A spin field effect transistor for low leakage current

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A spin field effect transistor for low leakage current
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    a  r   X   i  v  :  c  o  n   d  -  m  a   t   /   0   4   0   7   1   7   7  v   2   [  c  o  n   d  -  m  a   t .  m  e  s  -   h  a   l   l   ]   9   S  e  p   2   0   0   4 A spin field effect transistor for low leakagecurrent S. Bandyopadhyay a and M. Cahay b a Department of Electrical and Computer Engineering and Department of Physics,Virginia Commonwealth University, Richmond, VA 23284, USA b Department of Electrical and Computer Engineering and Computer Science,University of Cincinnati, OH 45221, USA Abstract In a spin field effect transistor, a magnetic field is inevitably present in the channelbecause of the ferromagnetic source and drain contacts. This field causes randomunwanted spin precession when carriers interact with non-magnetic impurities. Therandomized spins lead to a large leakage current when the transistor is in the “off”-state, resulting in significant standby power dissipation. We can counter this effectof the magnetic field by engineering the Dresselhaus spin-orbit interaction in thechannel with a backgate. For realistic device parameters, a nearly perfect cancella-tion is possible, which should result in a low leakage current. Key words:  Spintronics, Spin field effect transistors, spin orbit interaction PACS:  85.75.Hh, 72.25.Dc, 71.70.Ej Preprint submitted to Elsevier Science 2 February 2008   Much of the current interest in spintronic transistors is motivated by a well-known device proposal due to Datta and Das [1] that has now come to beknown as a Spin Field Effect Transistor (SPINFET). This device consistsof a one-dimensional semiconductor channel with half-metallic ferromagneticsource and drain contacts that are magnetized along the channel (Fig. 1).Electrons are injected from the source with their spins polarized along thechannel’s axis. The spin is then controllably precessed in the channel with agate voltage that modulates the Rashba spin-orbit interaction [2]. At the drainend, the transmission probability of the electron depends on the component of its spin vector along the channel. By controlling the angle of spin precessionin the channel with a gate voltage, one can control this component, and hencecontrol the source-to-drain current. This realizes the basic “transistor” action[3].In their srcinal proposal [1], Datta and Das ignored two effects: (i) the mag-netic field that is inevitably present in the channel because of the ferromagneticsource and drain contacts, and (ii) the Dresselhaus spin orbit interaction [4]arising from bulk (crystallographic) inversion asymmetry. In the past, we an-alyzed the effect of the channel magnetic field and showed that it could causeweak spin flip scattering via interaction with non-magnetic elastic scatterers[5]. The flipped spins, whose precession angles have been randomized by thespin flip scattering events, will lead to a large leakage current when the deviceis in the “off”-state. This is a serious drawback since it will lead to an unac-ceptable standby power dissipation in a circuit composed of Spin Field EffectTransistors. In order to eliminate the leakage current, we must eliminate theunwanted spin flip scattering processes. In other words, we must find ways tocounter the deleterious effect of the channel magnetic field. The purpose of 2  Source DrainTop Gateconducting substratei-InAsn-AlAs2-DEGSplit gate(negative voltage)Top gate(positive voltage)Source Drain(a)(b)xyzBack gatexzd Fig. 1. A spin field effect device with a one-dimensional channel. (a) side viewshowing a top gate for modulating the spin precession via the Rashba interactionand a back gate for modulating the channel carrier concentration. The substratewill be p + if we want to deplete the channel with the back-gate, and n + if wewant to accumulate it. (b) top view showing the split gate configuration requiredto produce a one-dimensional channel, as well as the top gate. A positive voltageis applied to the top gate to increase the interface electric field that modulates theRashba interaction and produces the conductance modulation, whereas a negativevoltage is applied to the split gates to constrict the one-dimensional channel. this paper is to explore how this can be achieved.In a strictly one-dimensional structure, where transport in single channeled,there is no D’yakonov-Perel spin relaxation [6]. Therefore, the only agentsthat can cause spin randomization are hyperfine interactions with the nuclei3  and spin mixing effects caused by the channel magnetic field [5]. In order toeliminate the latter (which is the stronger of the two agents), we can adoptone of two options: either eliminate the magnetic field by using non-magneticspin-injector (source contact) and detector (drain contact) [7], or counteractthe effect of the magnetic field with some other effect. The former approachpresents a formidable engineering challenge. The latter can be implementedmore easily, and, as we show in this paper, is achieved by countering the effectof the magnetic field with the Dresselhaus interaction. Calculations basedon realistic parameters for InAs transistor channels show that this is indeedpossible.Consider the one-dimensional channel of the device in Fig. 1. Because of themagnetized source and drain contacts, a magnetic field exists along the wirein the x-direction. We will assume that the channel (x-direction) is along the[100] crystallographic axis.The effective mass Hamiltonian for the wire, in the Landau gauge  A  = (0, − Bz  , 0), can be written as H   =(  p 2 x  +  p 2 y  +  p 2 z ) / (2 m ∗ ) + ( eBzp y ) /m ∗ + ( e 2 B 2 z  2 ) / (2 m ∗ ) − ( g/ 2) µ B Bσ x + V   ( y ) +  V   ( z  ) + 2 a 42 [ σ x κ x  +  σ y κ y  +  σ z κ z ] +  η [(  p x /   ) σ z − (  p z /   ) σ x ](1)where  g  is the Land`e g-factor,  µ B  is the Bohr magneton,  V   ( y ) and  V   ( z  ) arethe confining potentials along the y- and z-directions,  σ -s are the Pauli spinmatrices, 2 a 42  is the strength of the Dresselhaus spin-orbit interaction ( a 42  is amaterial parameter) and  η  is the strength of the Rashba spin-orbit interaction.The quantities  κ  are defined in ref. [8]. We will assume that the wire is narrowenough and the temperature is low enough that only the lowest magneto-4  electric subband is occupied. Since the Hamiltonian is invariant in the  x -coordinate, the wavevector  k x  is a good quantum number and the eigenstatesare plane waves traveling in the x-direction. Accordingly, the spin Hamiltonian(spatial operators are replaced by their expected values) simplifies to H   = (   2 k 2 x ) / (2 m ∗ ) +  E  0  + ( αk x − β  ) σ x  +  ηk x σ z  (2)where  E  0  is the energy of the lowest magneto-electric subband,  α ( B ) = 2 a 42 [ <k 2 y  >  −  < k 2 z  >  +( e 2 B 2 < z  2 > /   2 )],  ψ ( z  ) is the z-component of the wave-function,  φ ( y ) is the y-component of the wavefunction,  < k 2 y  >  = (1 /   2 )  <φ ( y ) |− ( ∂  2 /∂y 2 ) | φ ( y )  > ,  < k 2 z  >  = (1 /   2 )  < ψ ( z  ) |− ( ∂  2 /∂z  2 ) | ψ ( z  )  > , and β   = ( g/ 2) µ B B .Since the potential  V   ( z  ) is parabolic ( V   ( z  ) = (1 / 2) m ∗ ω 20 z  2 ), it is easy to showthat  < k 2 z  > =  m ∗ ω/ (2   ) and  < z  2 > =    / (2 m ∗ ω ) where  ω 2 =  ω 20  +  ω 2 c  and  ω c is the cyclotron frequency ( ω c  =  eB/m ∗ ). Furthermore,  E  0  = (1 / 2)   ω  +  E  ∆ where  E  ∆  is the energy of the lowest subband in the triangular well  V   ( y ).Diagonalizing this Hamiltonian in a truncated Hilbert space spanning the twospin resolved states in the lowest subband yields the eigenenergies [9] E  ±  =    2 k 2 x 2 m ∗  +  E  0 ±   ( η 2 +  α 2 )  k x −  αβ η 2 +  α 2  2 +  η 2 η 2 +  α 2 β  2 (3)and the corresponding eigenstatesΨ + ( B,x ) =  cos ( θ k x ) sin ( θ k x )  e ik x x Ψ − ( B,x ) =  sin ( θ k x ) − cos ( θ k x )  e ik x x (4)where  θ k x = (1 / 2) arctan [( αk x − β  ) /ηk x ].5
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