APPLIED PHYSICS LETTERS
VOLUME 82, NUMBER 16
21 APRIL 2003
Datta–Das transistor with enhanced spin control
J. Carlos Egues,a) Guido Burkard, and Daniel Loss
Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82,
CH-4056 Basel, Switzerland
共Received 1 October 2002; accepted 5 February 2003兲
We consider a two-channel spin transistor with weak spin-orbit induced interband coupling. We
show that the coherent transfer of carriers between the coupled channels gives rise to an additional
spin rotation. We calculate the corresponding spin-resolved current in a Datta–Das geometry and
show that a weak interband mixing leads to enhanced spin control. © 2003 American Institute of
Physics. 关DOI: 10.1063/1.1564867兴
The pioneering spin-transistor proposal of Datta and
Das1 best exemplifies the relevance of electrical control of
magnetic degrees of freedom as a means of spin modulating
charge flow. In this device,2 a spin-polarized current3,4 injected from the source is spin modulated on its way to the
drain via the Rashba spin-orbit5 共s-o兲 interaction, Fig. 1共a兲.
The spin transistor operation relies on gate controlling6 the
strength ␣ of the Rashba interaction which has the form
H R ⫽i ␣ ␴ y ⳵ / ⳵ x in a strictly one-dimensional channel.5 Upon
crossing the Rashba-active region of length L, a spin-up incoming electron emerges in the spin-rotated state
冉冊 冋
册
1
cos共 ␪ R /2兲
→
,
⫺sin共 ␪ R /2兲
0
共1兲
where ␪ R ⫽2m * ␣ L/ប ⬅2k R L is the rotation angle and m*
is the electron effective mass.1 The corresponding spinresolved conductance is found to be G ↑,↓ ⫽e 2 (1⫾cos ␪R)/h.
Here we extend the above picture by considering a geometry with two weakly coupled Rashba bands in the
quasione-dimensional channel, Fig. 1共b兲. We treat the degenerate k states near the band crossings perturbatively in analogy to the nearly free electron model.7 This approach allows
for a simple analytical description of the problem. We calculate the spin-resolved current by extending the usual procedure of Datta and Das1 to account for weakly coupled bands.
Our main finding is an additional spin rotation for injected
electrons with energies near the band crossing 共see shaded
region around ␧ F in Fig. 2兲. As we derive later on, an incoming spin up electron in channel a emerges from the Rashba
region in the rotated state
G ↑,↓ ⫽
冉冊 冋
→
1
2
册
cos共 ␪ d /2兲 e ⫺ik R L ⫹e ik R L
⫺i cos共 ␪ d /2兲 e ⫺ik R L ⫹ie ik R L
,
⫺i sin共 ␪ d /2兲 e ⫺ik R L
sin共 ␪ d /2兲 e ⫺ik R L
共3兲
H R⫽
冋
a
␧⫹
共k兲
0
0
⫺␣d
0
a
␧⫺
共k兲
␣d
0
0
␣d
b
␧⫹
共k兲
0
⫺␣d
0
0
b
␧⫺
共k兲
册
,
共4兲
where d⬅ 具 ␾ a (y) 兩 ⳵ / ⳵ y 兩 ␾ b (y) 典 , ␧ sn (k)⫽ប 2 (k⫺sk R ) 2 /2m *
⫹ ⑀ n ⫺ ⑀ R , ⑀ R ⬅ប 2 k R2 /2m * , (s⫽⫾,n⫽a,b) and we have
considered 兩␴典 to be the eigenbasis of ␴ y . For d⫽0 the
Hamiltonian in Eq. 共4兲 is diagonal and yields uncoupled
Rashba dispersions ␧ sn (k) 共thin lines in Fig. 2兲; the corresponding wave functions are ␸ k,n,s (x,y) 共here 兩 ␴ 典 → 兩 s
⫽⫾ 典 ⫽ 关 兩 ↑ 典 ⫾i 兩 ↓ 典 ]/&). Note that for d⫽0 the bands cross
for some values of k. For instance, for k⬎0 a crossing occurs
at k c ⫽( ⑀ b ⫺ ⑀ a )/2␣ . For nonzero interband coupling d⫽0,8
we can diagonalize H R exactly 共see Mireles and Kirczenow
in Ref. 8兲 to find the dispersions 共thick lines in Fig. 2兲.
共2兲
where ␪ d ⫽ ␪ R d/k c is the additional spin rotation angle, d is
the interband matrix element, and k c is the wave vector at the
band crossing, Fig. 2. From Eq. 共2兲 we can find the spinresolved conductance
a兲
册
We now proceed to derive Eqs. 共2兲 and 共3兲.
Model. We consider a quasione-dimensional wire of
length L with two bands a and b described by ␧ n, ␴ z (k)
⫽ប 2 k 2 /2m * ⫹ ⑀ n , n⫽a,b and eigenfunctions ␸ k,n, ␴ (x,y)
⫽e ikx ␾ n (y) 兩 ␴ 典 / 冑L, ␴ ⫽↑,↓ where the ␾ n (y)’s denote the
transverse confinement wave functions. In the presence of
the Rashba s-o interaction, we can derive a Hamiltonian for
the system in the basis of the uncoupled wave functions
关 ␸ k,n, ␴ z (x,y) 兴 . This reads
2
1
0
0
0
冋
e 2 1⫹cos共 ␪ d /2兲 cos ␪ R
.
h 1⫺cos共 ␪ d /2兲 cos ␪ R
Also at: Department of Physics and Informatics, University of São Paulo at
São Carlos, 13560-970 São Carlos/SP, Brazil; electronic mail:
egues@if.sc.usp.br
FIG. 1. Spin transistor geometry with a two-band channel. 共a兲 The length L
of the Rashba region is smaller than the total length L 0 of the wire. 共b兲
Sketch of energy dispersions in the s-o active region with and without interband coupling 共Rashba bands兲 and away from it 共parabolic bands兲. Note
the small band offsets between adjacent regions in the wire.
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© 2003 American Institute of Physics
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Appl. Phys. Lett., Vol. 82, No. 16, 21 April 2003
冉冊 冋
kF
0
0
0
FIG. 2. Band structure in the presence of spin-orbit coupling. In absence of
interband mixing the Rashba dispersions are uncoupled 共thin solid lines兲 and
cross at, e.g., k c . For nonzero interband coupling the bands anti cross 共thick
solid lines兲. The inset shows a blowup of the dispersion region near the
crossing: the approximate solution 关dotted lines, perturbative approach, Eq.
共6兲兴 describes well the energy dispersions near k c .
Bands near k c . Since we are interested in transport with
injection energies near the crossing, we follow here a simpler
perturbative approach7 to determine the energy dispersions
and wave functions near k c . Near the crossing we can solve
the reduced Hamiltonian
H̃ R ⫽
冋
a
␧⫺
共k兲
␣d
␣d
b
␧⫹
共k兲
册
共5兲
,
1
⫽
2
ប 2k 2 1
1
⫹ ⑀ ⫹ ⑀ ⫾ ␣ d.
2m * 2 b 2 a
⌿ ↑,L ⫽
共6兲
1
4
兩 ␺ ⫾典 ⫽
&
关 兩 ⫺ 典 a ⫾ 兩 ⫹ 典 b ]⫽
1
&
冋冉 冊 冉 冊 册
1
⫺i
1
⫾
i
a
,
⌬⫽
2m * ␣ d
kR
⫽2 d.
2
ប kc
kc
共7兲
共8兲
兩 ⌿ 典 ⫽ 兩 ␺ ⫹典 e
ik c1 x
⫹ 兩 ␺ ⫺典 e
1
2
1
i
0
0
⫹
e iL⌬/2
⫺ie iL⌬/2
⫺e iL⌬/2
⫺ie iL⌬/2
ik c2 x
⫹ 兩⫹ 典 ae
1
&
ik 2 x
.
共9兲
The above ansatz satisfies the boundary conditions for both
the wave function and 共to leading order兲 its derivative at x
⫽0. More explicitly, the velocity operator condition9 at x
⫽0 for an electron with k⫽k F yields
,
共10兲
e ik c L
e ik 2 L
cos共 ␪ d /2兲 e ⫺ik R L ⫹e ik R L
⫺i cos共 ␪ d /2兲 e ⫺ik R L ⫹ie ik R L
,
⫺i sin共 ␪ d /2兲 e ⫺ik R L
⫺ik R L
sin共 ␪ d /2兲 e
共11兲
which is essentially Eq. 共2兲. Observe that in absence of interband coupling 共i.e., ␪ d ⫽0) Eq. 共11兲 reduces to the Datta–
Das state in Eq. 共1兲. An expression similar to Eq. 共11兲 holds
for the case of an incoming spin-down electron.
Spin-resolved current. For x⭓L we have
⌿ ↑ 共 x⭓L,y 兲
Note that to the lowest order used here the horizontal splitting ⌬ is constant and symmetric about k c .
Boundary conditions. We now consider a spin-up electron entering the Rashba-active region of length L in the
wire. Following the usual approach, we expand this incoming state in terms of the coupled Rashba states in the wire.
We consider only the states k c1 , k c2 , and k 2 in the expansion
1
2
e ⫺iL⌬/2
⫺ie ⫺iL⌬/2
e ⫺iL⌬/2
ie ⫺iL⌬/2
1
⫽ e i 共 k c ⫹k R 兲 L
2
b
where the subindices indicate the respective channel. The
analytical form in Eq. 共6兲 allows us to determine the wave
vectors k c1 and k c2 in Fig. 2 straightforwardly: we assume
approx
(k c1 )
k c1 ⫽k c ⫺⌬/2 and k c2 ⫽k c ⫹⌬/2 and solve ␧ ⫹
approx
⫽␧ ⫺ (k c2 ) 共assumed ⬃␧ F ) to find
k c ⫹k 2
0
⫺⌬/2
⫺i⌬/2
冋冉 冊 冉 冊册
冉冊
冋
册
1
⫹
2
As shown in the inset of Fig. 2, Eq. 共6兲 describes very well
the anticrossing of the bands near k c . The corresponding
zero-order eigenstates are
1
册冉 冊
k c ⫹k 2
1
⫺i 共 k c ⫺k 2 ⫺2k R 兲
⫽
⫺⌬/2
2
⫺i⌬/2
2659
where we used k 2 ⫺k c ⫽2k R 共still valid to leading order兲.
The ‘‘four-vector’’ notation in Eq. 共10兲 concisely specifies
the spin states in channels a 共upper half兲 and b 共lower half兲.
Note that Eq. 共10兲 is satisfied provided that ⌬Ⰶ4k F . This
inequality is satisfied in our system for realistic parameters.
Underlying the ansatz in Eq. 共9兲 is the assumption of
unity transmission through the Rashba region. Here we have
in mind the particular spin-transistor geometry sketched in
Fig. 1共a兲: a gate-controlled Rashba-active region of extension L smaller than the total length L 0 of the wire. In this
configuration, there are only small band offsets 共which we
neglect兲 of the order of ⑀ R Ⰶ␧ F at the entrance (x⫽0) and
exit (x⫽L) of the Rashba region. Hence, transmission is
indeed very close to unity, see Ref. 10. The boundary conditions at x⫽L are also satisfied.
Generalized spin-rotated state. From Eq. 共9兲 we find that
a spin-up electron entering the Rashba region at x⫽0
emerges from it at x⫽L in the spin-rotated state
which to lowest order yields
approx
␧⫾
共 k 兲⫽
Egues, Burkard, and Loss
⫽
冋
册
1 e ⫺i ␪ R /2 cos共 ␪ d /2兲 ⫹e i ␪ R /2
e i 共 k c ⫹k R 兲 x ␾ a 共 y 兲
2 ⫺ie ⫺i ␪ R /2 cos共 ␪ d /2兲 ⫹ie i ␪ R /2
⫹
冋
册
1 ⫺ie i ␪ R /2 sin共 ␪ d /2兲 i 共 k ⫺k 兲 x
e c R ␾ b共 y 兲 ,
2 e i ␪ R /2 sin共 ␪ d /2兲
共12兲
which describes planes waves in the uncoupled channels a
and b arising for an incoming spin-up electron in channel a.
The total current follows straightforwardly 共Landauer–
Büttiker兲 from Eq. 共12兲:
e
I ↑,↓ ⫽ eV 关 1⫾cos共 ␪ d /2兲 cos ␪ R 兴 .
h
共13兲
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2660
Appl. Phys. Lett., Vol. 82, No. 16, 21 April 2003
Egues, Burkard, and Loss
⬃0.05 关validity of Eq. 共10兲兴 for the previous parameters.
Finally, we note that the most relevant spin-flip mechanism
共Dyakonov–Perel兲 should be suppressed in quasionedimensional systems such as ours.11 In addition, thermal effects are irrelevant in the experimentally feasible linear
regime12 we consider here.
FIG. 3. Angular dependence of the spin-down conductance. The additional
modulation ␪ d due to s-o interband mixing and ␪ R can be varied independently.
where eVⰆ␧ F the applied bias between the source and drain.
The spin-dependent conductance in Eq. 共3兲 follows immediately from Eq. 共13兲. Equation 共13兲 clearly shows the additional modulation ␪ d of the spin-resolved current due to s-o
induced interband coupling. Figure 3 illustrates the angular
dependence of G ↓ as a function of ␪ R and ␪ d . The s-o mixing angle ␪ d enhances the possibilities for spin control in the
Datta–Das transistor.
Realistic parameters. For concreteness, let us consider
infinite transverse confinement 共width w兲. In this case, ⑀ b
⫺ ⑀ a ⫽3ប 2 ␲ 2 /2mw 2 and the interband coupling constant d
⫽8/3w. We choose ⑀ b ⫺ ⑀ a ⫽16⑀ R , which implies 共i兲 ␣
⫽() ␲ /4)ប 2 /mw⫽3.45⫻10⫺11 eV m 共and ⑀ R ⬃0.39 meV)
for m⫽0.05m 0 and w⫽60 nm, 共ii兲 ␧(k c )⫽24⑀ R 关 ␧ F should
be tuned to ⬃␧(k c )], and 共iii兲 k c ⫽8 ⑀ R / ␣ . Assuming L
⫽69 nm 关Rashba region length, Fig. 1共a兲兴, we find ␪ R ⫽ ␲
and ␪ d ⫽ ␪ R d/k c ⫽ ␲ /2, since d/k c ⬃0.5. This is a conservative estimate. In principle, ␪ d can be varied independently of
␪ R via lateral gates which alter w. Note also that ⌬/4k F
This work was supported by NCCR Nanoscience, the
Swiss NSF, DARPA, and ARO. The authors acknowledge
useful discussions with D. Saraga.
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