Effect of modified periodic waveforms on current-induced spin polarization measurements

Much effort has been dedicated to the study of spin-orbit interactions in semiconductors.^1,2 Specifically, there is interest in understanding the electrical generation, manipulation, and detection of spin-polarized carriers, towards the goal of an all-electrical spin-based device.^3,4 One way forward is to make use of current-induced spin polarization (CISP), a phenomenon in which an electrical current generates a bulk, homogeneous, steady-state electron spin polarization.⁵ CISP has been observed in various materials,^6–14 with both electrical generation and manipulation achieved in an n-InGaAs heterostructure,¹⁵ and recent theoretical work has produced a model that qualitatively agrees with the measured dependence of CISP on sample parameters.^14,16

A majority of previous work has observed that CISP increases linearly with applied electric field.^5,8,9,11,12 However, previous investigations have also reported being limited to using smaller electric fields to avoid signal degradation due to heating.⁸ Hernandez et al. also attributed a decrease in observed Kerr rotation at higher electric field to heating.¹³ In our previous work, we found the CISP spin generation rate to deviate slightly from linear behavior as electric field was increased.¹⁴ Naturally, we question whether this deviation is the result of heating or a new regime in which the physics changes. To settle the matter, we offer a method for reducing the effect of heating. In this letter, we investigate CISP using a modified time-varying voltage. We find that when heating is reduced through our method we confirm the expected linear dependence on electric field.

The sample under study is a 500 nm In_0.026Ga_0.974As epilayer grown via molecular beam epitaxy on a (001) GaAs substrate, doped with silicon to achieve a doping density n = 1.55 × 10¹⁷ cm^-3 (same wafer as Sample B studied by Luengo-Kovac et al.).¹⁴ Optical lithography and wet etching were used to pattern a cross-shaped channel onto the sample with arms along the [110] and [1$1¯$0] crystal axes, following Norman et al.¹² We apply a voltage to Au/Ge/Ni contacts to generate a current along the [1$1¯$0] direction. The sample is mounted on the cold finger of a Janis ST-300 helium-flow cryostat, with the cold finger situated between two poles of an electromagnet. The following measurements were performed at 30 K.

The electrical current generates a homogeneous electron spin polarization in the plane of the sample. For current along [1$1¯$0], the spin polarization is perpendicular to the current direction.^5,12 We apply an external magnetic field along the current direction, causing the spin polarization to precess. We direct the linearly-polarized output of a mode-locked tunable-wavelength Ti:Sapph laser (tuned to 836 nm) through our sample to optically detect the out-of-plane spin polarization. The polarization of the transmitted beam is rotated by an amount proportional to this spin polarization,¹⁷ which we measure with a Wollaston prism and balanced photodiode bridge. This Faraday rotation θ_F is related to the applied magnetic field by⁵ 

where θ_el is the electrically-induced Faraday rotation, τ is the transverse spin lifetime, and ω_L = gμ_BB/ℏ is the Larmor precession frequency corresponding to the applied field B. For this sample, the g-factor was determined via time-resolved Faraday rotation to be -0.405. In this paper, we are interested in the quantity γ ≡ θ_el/τ, which we will refer to as the spin generation rate, with units of (Faraday) rotation per unit time. This γ is proportional to the density of spins oriented per unit time,⁵ but in contrast to Kato et al., we omit the corresponding proportionality factor of spin density per (Faraday) rotation angle. Previous studies have determined this factor to be 1.42 μm⁻³μrad⁻¹ for our material.¹⁴ 

The Faraday rotation is small (order tens of microradians), so we make use of lock-in detection. The basic principles of lock-in detection are described in the supplementary material. The applied voltage is modulated as a bipolar square wave with frequency 1.167 kHz. By scanning the applied magnetic field, we obtain data such as that in the left plot of Fig. 1a.

Rather than utilize a standard bipolar square wave, we can modify the voltage further by reducing the amount of time the voltage is nonzero. We define the off/on ratio for a square wave as the ratio of time spent at zero voltage to the amount of time spent at peak voltage. An example waveform with off/on ratio 5 is plotted in Fig. 1b. This change preserves the peak voltage V_pk while reducing the root-mean-square voltage:

where n is the off/on ratio, as derived in the supplementary material. This is vital as the power dissipated through resistive heating by a time-varying voltage is given by¹⁸ 

The resistance of our sample is approximately 500 Ω at 30 K. Larger off/on ratios will reduce the amount of sample heating, as shown for a square wave with V_pk = 1 V in Fig. 1c.

As the off/on ratio is increased, the overlap of the square wave with a sine at the same frequency is reduced. That is, the Fourier sine coefficient decreases as off/on ratio increases (Fig. 1c). Therefore, for a constant CISP amplitude, the lock-in amplifier will measure a smaller signal with increasing off/on ratio. To compensate for this drop-off in measured Faraday rotation due to measurement technique, we apply a correction factor to each of our data sets, normalizing by the Fourier coefficients (see the supplementary material). This processing step allows us to recover the actual Faraday rotation. We demonstrate the effect of this processing in Fig. 1a. Each curve corresponds to 1 V peak voltage. Note, however, that signal noise is also adjusted, preserving the signal-to-noise ratio. Measurements taken using square waves with larger off/on ratios should then yield the same data with decreased signal-to-noise ratio.

This is not observed for larger voltages, however. We measure CISP at various applied voltages and fit to Eq. 1, allowing us to extract θ_el and τ and thus γ. We plot γ as a function of voltage in Fig. 2. For a standard bipolar square wave (black squares, off/on ratio 0), γ reaches a maximum at V_pk = 4 V before decreasing at higher applied voltages. This is a clear deviation from the nearly-linear dependence on voltage seen in previous experiments.

We determine the role of heating in this change by carrying out the measurements again using square waves with nonzero off/on ratios. Only V_RMS changes while all other experimental parameters remain the same. The functional dependence of CISP on applied voltage becomes linear as off/on ratio is increased. Data is shown for five ratios in Fig. 2. It is observed that the extracted values for γ are more sensitive to off/on ratio at larger voltages. This latter point is shown in Fig. 3a, in which we plot γ values as a function of off/on ratio for three selected applied voltages. At 1 V applied, γ does not appreciably change with off/on ratio.

Ideally we would choose an arbitrarily large off/on ratio to eliminate heating altogether. However, increasing the off/on ratio decreases the signal-to-noise ratio, compromising the ability to obtain accurate values of γ at off/on ratios of 25 and above. Furthermore, the voltage is supplied by a function generator that utilizes a finite number of points to represent the waveform. For sufficiently large ratios, it will no longer be possible to form a square wave with sharp rising and falling edges. Referring back to Fig. 3a, γ appears to approach a constant value as off/on ratio is increased. The ideal ratio yields a γ close to the asymptotic value while preserving as large a signal-to-noise ratio as possible. For this sample, an off/on ratio between 5 and 10 (corresponding to linear behavior in Fig. 2) is ideal.

Thus far we have assumed that CISP depends on V_pk, but in previous measurements using standard bipolar square waves, V_pk and V_RMS are identical. We consider the possibility that γ actually depends on V_RMS rather than V_pk by recasting the measurements of Fig. 3a as a function of V_RMS in Fig. 3b. A given V_RMS can be achieved with multiple combinations of V_pk and off/on ratios, so we are interested in whether two similar V_RMS based on different V_pk have the same γ. Instead, we observe three curves corresponding to the three peak voltages. For a given V_RMS, the measured γ are distinguished by V_pk, implying that the latter quantity sets the asymptotic γ while V_RMS determines the deviation from that ideal value. Because dissipated power is proportional to $VRMS2$⁠, we attribute this dependence on V_RMS to heating effects.

In order to understand these results, we appeal back to dissipated power in Fig. 1c. That plot displays the power for a waveform with V_pk = 1. For other applied peak voltages, one must scale the dissipated power by $Vpk2$⁠. For example, applying 6 V via a standard bipolar square wave should dissipate 72 mW, an order of magnitude difference from 2 mW in the 1 V case. As expected, application of a 6 V standard bipolar square wave noticeably affects the behavior of our cryostat’s temperature controller. When peak voltages above 2 V are applied, we observe a drop in the amount of power required by the cryostat heater to maintain a 30 K temperature, sometimes by as much as 30%. Predictably, then, the greater the expected dissipated power, the greater the deviation from linear dependence of γ. The temperature controller heater power is less affected when larger off/on ratios are used.

Sample heating changes the temperature of our material, changing with it optical and electronic properties that can impact our measured spin generation rate.¹⁹ For example, carrier concentration and mobility are electronic properties that can change with temperature, and CISP has been observed to depend on both of them.¹⁴ One of the chief spin relaxation mechanisms in gallium arsenide and its alloys is the D’yakonov-Perel’ mechanism, with relaxation rate proportional to the momentum scattering time.²⁰ If sample heating changes the momentum scattering time, the spin relaxation time and consequently the spin generation rate will be affected as well.

Sample heating will also change the optical absorption profile of the sample and shift the wavelength at which maximum Faraday rotation occurs. To verify this, we measured γ as a function of laser wavelength. For all-optical measurements in which a second laser pulse is used to generate a spin polarization instead of an applied voltage, the maximum Faraday rotation occurs around 836 nm. We measured CISP as a function of laser wavelength under application of both 1 V and 6 V standard bipolar square wave. For 1 V, both Faraday rotation and γ are maximized around 836 nm as well. For 6 V we observe the peak wavelength redshift to approximately 838 nm. This implies that sample heating detunes our laser from the peak Faraday rotation wavelength by altering the absorption profile of our sample. For gallium arsenide, this shift in the absorption edge is roughly equivalent to a temperature increase of 20 K.¹⁹ This will reduce the magnitude of the signal we measure, even when the magnitude of the electrically-generated spin polarization is unaffected.

Thus we have demonstrated that observed deviations from linear CISP behavior are attributable to heating and can be reduced in an $In0.026Ga0.974$As epilayer by utilizing a modified square wave voltage and applying a Fourier correction factor to the measurement data. Selecting the proper off/on ratio must balance the magnitude of heating with the desired signal-to-noise ratio. Future CISP studies can utilize this technique to reduce or eliminate heating effects, removing the limitations of experiments performed at large applied electric fields.

See supplementary material for a brief description of lock-in amplifier operation followed by a mathematical definition of a square wave with nonzero off/on ratio (as depicted in Fig. 1b). We derive V_RMS for off/on ratio n (Eq. 2) as well as the appropriate Fourier coefficients used to rescale the CISP data.

The authors would like to thank Caleb Zerger and Dr. Marta Luengo-Kovac for their contributions. J.R.I. was supported by the Department of Defense through the National Defense Science and Engineering Graduate Fellowship (NDSEG) Program. S.H. and R.S.G. were supported in part by the National Science Foundation, Grant No. DMR 1410282. D.D.G. was supported in part by the National Science Foundation, Grant No. EECS 1610362. V.S. was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, Award DE-SC0016206. Sample fabrication was performed in part at the University of Michigan Lurie Nanofabrication Facility.