Field-induced negative differential spin lifetime in silicon
aa r X i v : . [ c ond - m a t . m t r l - s c i ] S e p Field-induced negative differential spin lifetime in silicon
Jing Li, Lan Qing, Hanan Dery,
2, 3 and Ian Appelbaum ∗ Department of Physics and Center for Nanophysics and Advanced Materials,University of Maryland, College Park, MD 20742, USA Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627 Department of Electrical and Computer Engineering, University of Rochester, Rochester, NY 14627
We show that the electric field-induced thermal asymmetry between the electron and latticesystems in pure silicon substantially impacts the identity of the dominant spin relaxation mechanism.Comparison of empirical results from long-distance spin transport devices with detailed Monte-Carlo simulations confirms a strong spin depolarization beyond what is expected from the standardElliott-Yafet theory already at low temperatures. The enhanced spin-flip mechanism is attributedto phonon emission processes during which electrons are scattered between conduction band valleysthat reside on different crystal axes. This leads to anomalous behavior, where (beyond a criticalfield) reduction of the transit time between spin-injector and spin-detector is accompanied by acounterintuitive reduction in spin polarization and an apparent negative spin lifetime.
In compound semiconductors, the eventual reductionin drift velocity of conduction electrons with increasingapplied electric field is known as negative differential mo-bility or the Gunn effect [1, 2]. In this field regime (typi-cally several kV/cm), hot electrons scatter into low-lyingsecondary energy minima in the conduction band wherethe effective mass is larger, reducing their kinetic energy.The multivalley band structure of silicon also allows forthe existence of this phenomenon but only at low temper-atures; for all
T >
30 K, the drift velocity increases andeventually saturates with increasing applied field [3],[4].Therefore, at elevated temperatures the time-of-flight ofconduction electrons across the Si channel of a transportdevice drops monotonically with increasing electric field.If electrons are initially spin polarized, then the acceptedElliott-Yafet spin relaxation theory suggests that the spindepolarization during transport is dependent only on thetime-of-flight. In this theory the spin and momentumrelaxation times are proportional [5, 6], so the result-ing spin polarization increases with electron drift veloc-ity. Indeed, we have confirmed this expectation in previ-ous experiments where low and moderate applied fields( < T ≥
30 K in high electric fields. Withincreasing field, the spin polarization of detected elec-trons first increases as expected from the Elliott-Yafetstatic lifetime model; however, above 2 kV/cm it startsto decrease, showing a Gunn-effect dependence akin toa negative differential spin lifetime without any simul-taneous negative differential charge/spin mobility. Theorigin of this counterintuitive behavior is then elucidatedby Monte Carlo simulations and a quantitative analyticaldescription. When the electron ensemble is out of ther-mal equilibrium with the lattice, an efficient spin relax- ation mechanism becomes accessible due to field-inducedintervalley scattering. We quantify the spin relaxationtime as a function of both the lattice and electron en-semble temperature. The latter provides a means to de-termine the dependence of spin relaxation in silicon onthe electric field and will enable the optimization of spin-tronics devices.Coherent spin precession and spin valve measurementswere performed to observe the nonequilibrium depolar-ization effect and to quantify the negative differentialregion of spin lifetime. In both experiments, we em-ployed all-electrical devices in which spin-polarized elec-trons (aligned with the in-plane magnetization directionof a ferromagnetic thin-film source) are tunnel injectedthrough a Schottky metal contact and into a 225 µ m -thick wafer of nominally undoped Si(100). The electronsthen drift across the wafer thickness due to an appliedelectric field, and are collected by a second ferromag-netic film where their spin is analyzed using a ballisticspin detection scheme. The results presented below donot depend on the injection and detection techniques butonly on the spin and charge transport characteristics ofthe Si channel. We therefore include all device-specificdescription in the supplemental material [11], and referthe interested reader to Refs. [7, 9, 10] for further details.The time-of-flight distribution of the electron currentcan be recovered from quasistatic spin precession mea-surements by applying an external magnetic field, B ,perpendicular to the injected spin direction but paral-lel to the electric field [12]. This magnetic field inducesspin precession at frequency ω = gµ B B/ ¯ h , where g is theelectron g-factor, µ B is the Bohr magneton, and ¯ h is thereduced Planck constant. We denote the time-of-flightdistribution by D ( t ) where its mean and standard devi-ation are, respectively, measures of the average transittime and of diffusion and dephasing effects in the chan-nel. The signal contribution from the spin componentparallel to the detector magnetization of an electron ar-riving at the detector in the time interval [ t, t + dt ] is T r a n s i t T i m e [ n s ] Accelerating Voltage [V] -1000 700800 Sp i n S i g n a l [ p A ] Magnetic Field [Oe] U = 80 V T = 61 K P r o b a b ili t y Transit Time [ns]
50 100
FIG. 1. Average transit time across a 225-micron-thick in-trinsic silicon device as a function of the applied voltage forvarious temperatures. Error bars indicate the transit timeuncertainty (extracted from the width of the time-of-flightdistribution; see lower inset). Top inset: Symmetrized spinprecession data at T = 61 K and 80 V (3.5 kV/cm). Thedata show high spin coherence with well-defined oscillationfield period. Lower inset: spin current transit time distribu-tion obtained by transforming the precession signal (see text). therefore D ( t ) cos ωtdt . The variation in quasistatic de-tected signal is then D ( ω ) ∝ R ∞ D ( t ) cos ωtdt ; by repeat-ing the measurement at various applied magnetic fieldsone can map the precession frequency dependence of thedetected signal [8]. Finally, the empirical time-of-flightdistribution is recovered without any model fitting by theinverse Fourier transform of D ( ω ). An example of thistransformation between D ( ω ) and D ( t ) is shown in thecoupled insets to Fig. 1. The main figure shows the aver-age transit time across the silicon channel as a functionof applied voltage for several temperatures. Clearly, in-creasing the internal electric field with applied voltagereduces the transit time until the onset of velocity satu-ration for voltages ?
60 V (electric field ∼ . ≈
20 Oe is applied along the source mag-netization axis and thus no spin precession is induced.The final spin polarization after transport is extractedby the ratio P = ( I P − I AP ) / ( I P + I AP ), where I P isthe measured signal current in a configuration where thein-plane injector and detector magnetization directions(and hence spin initialization and measurement axes) areparallel, and I AP is for antiparallel configuration. Theinset in Fig. 2(a) shows an example of this spectroscopytaken by interleaving P and AP measurements at eachapplied voltage to avoid signal drift from field-induced
10 10067891011 P o l a r i z a t i o n [ % ] Mean Transit Time [ns] Sp i n S i g n a l [ n A ] Accelerating Voltage [V]
PAP
Background T = 61 K FIG. 2. Measured current polarization in a ferromag-net/silicon/ferromagnet device as a function of transit timein the silicon channel. Solid lines indicate exponential fitsto the low-field (long transit time) data and indicate longspin lifetimes in that regime. Effects of spin depolarizationfrom electric-field-induced spin relaxation are evident at high-fields (short transit time; circled data). Inset: Example spec-troscopy at 61 K showing spin signal in a parallel (P) andantiparallel (AP) magnetic configuration, with backgroundsignal for subtraction. stress. We also include the background detector currenttaken under conditions of zero injection current after thesignal measurement. It has subsequently been subtractedin the polarization calculation to avoid misinterpreting aspurious dilution for spin depolarization.Figure 2(a) shows the measured polarization as a func-tion of the average transit time ( τ tr ). This ratio de-pends on the spin relaxation time in the Si channel by( I P − I AP ) / ( I P + I AP ) = P e − τ tr /τ s where P (limited bythe spin-injection and detection efficiencies of the device)is the optimal attainable value. The figure shows that atlong transit times, the polarization increases with reduc-ing the transit time, as expected from the Elliott-Yafettheory. However, at short transit times (circled data)the trend is unexpectedly opposite. This observation ofa nonmonotonic spin polarization Gunn effect is the mainexperimental result of this Letter.The origin of this phenomenon is a transition to apreviously-ignored regime where electric field directly en-ables a spin relaxation pathway. The field-induced mo-mentum relaxation enhancement, as implied by the satu-ration in charge transport data of Fig. 1, is not commen-surate with the spin relaxation enhancement. Applyingthe accepted Elliott-Yafet theory (proportionality of spinand momentum relaxation times) would therefore lead tothe false conclusion that the rising polarization with ini-tially increasing transit time is indicative of an unphysical negative spin lifetime.We have performed Monte-Carlo simulations in order Hot Valley (h)Cold Valley (c) × D r i f t V e l o c i t y [ c m / s ] P o l a r i z a t i o n [ % ] M e a n E n e r g y [ m e V ] Sp i n L i f e t i m e [ n s ] Electric Field [V/cm] Electric Field [V/cm] (a) (b) (c) (d)
FIG. 3. (a) Electron drift velocity, (b) mean energy, (c) spinrelaxation time and (d) final polarization as a function of elec-tric field, calculated from numerical integration of the distri-bution obtained from Monte Carlo simulation. The dottedlines in (c) denote Eq. (1). to elucidate the charge transport and spin relaxation ofconduction electrons heated by the electric field (“hot”electrons). A full description of the numerical procedureis provided in the supplemental material [11], and herewe summarize the important features. Ellipsoidal energybands are used to model the equivalent six conductionband valleys [13]. Momentum relaxation mechanismsare modeled by electron-phonon interactions (both in-travalley and intervalley processes) and intravalley elec-tron scattering from ionized impurities [14]. Betweenscattering events, electrons are treated as classical parti-cles accelerated by the electric field. Typically, an out-of-equilibrium electron distribution reaches its steady-state within 1 ns regardless of the initial condition. Fig-ure 3(a)-(b) show the corresponding drift velocity andmean energy as a function of applied electric field. Hot(cold) valleys refer to the four (two) valleys whose axis isperpendicular to (collinear with) the electric field. Themean energy in a hot (cold) valley is higher (lower) due tothe different projections of electric field on the ellipsoidalenergy bands.The spin relaxation due to electron-phonon interac-tions is calculated by integration of intravalley and in-tervalley spin-flip matrix elements [15] while using theMonte-Carlo hot-electron distributions. The solid linesin Fig. 3(c) show the results of this numerical proce-dure. Note that spin relaxation due to scattering withionized impurities is negligible in nearly intrinsic wafers.As a result the total spin lifetime in our devices is signif-icantly longer than in heavily doped Si channels [16–18].Figure 3(d) shows the spin polarization P exp( − τ tr /τ s ),where P = 0 .
125 is chosen to fit the experimental injec-tion and detection efficiencies. The average transit time τ tr is calculated from the drift velocity after transport across 225 µ m. At low fields, the polarization rises withelectric field since the increase of drift velocity surpassesthe decrease of spin relaxation time. As the drift ve-locity begins to saturate in high fields, the polarizationdrops slowly due to the enhanced reduction of the spinrelaxation time. This dependence of spin polarization onthe electric field agrees well with the experimental resultsand reproduces the Gunn-type behavior (here shown asa function of the field).We focus on the important underlying spin relaxationmechanism and analytically quantify the observed ef-fect. From the mean energy (Fig. 3(b)), one can see thatelectrons driven by the electric field become hot enoughto undergo intervalley electron-phonon processes duringwhich the electron delivers to the lattice a few tens ofmeV [4]. We consider f -processes at which electrons arescattered between valleys of different crystal axes. Thisprocess dominates the spin relaxation of hot electronssince it involves a direct coupling of valence and con-duction bands [15]. To conserve crystal momentum, thephonon wavevector resides on the Σ axis. The symmetry-allowed phonon modes for spin relaxation are Σ and Σ with respective phonon energies of Ω f, ≈
47 meV andΩ f, ≈
23 meV. The Σ mode allows for scattering be-tween all valleys and the Σ mode restricts them to thecase that one of the involved valley axes is not perpendic-ular to the spin quantization axis. To analytically quan-tify the spin relaxation we functionalize the hot electrondistributions. Fig. 4(a) shows the Monte-Carlo steady-state energy distributions in hot and cold valleys at 30 Kand 4 kV/cm. The distribution (in each of the valleys)can be described by a two-component heated Boltzmanndistribution. At the low energy part, the effective tem-perature of the electron distribution can be extractedfrom the mean energy k B T e , shown in Fig. 3(b). Atthe higher energy part, intervalley process tend to coolthe system. To simplify the analysis below we employan effective electron temperature, T ′ e = T + γ ( T e − T )where T is the lattice temperature and γ ≈ . y i = Ω f,i /k B T , y ′ i,µ = Ω f,i /k B T ′ e,µ where i denotes the phonon modesand T ′ e,µ is the effective temperature of the electrons ina cold ( µ = c ) or a hot ( µ = h ) valley. Using the above,we arrive at an analytical spin lifetime [11],1 τ s ≈ C µ = h,c X i =1 , A i,µ n µ exp( y i − y ′ i,µ )+1exp( y i ) − y ′− i,µ + √ , (1)where C = 0 .
036 ns − is a constant related to the spin-orbit coupling parameter of the X point at the edgeof the Brillouin zone. A ,h = 8 (12), A ,h = 1 . . A ,c = 8 (4) and A ,c = 0 . .
75) are symmetry relatedparameters when the electric field is collinear with (per-pendicular to) the spin-quantization axis. The n c and Device 1Device 2Monte Carlo P o l a r i z a t i o n [ % ] Electric Field [V/cm] (a) (b) (c) (d) R e p o pu l a t i o n R a t i o n c / n h Electron Energy [meV] D i s t r i bu t i o n F un c t i o n − − Temperature [K] D e p o l a r i z a t i o nS c a l e V [ k V ] Accelerating Voltage [V] . . . . . . Hot Valley (h)
Cold Valley (c) T = 30 K E = 4 kV/cm FIG. 4. (a) Electron distributions in hot (solid line) and cold(dashed line) valleys. The total electron density is 10 cm − ,the electric field is 4 kV/cm and the lattice temperature is30 K. (b) Ratio between electron densities in cold and hotvalleys as a function of the field. (c) Experimental depolar-ization at high-fields (extracted from Fig. (2)). (d) Charac-teristic scale of the drop in spin polarization as a function oftemperature. n h denote, respectively, the fractional population at coldand hot valleys where 2 n c + 4 n h = 1. Figure 4(b) showsthe repopulation ratio, n c /n h . The asymmetry in val-ley population is largest at intermediate fields since elec-trons that reside in hot valleys become energetic enoughfor intervalley scattering to cold valleys. At high fields,scattering in the opposite direction also becomes ac-cessible and the valley population is more symmetrical( n c /n h ≈ n µ and T ′ e,µ in Eq. (1) reproduces the spin re-laxation of hot electrons as can be seen from the compar-ison between the dotted and solid lines in Fig. 3(c). Atequilibrium conditions where T ′ e,µ = T this mechanism isgreatly suppressed (especially at low temperatures).We can quantitatively compare the results of our cal-culations with empirical data by extracting a character-istic voltage scale V for f -process-induced spin depolar-ization, where we approximate P ≈ P (1 − VV ) in thehigh electric-field regime. Fits to the measured spin po-larization data are shown in Fig. 4(c), and the consis-tent temperature dependence of V for several devicesis shown in Fig. 4(d). The Monte-Carlo prediction ex-tracted from the high-field regime in Fig. 3(d) closelyresembles the empirical values in both magnitude andlattice-temperature dependence due to the more efficientgeneration of intervalley scattering (and hence lower V )at lower temperatures. By taking the high-field limit ofEq. (1), one can write V ≈ v d / [ d ( τ − s ) /dE ] ≈ . v d is the saturated drift velocity. This close corre-spondence confirms our interpretation of field-induced f - process spin depolarization in the experiment, in a regimewhere acoustic-phonon-mediated scattering as well asscattering with states at the spin hot-spot [15, 19] aretoo small to account for this effect [11].Finally, we note that a recent theoretical proposal sug-gesting that stochastic polarization fluctuations can beamplified by spin-dependent mobility [20] has also beentermed a “spin Gunn effect”. Our experiment and the-ory differ from this scheme in that the mobility and dif-fusion constants are spin independent, electron-electroncollisions are negligible [21], and the physical origin ofthe effect is attributed to the signature of spin-orbit cou-pling on electron-phonon intervalley scattering. The phe-nomenon observed here, as in its charge-based counter-part, is due to a strong electric field-induced relaxationwhich leads to a qualitatively different spin transportregime distinct from expectations based on the Elliott-Yafet theory.In conclusion, high electric fields present in silicondevices can substantially change the dominant physi-cal mechanism of spin relaxation. In this regime, theElliott-Yafet mechanism mediated by intravalley acous-tic phonons is far outweighed by the depolarizing effectsof inelastic scattering with intervalley f -process phononscreated by the efforts of the system to recover thermalequilibrium. This behavior is expected to be criticallyimportant in the design of devices making use of spinsto transmit information, especially when strong staticelectric fields are required [22]. Similarly, the derived de-pendence of the spin lifetime on the electric field is offundamental importance to the design of semiconductordevices that make use of spin as an alternative degree offreedom [23–26].Work at UMD is supported by the Office of NavalResearch and the National Science Foundation. We ac-knowledge the support of the Maryland NanoCenter andits FabLab. Work at UR is supported by AFOSR andNSF (No. FA9550-09-1-0493 and No. DMR 1124601). ∗ [email protected][1] J. Gunn, IBM J. Res. Dev. , 141 (1964).[2] H. Kroemer, Proc. IEEE , 1736 (1964).[3] In T < ∼
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