Electric field modulation of spin transport

The ultimate goal of spintronics is the active control of spin-polarization within a solid-state environment.^1,2 Recently, spin–orbit coupling (SOC) emerged as a promising tool for implementing such a spin manipulation.³ A large SOC, however, leads to short spin lifetime and diffusion lengths, representing a severe limitation for the development of spintronic devices. Conversely, the small SOC in light semiconductors results in longer electron spin lifetimes, as observed in both lightly doped bulk⁴ and low-dimensional materials.^5,6 Among semiconductors, Si and Ge possess the longest lifetimes since, at variance from GaAs, the Dyakonov–Perel mechanism of spin relaxation is totally suppressed in a centrosymmetric lattice.⁷ Although Si-based spintronics would be desirable, optically generating non-equilibrium spin populations in lightly doped Si are inefficient compared to Ge.^8,9 Moreover, Ge-based platforms could host spintronic, electronic, and photonic devices, all integrated on a Si-compatible platform. Indeed, the Ge direct gap matches the conventional telecommunication window,^10,11 and the 4% lattice mismatch with Si does not prevent the ready integration of Ge with the mainstream Si-based technology, eventually enabling strain to engineer new functionalities.^12–14 

Within this frame, a relevant feature of a semiconductor spintronic device would be to carry the spin information over long distances. To this aim, spin drift helps in increasing the spin-transport length,^15–17 and variation of the spin-dependent signal upon the application of electric fields has been observed in various contexts.^18–24 Although experimental studies on spin drift in semiconductors^15–20 have shown that the spin-transport length can be modulated, most of them are performed at cryogenic temperatures^15,17,18 or investigate the effect of electric fields on spin transport only in an indirect way.^16–20 

We have recently proposed a nonlocal spin-injection/detection scheme which enables one to directly image spin transport in a diffusive regime^25,26 and obtain a direct measure of the spin-diffusion length. We employ the same paradigm to investigate drift-diffusive spin transport at room temperature with a device-oriented architecture, showing that a finite spin-diffusion length combined with the application of an electric field can be capitalized to manipulate spin accumulation. The investigated scheme could thus modulate spin transport over long distances and is expected to reach fast timescales. Preliminary simulations obtained from a time-dependent spin drift-diffusion model predict that, for realistic electric field amplitudes, carrier mobility, and spin lifetime, modulation frequencies reaching 10 GHz can be attained, which is much faster compared to other spin accumulation/modulation systems.²⁷ With respect to previously reported solutions,^15–20 our platform combines all the following characteristics, which make it extremely promising for the development of next generation spintronic devices: (i) room temperature operation; (ii) a large (about one order of magnitude) modulation of spin accumulation; (iii) spin-transport lengths that can be pushed above 40 µm; and (iv) application of moderate control electric fields, which is essential for low power consumption operation.

In particular, we report on the drift-diffusive spin-transport regime in lightly n-doped Ge at room temperature. We exploit the optical orientation technique^28,29 taking place below the edges of Pt stripes^30–32 to locally generate spin-polarized electrons in Ge. The spin detection is performed by measuring the voltage drop given by the inverse spin-Hall effect (ISHE) in a thin Pt pad^33,34 grown on Ge and spatially separated from the point where spin is generated. This nonlocal spin injection/detection scheme allows one to directly image spin transport either in the diffusive³¹ or the drift-diffusive regime. We find that the typical decay length of the spin polarization L_s is doubled (halved) compared with the diffusive case for spin-polarized electrons flowing downstream (upstream) with respect to the applied electric field E = 24 V/cm. In the best case scenario, we experimentally detect a spin loss of just the 40% within a 30 μm-long path, corresponding to L_s ≈ 40 µm. Comparable values of the spin-diffusion length in semiconductors have been predicted^13,35–38 and observed³⁹ only at cryogenic temperatures.

A sketch of the investigated sample is presented in Fig. 1(a). We etch a lightly n-doped Ge substrate (donor concentration N_d ≈ 2 × 10¹⁵ cm⁻³) to obtain a 1 μm-thick and 50 µm × 1 mm-wide mesa to loosely constrain the spin transport along one direction. We then lithographically define a set of Pt stripes on the mesa surface. When illuminated, these stripes participate in the optical spin orientation process, as described below. The stripes are 10 nm-thick and 40 × 3 µm²-wide, with a center-to-center pitch of 6 µm. One of them [the first and 5 μm-wide stripe on the left in Fig. 1(a)] is contacted with Au/Ti electrodes and serves as a spin detector. The flow of spin-polarized electrons entering this Pt detection pad is converted via the inverse spin-Hall effect (ISHE) in a charge flow, which we record as a voltage drop in open-circuit conditions across the ISHE electrodes. A SiO₂ layer [green in Fig. 1(a)] below the contacts on the ISHE detector prevents direct spin and charge absorption from the substrate. We apply a bias electric potential V_b across two AgSb Ohmic contacts⁴⁰ grown on Ge to generate a static electric field E in the semiconductor. For further details on sample growth and characterization, see the supplementary material.

The measurements are performed at room temperature with a confocal microscope (see Ref. 31). The light source is a laser diode tuned at the direct gap of Ge (hν = 0.8 eV) delivering an optical power equal to 270 µW. The circular polarization of the photons is modulated at 50 kHz by means of a photoelastic modulator (PEM). An objective with 0.7 numerical aperture focuses the light on the sample with a diffraction-limited spot of ≈1.3 µm. The measured ISHE signal ΔV_ISHE is obtained by demodulating at the PEM frequency the voltage drop recorded across the Au/Ti ISHE contacts, while the laser spot raster scans the surface of the sample. The optical reflectivity of the sample is simultaneously measured with a photodetector.

In our setup, only the electrons with a component of the spin polarization directed along x [reference frame of Fig. 1(a)] give rise to a detectable ΔV_ISHE (as detailed, e.g., in Ref. 25). When the light beam impinges on the edge of a metal stripe, the electromagnetic wavefront is altered³⁰ yielding a highly localized spin accumulation in the semiconductor, with a detectable in-plane positive (negative) component along the x axis below the left (right) edge of the Pt stripe. The details of the technique are reported in Refs. 30–32.

The spin-polarized electrons are generated at the Γ point of the Brillouin zone and within 300 fs they scatter to the L valleys, mostly preserving their spin polarization.^25,41 Within the explored range of applied electric fields $(E<24V/cm)$⁠, conduction occurs at L.^42,43 It is worth mentioning that, together with spin-polarized electrons, a net population of spin-polarized holes is also generated via optical orientation.⁸ However, any hole contribution to the spin current over the investigated length scales can be neglected since the hole spin lifetime is only a few picoseconds.⁴⁴ This limits the spin-diffusion length of holes to few tens of nanometers in the purely diffusive regime and to—possibly—100 nm in the drift-diffusion regime. Note that the minimum distance between generation and detection points in our setup is 3 µm. It is also worth mentioning that the presence of the Pt stripes employed in the spin-injection process is not expected to alter the spin-transport length in a significant way.³¹ 

In Figs. 1(b)–1(e), we report the experimental map acquired in the 40 × 2 µm² band highlighted in red in Fig. 1(a). Due to the higher reflectivity of Pt stripes compared to the Ge substrate, the former appear bright in the reflectivity map [Fig. 1(b)]. Panels (c)–(e) show the map of the electrical signal ΔV_ISHE recorded by the detector (located out of the maps at x = 0) for three values of the bias voltage V_b. In all cases, when the light beam illuminates the left (right) edges of Pt stripes, a positive (negative) ΔV_ISHE is detected, corresponding to the complementary spin populations generated in Ge below opposite Pt edges. This, together with the linear dependence of ΔV_ISHE on the optical power and on the degree of circular polarization of the light (data not shown), confirms the spin-related nature of the detected signal.²⁵ 

In Figs. 1(c)–1(e), the absolute value of the ISHE signal decreases with the increase in the distance of the injection point of the spin-polarized electrons, i.e., at larger x values. This is the fingerprint of the spin depolarization that electrons experience before reaching the detector. Such a decrease is less (more) pronounced for negative (positive) V_b values, corresponding to E_x > 0(E_x < 0) [see panels (e) and (c), respectively] compared to the zero field case [panel (d)]. This indicates that the transport of spin-polarized electrons can be modulated by the application of an electric field: Depending on the field direction, electrons acquire a drift velocity and cover the distance to the ISHE detector in less (more) time, thus reducing (increasing) their loss of spin polarization. It is worth mentioning that, despite the fact that high electric fields could contribute to a faster spin depolarization, the employed E values are not expected to significantly affect the spin-relaxation time.^21,45

In Fig. 2, we show the average along the y axis of the ΔV_ISHE maps similar to those reported in Figs. 1(c)–1(e) for a set of positive [panel (a)] and negative [panel (b)] bias voltages. Due to the modulation of the typical decay length of the spin-polarization—spin-transport length L_s, from now on—spin-polarized electrons generated far from the detection point are suppressed (preserved) as $Vb$ increases for positive (negative) values. In the most favorable scenario, corresponding to V_b = −5 V, a 30 μm-long path reduces the ΔV_ISHE signal by 40% only. Moreover, ΔV_ISHE is modulated by a factor 7 ± 2 when optical spin orientation is performed at the last Pt stripe (x ≈ 40 µm) by switching the bias voltage between V_b = + 5 V and V_b = −5 V.

It can be noticed that in Figs. 1(c)–1(e) and 2, the maximum value of ΔV_ISHE is obtained for V_b = 0. For V_b > 0, the lowering of ΔV_ISHE is due to the reduction of the spin-transport length in the drift-diffusion regime. Conversely, for V_b < 0, the applied field quickly removes electrons from below the detection pad, reducing their transit time and, consequently, the probability that spin-polarized electrons are injected into the detection pad and contribute to ΔV_ISHE.

We perform a quantitative analysis of the profiles in Fig. 2 by fitting the absolute value of the ΔV_ISHE peaks with a function $Ae−x/Ls$ that accounts for spin depolarization in the generation-to-detection path, with A being the amplitude related to the spin-sink efficiency of the detector and L_s being the spin-transport length.³¹ For V_b = 0, the latter corresponds to the spin-diffusion length $Ls,0=Dτs$⁠, with D being the diffusion coefficient and τ_s being the spin lifetime. The L_s and A values obtained from the fitting procedure are reported in Fig. 3 as a function of the applied electric field E_x. The latter has been estimated from the bias potential V_b by numerically simulating the three-dimensional field distribution in the whole structure. We obtain that a bias potential V_b = 1 V yields an almost uniform electric field E_x = −4.75 V/cm into the stripe. Further information about the electrical simulations is reported in the supplementary material.

While the L_s,0 = 20 ± 0.5 µm value obtained at E_x = 0 is comparable with previous results obtained with lightly n-doped Ge,⁴⁶ L_s is nearly doubled (halved) compared to L_s,0 at E_x = ±24 V/cm for spin-polarized electrons flowing downstream (upstream) [see panel (a)]. The largest value we obtain is L_s = 39 ± 2 µm, a distance previously achieved only under cryogenic conditions.³⁹ In Fig. 3(a), the experimentally inferred L_s values are compared to the spin-transport length obtained from the analytical solution of the one-dimensional (1D) coupled spin continuity and spin drift-diffusion equations (green dotted line, details are given in the supplementary material),^21,47

with k_B being the Boltzmann constant, e being the elementary charge, and T being the lattice temperature. Equation (1) well reproduces the experimental L_s vs E_x trend reported in Fig. 3(a). In the supplementary material, we derive the conditions at which a sizable modulation of the spin signal occurs. The key ingredients are a finite spin-diffusion length L_s,0, which needs to be comparable to the device length, and electric fields |E_x| ≲ 2k_BT/(eL_s,0). Equation (1) also allows predicting the dependence of the spin-transport length L_s on the doping concentration. The latter affects L_s by influencing the spin-diffusion length L_s,0, which can be addressed either experimentally^4,46 or theoretically.^4,13,25

To better understand the role of geometry in our device, we performed two-dimensional (2D) finite-element spin-transport simulations on the xz plane in Fig. 1(a). To keep the numerical simulations as simple as possible, we assumed a uniform electric field directed along x (details are given in the supplementary material). We then estimated the spin current density j_s entering the Pt detector, which is proportional to the experimentally measured ΔV_ISHE value.⁴⁸ Compared to the 1D model, the 2D simulations slightly improve the agreement with the experimentally inferred L_s value at positive E, as shown by the thick line in Fig. 3(a). The difference between the theoretical and experimental values of L_s is lower than 2 µm within the whole explored range of applied electric fields.

The 2D simulations also qualitatively reproduce the trend of the experimentally determined amplitude A [Fig. 3(b)], confirming the reduction of A both for positive and negative applied fields, as a consequence of the variation of the spin-diffusion length for E_x < 0 and of the transit time of spin-polarized electrons below the detection pad for E_x > 0. The main discrepancy between experimental data and 2D simulations is that the latter underestimate the decrease of A for E_x > 0. We ascribe this deviation to the presence of a z component of the electric field (not accounted for in the simulations), which modifies the built-in electric field of the Pt/Ge junction and thus modulates the transmission efficiency of electrons through the Schottky barrier.⁴⁹ We stress, however, that 2D simulations capture the fundamental physical mechanisms underpinning spin-transport.

In conclusion, we have directly imaged the transport of spin-polarized electrons in Ge in either the diffusive or drift-diffusive regime. We experimentally demonstrate that, thanks to the finite spin lifetime, an electric field can strongly modulate the distance traveled by spin carriers before they depolarize. In this way, we are able to modulate by about one order of magnitude the spin-polarized electrons reaching the detector. The largest spin-transport length we measure is 40 µm, which in a purely diffusive regime can only be attained at cryogenic temperatures. Model predictions are in agreement with the experimental findings. Our results open the route to the exploitation of electric fields for guiding and controlling spins across spintronic devices.

See the supplementary material for details about sample microfabrication, electrical simulations, spin drift-diffusion models (1D and 2D), and modulation of spin accumulation.