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A. N. M. Zainuddin, 1 S. Hong, 1 L. Siddiqui, 1 and S. Datta 1 

# Voltage controlled spin precession 

$^{1}$School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA ∗ 

A recent experiment reports a non-local spin-signal that shows oscillatory behavior as a function of gate voltage when the contacts are magnetized along the direction of current flow, but not when they are magnetized perpendicular to the current, in agreement with the predictions from a simple theory. In this paper we first present a straightforward extension of this theory to include the angular spectrum of electrons and the extended injecting and detecting contacts. The results are in good agreement with those from a non-equilibrium Green function (NEGF)- based model with contact parameters adjusted to fit the experimental contact conductances. They also describe certain aspects of the experiment fairly well, but other aspects deserve further investigation. 

Voltage controlled spin precession (see [1] and refer- ences therein), proposed in 1990 [2], posed two diffi- cult challenges namely (1) spin-polarized injection into a semiconducting channel, and (2) gate control of the Rashba spin-orbit interaction (RSO) in the channel [3]. The latter was demonstrated by Nitta et al. in 1997 us- ing an inverted InGaAs/InAlAs quantum well with a top gate [4]. But spin-polarized injection into a semiconduc- tor proved to be a more difficult challenge which has only recently been overcome through the combined efforts of many groups around the world [5]. Very recently, Koo et al. [1] have combined both ingredients, spin-polarized in- jection and gate-controlled RSO, into a single experimen- tal structure using a high mobility InAs heterostructure with a top gate interposed between the current contacts and the voltage contacts (Fig. 1(a)). The non-local volt- age signal [6] showed an oscillatory behavior when the contacts are magnetized along the direction of current flow, but not when they are magnetized perpendicular to the current flow (Fig. 1(b)), as expected from the theory presented in [2]. Furthermore, it was shown in [1] that the oscillation (see Fig. 1(b)) is described well by the expression 

where m ∗ is the effective mass, α ( V$_{G}$ ) is the RSO mea- sured independently from the Shubnikov-de Haas (SDH) beating pattern, and A and φ are fitting parameters. The oscillation period 2 m $^{∗}$α ( V$_{G}$ ) L/ ℏ 2 was derived by Datta and Das [2] for electrons with wavevectors that are purely along the direction of current flow (x-) with k$_{Y}$ = 0, not- ing that ‘in practice we have an angular spectrum of elec- trons’ and the ‘effect is reduced as ⃗ k turns away from the x-axis.’ 

PACS numbers: 

## I. INTRODUCTION 

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In this paper we will first (Section II) describe a straightforward extension of the theory in [2] to in- clude the sum over the angular spectrum k$_{Y}$ of electrons. The results closely follow those obtained earlier in ref. [7] which are more general since they include both electron and hole systems. We show that the results from this sim- ple model are in good agreement with those in (Fig.1(d)) from a non-equilibrium Green function (NEGF) based model with contact parameters adjusted to fit the ex- perimental contact conductances (Section III). We hope that a careful comparison of experiments will help re- fine model proposed here and establish this effect on a firm footing, so that it can be used both for fundamen- tal studies as well as for various proposed applications such as spin-filtering, magnetic recording and sensing or quantum computing [8]. 

We start from an effective mass Hamiltonian for a two- dimensional conductor with RSO interaction of the form ( ⃗ σ : Pauli spin matrices) 

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We neglect Dresselhaus spin-orbit (DSO) coupling since this is believed to be small in structures of this type [9]. Eq. 2 leads to the dispersion relation 

with the upper and lower signs corresponding to eigen- spinors of the form { 1 ± exp( iφ ) } $^{T}$, where tan φ ≡ − k$_{X}$/k$_{Y}$ . Here, x and y are the longitudinal (or trans- port) and transverse direction respectively following the co-ordinate system used in [1], which is different from that used in [2]. Assuming periodic boundary conditions in the transverse direction, both E and k$_{Y}$ are conserved in the absence of scattering and the two eigenmodes have different k$_{X}$ ’s so as to satisfy Eq. 3 with the upper and 

## II. SIMPLE MODEL 

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