature dependence of Rxx 2ω for By ≥ 8 T shows a characteristic hump as indicated by a down arrow. In addition, the peak temperature of the hump increases with By ( Fig. 4(b)). Since the excitation gap of the magnon increases with magnetic field, Rxx 2ω mediated by magnon decreases at high magnetic field and low temperature. This behavior also supports the magnon-mediated scattering mechanism in (Ge,Mn)Te. The similar temperature and magnetic field dependences were observed in the magnetic topological insulator bilayer [15].
We next investigate the EF dependence of the nonreciprocal response. We can successfully control the hole density in the sample from p = 1.8 × 10 19  where the magnon scattering mechanism works most effectively.
It is noteworthy that there may be another possible mechanism for the nonreciprocal electrical resistance other than the magnon-assisted spin scattering process. As demonstrated in a bulk Rashba semiconductor BiTeBr, the elastic scatterings on the deformed Fermi surface by in-plane magnetic field produces the nonreciprocal resistance [27]. However, the elastic mechanism alone cannot account for the magnetic field dependence of the observed nonreciprocal resistance, 10 because the nonreciprocal resistance caused by the elastic scattering is linear to magnetic field. It may play some role, especially in high magnetic field region, where the magnon scattering is suppressed.
In summary, we have investigated the nonreciprocal electrical transport in multiferroic

S1. Evaluation of nonreciprocal coefficient in pristine GeTe
Figure S1(a) shows the magnetic field dependence of nonreciprocal resistance Rxx 2ω for the GeTe thin film with excitation current of I = 100, 500, 1000, 1500, 2000, and 2500 μA. The data for 100 μA is identical to that shown in Fig. 1(c) in the main text. The linear magnetic field dependence of Rxx 2ω is discernible with high excitation current. Figure S1

S3. Transport properties of (Ge,Mn)Te with different growth conditions
We changed the hole density of (Ge,Mn)Te thin films by varying the equivalent beam pressure for Te (PTe) during the MBE thin film growth. We fixed the equivalent beam pressure for Ge PGe = 4.5 × 10 -6 Pa and changed that for Te from PTe = 2.0 × 10 -5 to 8.0 × 10 -5 Pa. Figure S3 shows the transport properties for samples with different PTe/PGe. The temperature dependence of ρxx becomes more metallic with increasing PTe/PGe ( Fig. S3(a)). Accordingly, hole density p, evaluated by the ordinary Hall effect for B > 1 T, monotonically increases from p = 1.8 × 10 19 cm -3 to 1.1 × 10 21 cm -3 ( Fig. S3(b)). All the samples show ferromagnetic hysteresis at T = 2 K.   Figure S5 shows the magnetic field dependence of nonreciprocal resistance Rxx 2ω for (Ge,Mn)Te samples with different hole densities. In order to measure Rxx 2ω in the low excitation regime where Rxx 2ω is linear to the current, we set different excitation currents I for samples. All the samples show a similar magnetic field dependence of Rxx 2ω ; a steep increase below B ~ 0.5-1 T and decrease in higher B region. The similar magnetic field dependence regardless of hole density indicates that the nonreciprocal electrical transport originates from the common mechanism.