A spin field effect transistor for low leakage current
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 nonmagnetic 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 spinorbit interaction in thechannel with a backgate. For realistic device parameters, a nearly perfect cancellation 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 wellknown 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 onedimensional semiconductor channel with halfmetallic 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 spinorbit 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 sourcetodrain current. This realizes the basic “transistor” action[3].In their srcinal proposal [1], Datta and Das ignored two eﬀects: (i) the magnetic 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 analyzed the eﬀect of the channel magnetic ﬁeld and showed that it could causeweak spin ﬂip scattering via interaction with nonmagnetic 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 unacceptable 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 substrateiInAsnAlAs2DEGSplit gate(negative voltage)Top gate(positive voltage)Source Drain(a)(b)xyzBack gatexzd
Fig. 1. A spin ﬁeld eﬀect device with a onedimensional 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 backgate, and n
+
if wewant to accumulate it. (b) top view showing the split gate conﬁguration requiredto produce a onedimensional 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 onedimensional channel.
this paper is to explore how this can be achieved.In a strictly onedimensional structure, where transport in single channeled,there is no D’yakonovPerel 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 nonmagneticspininjector (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 onedimensional channel of the device in Fig. 1. Because of themagnetized source and drain contacts, a magnetic ﬁeld exists along the wirein the xdirection. We will assume that the channel (xdirection) 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 gfactor,
µ
B
is the Bohr magneton,
V
(
y
) and
V
(
z
) arethe conﬁning potentials along the y and zdirections,
σ
s are the Pauli spinmatrices, 2
a
42
is the strength of the Dresselhaus spinorbit interaction (
a
42
is amaterial parameter) and
η
is the strength of the Rashba spinorbit 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 magneto4
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 xdirection. 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 magnetoelectric subband,
α
(
B
) = 2
a
42
[
<k
2
y
>
−
< k
2
z
>
+(
e
2
B
2
< z
2
> /
2
)],
ψ
(
z
) is the zcomponent of the wavefunction,
φ
(
y
) is the ycomponent 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