Switching in a reversible spin logic gate
Superlattices and Microstructures, Vol. 22, No. 3, 1997
Switching in a reversible spin logic gate
S. Bandyopadhyay
†
Department of Electrical Engineering, University of Nebraska, Lincoln, Nebraska, 685880511, USA
V. P. Roychowdhury
School of Electrical Engineering, Purdue University, West Lafayette, Indiana 47907, USA(Received 15 July 1996)
Theoretical results for the adiabatic switching of a reversible quantum inverter–realizedwith two antiferromagnetically coupled single electrons in adjacent quantum dots—arepresented. It is found that a large exchange interaction between the electrons favors fasterswitchingbutalsomakesthetimingofthereadcyclemorecritical.Additionally,thereexistsan optimal input signal energy to achieve complete switching. Only for this optimal signalenergy does the inverter yield an unambiguous, logically deﬁnite state. An experimentalstrategy for realizing circuits based on such gates in selfassembled arrays of quantum dotsis brieﬂy discussed.c
1997 Academic Press Limited
Key words:
quantum dots, single electronics, quantum computing.
1. Introduction
Research in nanoelectronic classical Boolean logic circuits derived from single electron interactions inquantum dots has been a busy ﬁeld for the last few years [1–9]. A number of ideas have appeared in the
literature[1–9]thatvisualizebuildingdissipative(nonreversible)logiccircuitsbasedonCoulomborexchange
interaction between single electrons in arrays of quantum dots. Some of these schemes (e.g. [2]), however,are not only ﬂawed, but they also violate the basic tenets of circuit theory. The individual logic devices haveno isolation between input and output so that the input bit cannot even uniquely determine the output bit! (fora discussion of this issue see [3–6,10]).
In this paper, we explore a different type of gate. It is a quantum mechanical gate that is reversible andnondissipative. It should be contrasted with ‘parametrontype’ constructs that dissipate less than
kT
ln2energyperbitoperation[11],butareotherwisenotentirelynondissipative.Whilethebitsinaparametronarecnumbers, the bits in the quantum gate to be described are true qubits and the time evolution of the systemis unitary. For the sake of simplicity, we consider the smallest quantum gate possible, namely an inverter.It is fashioned from two antiferromagnetically coupled single electrons in two closely spaced quantum dotsas envisioned in [3–5]. The equilibrium steadystate behavior of such a system has been investigated by
Molotkov and Nazin [7,8]. Here, we will explore the dynamic behavior and the unitary time evolution of this
system in a nondissipative and globally phasecoherent environment.
† On leave from the University of Notre Dame.
0749–6036/97/070411 + 06 $25.00/0 sm970365 c
1997 Academic Press Limited
412
Superlattices and Microstructures, Vol. 22, No. 3, 1997
Input OutputA B
Fig. 1.
Two adjacent quantum dots hosting single electrons. In the ground state, the spins of the two electrons are antiparallel. If spinpolarization is used to encode binary bits, the logic state of one dot is always the inverse of the other. This realizes an inverter in whichone dot acts as the input terminal and the other as the output.
2. Theory
Consider two single electrons housed within two closelyspaced quantum dots as shown in Fig. 1. It wasshownin[3]thatthepreferredorderingofthissystemisantiferromagnetic,i.e.thetwoelectronshaveoppositespins. If the spin polarization in one dot is considered to be the input ‘qubit’ and that in the other the output‘qubit’, then this system acts as an inverter since the spinpolarizations are antiparallel (logic complement)[3,8]. Note that an inverter is always logically reversible since one can invariably predict the input bit from a
knowledgeoftheoutputbit(inpractice,theinputbitisrecoveredbymerelypassingtheoutputthroughanotherinverter).However,suchagateisnotauniversalquantumgateunliketheToffoligate[12].Variousschemesforrealizing nondissipative and reversible quantum logic gates have recently appeared in the literature [13–18].
Experimental demonstrations of quantum logic gates have also been reported [19,20]. Almost all of theseschemes encode the qubit in a photon (rather than an electron) state thereby requiring optical componentsthat are incompatible with ultralargescale integration. In contrast, the spin gate based on single electrons inquantum dots is very appealing from the perspective of highdensity circuits.ToanalysethesysteminFig.1quantummechanically,wewillassumethatthereisonlyonesizequantizedlevelineachquantumdot.Then,theHubbardHamiltonianforthissysteminthepresenceofagloballyappliedmagnetic ﬁeld can be written following Molotkov and Nazin [7] as
H
=
i
σ
(ǫ
0
n
i
σ
+
g
µ
B
H
i
sign
(σ))
+
ij
t
ij
(
c
+
i
σ
c
j
σ
+
h
.
c
.)
+
i
U
i
n
i
↑
n
i
↓
+
ij
αβ
J
ij
c
+
i
α
c
i
β
c
+
j
β
c
j
α
+
H
z
i
σ
g
µ
B
n
i
σ
sign
(σ)
(1)wheretheﬁrsttermdenotestheelectronenergyinthe
i
thdot(
H
i
isa
z
directedlocalmagneticﬁeldselectivelyapplied at the
i
th dot), the second term denotes the hopping between dots, the third term is the Coulombrepulsion within the
i
th quantum dot, the fourth term is the exchange interaction between nearestneighbourdots and the last term is the Zeeman splitting energy corresponding to the globally applied magnetic ﬁeldoriented along the
z
direction.We can simplify the Hamiltonian in Eqn (1) to the Heisenberg model following Molotkov and Nazin [8] toyield
H
=
J
ij
σ
zi
σ
zj
+
J
ij
(σ
xi
σ
xj
+
σ
yi
σ
yj
)
+
input dots
σ
zi
h
input
zi
(
J
>
0
)
(2)where we have neglected the global magnetic ﬁeld. The quantity
h
input
zi
is the Zeeman energy caused by alocal magnetic ﬁeld applied to the
i
th dot in the
z
direction which will orient the spin in the
i
th dot along thatﬁeld. Such a local ﬁeld can be applied via a spinpolarized scanning tunneling microscope (SPSTM) tip asvisualized in [3] and serves to provide an input signal to the gate.
Superlattices and Microstructures, Vol. 22, No. 3, 1997
413
Table 1:
Eigenenergies and eigenstates of the Hamiltonian for an inverter.Eigenenergies Eigenstates
h
A
+
J
↓↓−
J
+
h
2
A
+
4
J
2
12
1
+
h
A
h
2
A
+
4
J
2
↑↓+
12
1
−
h
A
h
2
A
+
4
J
2
↓↑−
J
−
h
2
A
+
4
J
2
12
1
−
h
A
h
2
A
+
4
J
2
↑↓−
12
1
+
h
A
h
2
A
+
4
J
2
↓→−
h
A
+
J
↑↑
In the basis of states

σ
A
σ
B
(
A
and
B
are the two electrons), the Hamiltonian in Eqn (2) can be written as
h
A
+
J
0 0 00
h
A
−
J
2
J
00 2
J
−
h
A
−
J
00 0 0
−
h
A
+
J
(3)where
h
A
is the interaction with the input magnetic ﬁeld selectively applied to quantum dot
A
. The twoelectron basis states can be denoted as
↓↓
,
↑↓
,
↓↑
and
↑↑
; they form a complete orthonormal set.The ‘upspin’ polarization is oriented along the direction of the locally applied external magnetic ﬁeld in thisrepresentation.The eigenenergies and corresponding eigenvectors of the above Hamiltonian are given in Table 1.It is obvious that the third row in Table 1 corresponds to the ground state. In the absence of any externalmagneticﬁeld
(
h
A
=
0
)
,thegroundstateenergyis
−
3
J
andthegroundstatewavefunctionis
1
√
2
(
↑↓−↓↑
)
.note that the ground state in the absence of any external magnetic ﬁeld is an entangled state in which neitherthe quantum dot
A
nor the quantum dot
B
has a deﬁnite spin polarization.
3. Adiabatic switching
We now wish to study the following switching problem. Assuming that the inverter is in its ground statewithout any applied magnetic ﬁeld, we will calculate how long it takes after a magnetic ﬁeld is applied toquantum dot
A
for the spin in
A
to orient along the ﬁeld and the spin in
B
to orient in the opposite direction(as required by the inversion operation).After the external ﬁeld is applied at time
t
=
0, the inverter evolves in time according to the unitaryoperation[
c
(
t
)
]
=
exp[
−
i
H
t
/
¯
h
][
c
(
0
)
] (4)where
H
isgivenbyEqn(3)and[
c
]isafourelementunitvector[
c
1
,
c
2
,
c
3
,
c
4
]thatdescribesthewavefunction
ψ(
t
)
according to
ψ(
t
)
=
c
1
(
t
)
↓↓+
c
2
(
t
)
↑↓+
c
3
(
t
)
↓↑+
c
4
(
t
)
↑↑
.
(5)The initial conditions are described by
c
1
(
0
)
c
2
(
0
)
c
3
(
0
)
c
4
(
0
)
=
0
1
√
2
−
1
√
2
0
.
(6)
414
Superlattices and Microstructures, Vol. 22, No. 3, 1997
The solution of Eqn (4) subject to the initial condition given by Eqn (6) is
c
1
(
t
)
=
c
4
(
t
)
=
0
c
2
(
t
)
=
e
i
Jt
/
¯
h
√
2
cos
(ω
t
)
−
i
h
A
¯
h
ω
+
1
−
h
2
A
¯
h
2
ω
2
sin
(ω
t
)
(7)
c
3
(
t
)
=−
e
i
Jt
/
¯
h
√
2
cos
(ω
t
)
−
i
h
A
¯
h
ω
−
1
−
h
2
A
¯
h
2
ω
2
sin
(ω
t
)
where
ω
=
h
2
A
+
4
J
2
/
¯
h
.Therefore, the wavefunction at an arbitrary time
t
is given by
c
2
(
t
)
↑↓+
c
3
(
t
)
↓↑
(8)with
c
2
and
c
3
given by Eqn (7).After the switching is complete, the system should be in the state
↑↓
. Therefore, the switching delay
t
d
can be deﬁned as the time taken for

c
2
(
t
)

to reach its maximum value and, correspondingly, for

c
3
(
t
)

toreach its minimum value.This yields
t
d
=
h
4
h
2
A
+
4
J
2
.
(9)It should be understood that the system
does not
reach a steady state at time
t
=
t
d
, but instead continues toevolveinaccordancewithEqn(4).Thecomputation(inversion)canbehaltedbyreadingthespinpolarization(logic bit) in the output dot (dot
B
) with a SPSTM tip at time
t
=
t
d
since the reading operation is dissipativeand collapses the wavefunction. Note that the higher the frequency
ω
, the more critical is the timing for theread cycle that halts the quantum computation. Since
ω
increases with the exchange energy
J
, a larger
J
willmandate a greater accuracy in the read cycle.To achieve complete switching, the magnitude

c
2
(
t
d
)

should be unity and

c
3
(
t
d
)

should vanish. FromEqns (7) and (9), we obtain

c
2
(
t
d
)
=
h
A
+
2
J
2
h
2
A
+
8
J
2
.
(10)Themagnitude

c
2
(
t
d
)

2
asafunctionofthenormalizedinputsignalenergy
h
A
/
J
isshowninFig.2.Itreachesa maximum value of unity (corresponding to complete switching) when
h
A
=
2
J
. Therefore, there exists anoptimal value of the input signal energy
h
A
for which complete switching can be obtained.It should be noted from Eqn (9) that the switching delay decreases with increasing exchange energy
J
. Forthe optimal case
(
h
A
=
2
J
)
, the switching delay is
h
/(
8
√
2
J
)
. We can estimate the order of magnitude for
t
d
.Presumably, the maximum value of local magnetic ﬁeld that can be applied to a dot with a SPSTM tip is about1 T. Since
h
A
≈
g
µ
B
B
(
µ
B
is the Bohr magnetron), this means that the maximum value of
h
A
that we canhope to obtain is about 0.1 meV if we assume the Land´e
g
factor to be 2. Consequently,
J
optimal
=
0
.
05 meV.This gives a value of
t
d
≈
7 ps. Therefore, these inverters are capable of quite fast switching.We can also estimate the temperature of operation for such inverters. Since the exchange energy shouldexceed the thermal energy
kT
for stable operation, the ambient temperature should be restricted to below
T
=
J
/
k
=
570 mK. Because the operation of the inverter requires global phase coherence (i.e. the phasebreakingtimeshouldbesigniﬁcantlylongerthan
t
d
),alowtemperatureisalsootherwiserequired.Toincreasethe temperature to a more practical value of 4.2 K,
J
optimal
should be 0.364 meV and therefore
h
A
should beas large as 0.728 meV. This requires the ability to generate a local magnetic ﬂux density in excess of 7 T withan SPSTM tip as an input to cell A. This is not possible with present state of SPSTM technology, but couldbecome feasible in the future.
Superlattices and Microstructures, Vol. 22, No. 3, 1997
415
10.80.60.40.200 5
h
A
/
J
 c
2
(
t
d
) 
2
10
Fig. 2.
The magnitude of

c
2
(
t
d
)

2
as a function of the normalized unput signal energy
h
A
/
J
.100nm50010000 500 1000
Fig. 3.
Atomic force micrograph of a selfassembled mask to create a periodic array of quantum dots. Details can be found in [4,6].
We conclude this paper with a brief discussion of experimental strategies undertaken by us in our efforts tofabricate such gates. We believe that the optimal technique is ‘gentle’ selfassembly of quantum dots ratherthannanolithographywhichcausesprocessingdamageandhasaslowthroughput.Wefabricatearegulararrayof the dots using a selfassembled mask for mesaetching. The selfassembled mask is created by evaporatingaluminum on the chosen semiconductor structure and then electropolishing it in a solution of perchloric acid,butyl cellusolve and ethanol at 60 V for 30 s at room temperature. Figure 3 shows the raw atomic forcemicrograph of a selfassembled mask of aluminum with a dimpled surface that consists of a periodic array of crests and troughs with hexagonal packing. The troughs are etched away by an appropriate etchant leaving aregular pattern of isolated crests on the surface of the semiconductor structure that serve as a mask throughwhichmesasareetched.Owingtospacelimitations,wewillomitdetailsofthefabricationprocess,butinsteadrefer the reader to [4,6].
Acknowledgements
—The work on selfassembly of masks for fabricating quantum dots was performed incollaborationwithProf.A.E.Miller,Prof.H.C.Chang,Dr.G.BanerjeeandMr.V.YuzhakovoftheDepartment