Proposal for Efficient Generation of Spin-Polarized Current in Silicon
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Proposal for Efficient Generation of Spin-Polarized Current in Silicon
L. K. Castelano ∗ and L. J. Sham Department of Physics, University of California-San Diego, La Jolla, California 92093-0319
We propose a spin-dependent resonant tunneling structure to efficiently inject spin-polarized cur-rent into silicon (Si). By means of a heavily doped polycrystalline Si (Poly-Si) between the ferro-magnetic metal (FM) and Si to reduce the Schottky barrier resistance, we estimated raising thetunneling current density up to 10 Am − . The small Fermi sea of the charge carriers in Si focusesthe tunneling electrons to the resonant spin states within a small region of transverse momentum inthe ferromagnet which creates the spin polarization of the current. Because of the large exchangesplitting between the spin up and down bands, the decay of the spin current is explained in termsof scattering out of the focused beam. The spin polarization in the current survives only if thethickness of the FM-layer is smaller than the spin-diffusion length estimated from that cause.(October 30, 2018) Silicon-based devices are responsible for the process-ing of high volume information and for communicationdevices, thereby playing a fundamental role in existingelectronics. Information storage, on the other hand, isin separate magnetic media. The idea of combining bothfunctions in the same device spurred vast interest in spin-tronics with semiconductors. Many advantages of suchdevices were pointed out in recent years, e.g. , memorynonvolatility, increased data processing speed, decreasedpower consumption, and increased integration densities[1]. The first step to implement spintronics in semicon-ductors consists in the ability to inject and to detect spin-polarized carriers in such materials. Such a task can beachieved by making a contact between a semiconductorand a FM, since the electrical conductivity for majorityspin and minority spin is different in the FM. However,this type of contact produces a very low efficiency spininjection. The difficulty is due to the difference betweenthe conductivities of these materials [2] and to circum-vent such a problem, a barrier must be included betweenthe two media [3]. Thus, either an insulator or a Schot-tky barrier can be used to raise the efficiency of the spininjection into semiconductors; on the other hand, the cur-rent is drastically reduced by the barriers. The problemof spin injection efficiency into semiconductors is well un-derstood and many experimental results have shown itsfeasibility [4–6]. The current challenge is to create a sig-nificant electrical output signal that captures the spininformation. In other words, to build an efficient silicon-based spintronic device, we need to find a system wherehigh current and spin-polarization coexist.In this letter, we propose a structure composed of aFM between two heavily doped semiconductor to achieveboth spin-polarization and high total current. The op-eration of such a device is in a sense analogous to theoperation of the resonant tunneling diode (RTD) [7],therefore we will call it spin-RTD hereafter. Currentin a RTD flows only when the applied voltage reachesspecific values that correspond to resonant states withinthe barriers. If these resonant states are spin-dependent, each spin component of the current will behave differ-ently as a function of the applied bias. For instance,spin-down (spin-up) electrons will flow when the appliedvoltage reaches the specific resonant energy E ↓ r ( E ↑ r ) [seeFig. 1], while electrons with opposite spin will be filtered.The low density electrons in the conduction valleys of Siserve to focus the transport electrons in the FM regionto a small part of the Fermi surface [8]. These regions asfilamentary channels in k space together with the ferro-magnet subbands enable the relevant resonance tunnelingphenomenon. In this work we adopt the effective massapproximation for both metal and semiconductor. Thecrystal symmetry effects found important in epitaxiallygrown magnetic tunnel junctions [9, 10] are averaged outby the disorder in strongly doped Poly-Si. The I-V char-acteristics in Schottky barriers appear in general quali-tatively accounted for by the effective mass theory [11]. E F E SiF d P-Si (III)(II) V - s (z) V + s (z) m FM m Si Si SiPoly-Si FM Poly-Si d w + E P-SiF E r E r m Si (I) E=0 w - z eV d P-Si
FIG. 1: (Color online) Schematic conduction band diagramfor spin-up electrons (solid curve) and spin-down electrons(dashed curve) at bias eV . The Fermi energy of Poly-Si (Si)is represented by E P − SiF ( E SiF ). d (d P − Si ) is the thicknessof the FM (Poly-Si) layer. w + and V + s ( z ) ( w − and V − s ( z ))designate the width and potential of the Schottky barrier inthe right (left) side, respectively. The effective mass of the FM(Si) is given by m F M (m Si ). Also, resonant state with energy E ↑ r ( E ↓ r ) for spin-up (spin-down) electrons is schematized. ∆is the exchange energy, which also gives the difference betweenthe bottom of the two spin conduction bands of the FM. The band-structure of the FM is approximated by theStoner model, with a spin-up band and a spin-down bandsplit by a constant exchange ∆. In Fig. 1, we present aschematic band diagram for the proposed spin-RTD. Thetwo thin Schottky barriers are formed by the contact be-tween the FM and Poly-Si. Such a material has beenused industrially as the conducting gate for MOSFET[12], CMOS [13] and in thin-film transistors [14] applica-tions. Because Poly-Si can reach very high densities ofdopants, we shall make use of it as an intermediate ma-terial between the FM and Si. Also, this material formsan ohmic contact with Si, which is located at the edges ofour spin-RTD. In the middle of Fig. 1, we represent thetwo spin bands of the FM, shifted by the exchange energy∆. Experimentally, we believe that the techniques devel-oped for magnetic nanopillars [15] might be very usefulto build the proposed device.The current density for each spin component at zerotemperature is determined by [11] j σ = em Si, k ¯ h Z E SiF E min dE Z E dE k (2 π ) D σ ( eV ; E − E k ) , (1)where σ = ( ↑ , ↓ ), E min = ( E SiF −| eV | / E SiF −| eV | / E k = ¯ h k k / m Si, k , m Si, k is the effective mass of Si par-allel to the transport direction, and D σ ( eV ; E − E k ) de-notes the transmission probability of the electron of spin σ through the spin-RTD. Θ( x ) is the step function.To treat the spin relaxation of the current in the para-magnet regions of Si and Poly-Si, we use the usual spin-flip term in the spin diffusion theory for paramagneticmetals [16, 17]. Although the spin flip term is also usedin the multilayers of para- and ferromagnets [18], thelarge exchange splitting in the ferromagnet means that s − d electron scattering cannot satisfy energy conser-vation without concomitant change of the spatial states.Consider the common case of Si interface in the (001) di-rection with two pockets of conduction electrons. Theirlow density limits the tunneling current in the FM re-gion to be less than 0.1 % of the FM Fermi surface cross-section normal to (001), increased to at most 1 % in thePoly-Si regions. We suggest that, for the electron sub-bands in the quantum well of the FM, the confinementof the transport electrons near the Fermi level to smalltransverse momenta causes the current to be spin-filteredin resonance with a subband edge of a particular spin.The case of two different spin paths is analogous to thecase of strong spin-orbit split bands. [19]. The polariza-tion decay in the tunneling current is due to scatteringwith the electrons outside the tunneling current.To obtain a rough estimate of the spin currents, weuse, in FM, the exchange splitting of the two bands forthe difference in the real parts of V σ ( z ) and the con-stant imaginary parts of W σ for the current decays inthe d tunneling electrons in the different spin channels.Then the spin-dependent transport relaxation time is k- independent, τ σ = ¯ h/ W σ . We argue that the inhibit-ing effect of disparate energy levels of the exchange-splitbands in ferromagnetic metals is present at the inter-face, even without the semiconductor focussing effect andthus interpret the bulk and interface measurement re-sults in ferromagnetic metals which are generally pre-sented as the spin-diffusion length [20]. Without theadditional data for the spin-independent component ofthe transport time of the electron in the tunneling chan-nel, we cannot unentangle the conductivity and the dif-fusion coefficient for each spin channel, as was done inthe semiconductor case with optical excitation [19]. Wesimply approximate the two independent exponential de-caying spin-components in the accumulation layer by thesame measured [20] “spin-diffusion length” ℓ σsd . Thus, τ σ = ℓ σsd /v σF , where v σF = p E σF /m F M is Fermi velocityin the ferromagnet.The shape of the Schottky barrier in the Poly-Si semi-conductor region ( s ) is calculated in the depletion layerapproximation, with a constant donor concentration, N P-Si d , in − ( w − + a ) < z < − a and a < z < w + + a ,where a = d/ z = 0 is set in the middleof the FM. The electrostatic potential of the barriers is, V ± s ( z ) = 2 πe N P-Si d ǫ s (cid:0) z ∓ z ( a + w ± ) + 2 aw ± + a (cid:1) + E Si F + V s − eV / , (2)where ǫ s is the dielectric constant of the semiconductorand plus (minus) signal refers to the right (left) side. Theheight of the Schottky barriers is V s and the widths are w ± = h ǫ s πe N P − Sid ( E P-Si F + V s ∓ eV / i / .In our numerical calculation, we use the following pa-rameters: v ↑ F = 1 . × cm/s, v ↓ F = 4 . × cm/s (cor-responding to the two exchange split bands E ↑ F = 4 . = = 59 nm = 12 nm= 5 nm Bias [V] j ( A m - ) FIG. 2: (Color online) The black (red or light gray) curverepresents the current density for spin-up (spin-down) elec-trons as a function of applied bias. Here, we consider a fixedthickness for the FM-layer d = 8 . nm and T=0 K. The re-sults considering different “spin-diffusion length” are also in-dicated. eV, E ↓ F = E ↑ F − ∆ = 0 .
66 eV), and the effective mass ofelectrons inside the FM equal to the free-electron mass(m FM =m ) [21]. For silicon, we adopt m Si,z = 0 .
91 m and m Si, k = 0 .
19 m . For simplicity, we assume the sameeffective mass of silicon for Poly-Si. The carrier concen-trations are N P-Si d = 10 cm − and N Si d = 5 × cm − for heavily doped Poly-Si and Si, respectively. Also, weconsider a Schottky barrier height V s = 0 .
65 eV, theFermi energy of Poly-Si and Si equal to E P-Si F ≈
80 meV[22], and E Si F ≈
10 meV, respectively. The Schottky bar-rier width for zero bias is w − = w + ≈ . nm and thethickness of the Poly-Si layer is fixed at d P-Si = 5 nm .The current density for spin-up (spin-down) electronscalculated by Eq. (2) is shown by the black (red or lightgray) solid curve in Fig. 2 for a fixed thickness of theFM-layer d = 8 . nm . The spin-up current density showsa maximum (2.4 × Am − ) for an applied bias of 0.23Volts. A similar phenomenon is observed for the spin-down current density, although two peaks are observedin this case. Such behavior is the signature of the RTDdevices, where the current flows only after the appliedbias tunes the resonant states. Fig. 2 also shows theeffects of partial relaxation on spin components of thecurrent density for different values of the “spin-currentdecay length” corresponding to three different FM, Co( ℓ Cosd =59 nm), CoFe ( ℓ CoFesd =12 nm), and Fe ( ℓ Fesd =5 nm)[20]. As expected, we see that decreasing the spin currentdecay length reduces and broadens the current densitypeaks. When the spin-decay length is smaller than thethickness of the FM-layer ( ℓ Fesd =5 nm) we observe a largesuppression in the current density peaks. The changes inthe spin-dependent density current as a function of theapplied bias can be observed by the spin-polarization,defined as p = ( j ↑ − j ↓ ) / ( j ↑ + j ↓ ). Fig. 3 shows the spin-polarization as a function of the bias for a fixed thicknessfor the FM-layer ( d = 8 . = = 59 nm = 12 nm= 5 nm Bias [V] s p i n po l a r i z a t i on FIG. 3: Spin-polarization as a function of applied bias. Thethickness for the FM-layer and the temperature are the sameadopted earlier ( d = 8 . nm and T=0 K). The results consid-ering different “spin-diffusion length” are also indicated. decay lengths. In this case of ℓ sd >> d , i.e., sufficient preservation of the spin current, the current polarizationoscillates and can reach full spin polarization. Thus, ahighly spin-polarized current with either spin directionmay be obtained by appropriate choice of the appliedbias on the spin-RTD.In conclusion, we show the possibility of injection ofhighly spin-polarized and strong current density into sil-icon by employing a spin-RTD. The focusing effect ofthe tunnel current by the semiconductor enables resonanttunneling dominated by a single spin subband of the FMquantum well at a given bias voltage. The possibility ofpolarization switching by electrical bias control may be ofimportance to spin devices. The spin decay through theferromagnet is avoided by short layer width. The lowpolarization generation due to the resistance mismatchbetween the metal electrode and the semiconductor ismitigated by the intervening heavily doped electrodes(Poly-Si). Finally, the fabrication of such spin-RTD iswithin the capabilities of current nanomagnet plus semi-conductor stack fabrication.This work is supported by U.S. Army Research OfficeMURI W911NF-08-2-0032 and by CNPq (Brazil). ∗ Electronic address: [email protected][1] S. A. Wolf et al. , Science , 1488 (2001).[2] G. Schmidt et al. , Phys. Rev. B , R4790 (2000).[3] E. I. Rashba, Phys. Rev. B , R16267 (2000); A. Fertand H. Jaffr`es Phys. Rev. B , 184420 (2001).[4] Y. Ohno et al. , Nature , 790 (1999).[5] B. T. Jonker et al. , Nature Phys. , 542 (2007).[6] I. Appelbaum, B. Huang, and D. J. Monsma, Nature , 295 (2007).[7] R. Tsu and L. Esaki, Appl. Phys. Lett. , 562 (1973).[8] D. A. Pearson and L. J. Sham, Phys. Rev. B , 125408(2001).[9] W. H. Butler et al. , Phys. Rev. B , 054416 (2001).[10] P. Mavropoulos, Phys. Rev. B , 054446 (2008).[11] Tunneling in solids , C. B. Duke (Academic Press, NewYork, 1969).[12] S.-H. Lo, D. A. Buchanan, and Y. Taur, IBM J. Res.Develop. , 327 (1999).[13] E. P. Gusev, V. Narayanan, and M. M. Frank, IBM J.Res. Develop. , 385 (2006).[14] S. W. Lee and S. K. Joo, IEEE Electron Device Lett. ,160 (1996).[15] A. A. Tulapurkar et al. , Nature , 339 (2005).[16] P. C. van Son, H. van Kempen, and P. Wyder, Phys.Rev. Lett. , 2271 (1987).[17] M. Johnson, Phys. Rev. Lett. , 2142 (1993).[18] T. Valet and A. Fert, Phys. Rev. B , 7099 (1993).[19] A. Vinattieri et al. , Phys. Rev. B , 10868 (1994).[20] J. Bass and W. P. Pratt, J. Phys.: Condens. Matter ,183201 (2007); A. Sharma et al. , Phys. Rev. B , 224438(2008).[21] J. C. Slonczewski, Phys. Rev. B , 6995 (1989).[22] D. L. Young et al. , J. Appl. Phys.105