Analysis and Geometric Optimization of Single Electron Transistors for Read-Out in Solid-State Quantum Computing

0.46 MB·English

by  Vincent I. Conrad

Analysis and Geometric Optimization of Single Electron Transistors for Read-Out in
Solid-State Quantum Computing.
Vincent I. Conrad∗, Andrew D. Greentree, David N. Jamieson, Lloyd C.L. Hollenberg
Centre for Quantum Computer Technology, School of Physics, University of Melbourne, Vic. 3010, Australia
(Dated: February 2, 2008)
5
0 The single electron transistor (SET) offers unparalled opportunities as a nano-scale elec-
0 trometer,capableofmeasuringsub-electronchargevariations. SETshavebeenproposedfor
2
read-out schema in solid-state quantum computing where quantum information processing

n outcomesdepend onthe locationofa singleelectrononnearbyquantumdots. Inthis paper
a we investigate various geometries of a SET in order to maximize the device’s sensitivity to
J
chargetransferbetweenquantumdots. Throughtheuseoffiniteelementmodelingwemodel

9 thematerialsandgeometriesofanAl/Al O SETmeasuringthestateofquantumdotsinthe
2 3
1
Si substrate beneath. The investigation is motivated by the quest to build a scalable quan-

tum computer, though the methodology used is primarily that of circuit theory. As such
]
l we provide useful techniques for any electronic device operating at the classical/quantum
l
a interface.
h
Keywords: single electron transistor, quantum computing, capacitance
-
s
e
m I. INTRODUCTION searching. This has spurred further interest in the de-
. velopment of architectures for quantum computers, and
t
a The single electron transistor (SET)1,2 has presented other useful quantum algorithms. Since that time the
m race to build a workingquantum computer has takenoff
itself as a very interesting device in the field of classical
across the globe.
- computing. SystemsusingSETsaslogicdeviceshaveal-
d
ready been proposed3. SETs also present the possibility The essential difference between classical computers
n
o of utilizing the spin orientation of electrons in the cur- andquantumcomputersisthereplacementofthedualis-
c rents passing through them. This leads to their use in ticon/offbit(0or1),withthequantum-bit(qubit)which
[ the field of spintronics4,5, opening a plethora of possibil- is capable of being in both the on and off state simulta-
itiesinwhichthisextradegreeoffreedomcanbeusedto neously (α0 +β 1 with α2 + β 2 = 1 and α,β C).
1 | i | i | | | | ∈
v encode information. A SET can also be used as a highly Qubits may also make use of the purely quantum me-
7 sensitiveelectrometercapableofmeasuringchargevaria- chanicalpropertyofentanglement,whichisthesourceof
3 tionsassmallas10−6q /√Hz whichiscloseto thequan- theuniquecomputationalpowerofquantuminformation
e
4 tum limit2. Our research is motivated by the use of the processing10. The operation of any quantum computer
1 SET to perform solid-state quantum computer read-out, requiresthatqubitscanbeindividually‘rotated’(i.e. the
0 though the techniques employed here are applicable to valuesofαandβ canbemanipulatedforeachqubit)and
5
any application using a SET as an electrometer. that two qubit interactions can be controlled (to allow
0
formulti-qubit logicaloperationssuchas CNOT).While
/ The field of quantum computing began in the 1980s
t these are the essential qubit requirements for universal
a when researchers began to theoretically investigate how
m to apply quantum mechanics to information processing. information processing, an actual physical implementa-
tion requires further considerations11.
- The first application envisioned for such quantum infor-
d mation processing was as a physics simulator, presented One of the biggest difficulties in quantum information
n byFeynmanin1982whenhenotedthatclassicalsystems processing is that the encoded information is extremely
o
can not simulate quantum systems efficiently6. In 1985 fragile. A delicate balance between optimizing the de-
c
: Deutsch developed a quantum algorithm7 demonstrat- sired entanglement between qubits, and minimizing the
v ing the potential for quantum information processing in undesired entanglement between the qubit and its envi-
i
X its own right. However it was not until the discovery by ronmentmustbeachieved. Anyinteractionaqubitmight
Shorin1994ofaquantumalgorithmforpolynomial-time have with its environment causes the information stored
r
a factorizationof prime numbers8 that the possibilities for onthequbitto“leak”out. Thisisreferredtoasdecoher-
quantum information processing other than simulating ence. Untilthe developmentofquantumerrorcorrection
physics began to capture the mainstream imagination. algorithms12,13 it was believed by many that this deco-
Shor’sworkprovidedmuchimpetustothefieldduetothe herence would prevent the construction of any practical
ramifications to RSA encryption. Grover developed an- quantum computer. Even though constructing a quan-
otherquantumalgorithm9in1997forimproveddatabase tum computer is still at the limits of current fabrication
technologies,these errorcorrectionalgorithmsshowthat
a working quantum computer is (in principle) possible,
as longas the errorrate onthe qubits canbe kept below
∗Authortowhomcorrespondence shouldbeaddressed. some threshold (a commonly used estimate14 is 10−4).
2
Inthispaperweareconcernedwiththeread-outstage
of a silicon based quantum computers. Read-out con-
stitutes projecting the quantum information stored in a
qubit onto the sub-space of classical information of the
macroscopic measuring device. Qubit read-out will, in
general, be probabilistic. The read-out process for the
state α0 +β 1 will determine a 0 or 1 with respective
probabi|litiies |ofi α2 and β 2. We consider the read-out
| | | |
process for two proposed Si:P solid state quantum com-
puters(SSQC),theKaneSSQC15(Fig.1)andthecharge-
qubit SSQC16 (Fig.2). Much theoretical work has been
done on optimizing the read-out process based on how
to encode the quantum information and which degree of
freedom of the qubit we choose to measure17,18,19. Most
of these proposals essentially require the determination FIG. 1: Schematic of the Kane quantum computer. The
of the locationofa single electronfor read-out. We have qubit states are encoded in the orientation of 31P nuclear
focused on the SET design in order to optimize it as an spin. Quantuminformation ismediatedbythedonorvalence
electrons. A gates control single qubit rotations via the hy-
electrometer for determining this electron location.
perfine coupling between a valence electron and its nucleus.
The Kane quantum computer (Fig.1) uses the nuclear
Jgatescontrolqubitinteraction. Read-outisviaaSET(not
spinorientation(with respectto anexternalglobalmag- shown) in thevicinity of a P donor in a known state.
netic field) of a single P dopant in silicon as its qubit.
The relaxation time of the nuclear magnetic moment of
Pinsiliconwasextensivelystudiedinbulkresonanceex- Kane SSQC read-out (Fig.3 (a)) and allied electron-spin
periments in the 1960s20, and is known to be in excess proposals17,18,19 consists in transferringthe valence elec-
of 10 hours at low temperatures. This makes the sys- tronofaPdopanttoaneighboringP-dopant(forminga
tem highlyattractivewith respectto decoherenceissues. doubly occupied D− state on the latter site). Originally
Since the nuclei we wish to store our information on are this scheme involved transfer by a DC field. To avoid
so well isolated, read-out is extremely difficult. ionization of the D− state a resonant transfer technique
This difficulty in read-out can (in principle) be over- hasbeenproposed24. ThisD− statecanoccuronlyifthe
come using spin to charge transduction15 as follows. A twoP donorshaveopposite nuclearspins. This is due to
P atom has five electrons in its outermost shell. Four carefullycorrelatingthevalenceelectron’sspintothenu-
of these are used to bond to the surrounding Si lattice, clearspinby the hyperfine interaction,andthe D− state
leaving a single valence electron loosely bound to the P being Pauli limited. Spin to charge transduction is thus
nucleus. Thespinorientationofthis electroncanbe cor- achieved, converting the difficult problem of single spin
relatedwiththenuclearspinthroughthehyperfineinter- detection to charge detection. In this way, by preparing
action. In this way the electron can be used to address a read-out qubit in a known spin orientation, the ori-
and infer information about the nuclear spin state. The entation of the qubit being measured can be inferred.
strengthofthehyperfineinteractionisdeterminedbythe
electron’swavefunctionatthe nucleus. By carefullycon- The charge qubit SSQC (Fig.2) also uses P dopants
trolledvoltagesongatesplacedontopofthe Sisubstrate in Si for qubits, however a P-P+ pair constitutes a sin-
(A gates in Fig.1), the electron/nucleus overlap can be gle qubit. In this way a single valence electron is shared
controlled to bring the P nuclear magnetic moment into between the dopants. The 0 and 1 states are now en-
| i | i
andout-ofresonancewithanexternal,global,oscillating codedontothe locationofthe sharedelectron. Read-out
magnetic field. This allows for single qubit rotations. consists of determining this location (Fig.3 b). Geome-
The electron’s wavefunction extends across many lat- tries have also been proposed that may be sensitive to
tice sites. By placing appropriate voltages on surface the superposition basis, by either adding an additional
gatesplacedbetweenqubits (J gatesinFig.1),the wave- ionized donor25, or placing the SET centrally between
functionoverlapofneighboringPdopants’electronscan, the donors26.
in principle, be controlled, allowing for controlled two Singlequbitrotationsforthechargequbitareachieved
qubit interactions. In this way multi-qubit logic gates byvaryingvoltagesoncontrolgatesabovethesiliconsub-
can be achieved. In practice such control will require strate, in order to force the shared electronfrom the left
precise atomic placement of the P donors21,22. This use (L )totheright(R )dopantnuclei. Multi-qubitopera-
of the valence electron as a ‘messenger’ between qubits t|ioniscanbeachiev|edibycarefullyplacingtheP-P+pairs,
can only occur if the decoherence time of the electron such that the dynamics of one qubit becomes dependent
spin system is long enough to allow for information pro- on the other. The silicon band-structure implies atomic
cessing operations to occur. At sufficiently low temper- precision placement of the P-P+ pairs is required27.
atures, electron dephasing times of greater than 60ms The disadvantage of the charge-qubit is that it is
have been reported23, which is more than ample. The more susceptible to decoherence by local charge fluctua-
3
FIG. 4: SEM image (courtesy of F. Hudson et al, UNSW)
FIG. 2: Schematic of the charge-qubit quantum computer. of atypicalexperimental deviceprovidingmotivation for the
The qubit states are encoded in the location of a shared P- SET geometries underconsideration.
donor valence electron. S gates control the symmetry of po-
tential wells about P+ donors. The B gate creates a barrier
potentialtocontroltunnelingbetweenthedonorpair. Read-
sists of donors placed 20nm below a 5nm silicon oxide
out is via a SET (not shown) used to determine the location
of the shared electron on theP-P+ donor pair. layer. We also consider clusters of donors initially (see
sectionIVA),ratherthansingledopants,withthesingle
dopant case considered in section VD.
We begin our discussion by presenting some introduc-
tions due to the long range of the electromagnetic force.
torySETtheory,alongwithadescriptionofthe induced
This has been the subject of a number of theoretical
islandchargeasameasureofSETsensitivity. The inter-
investigations28,29. Current estimates put the decoher-
playbetweenquantizedelectrons,andcontinuouselectric
ence rate in the order of 10ns28. This short decoherence
fieldsinrelationtounderstandingmesoscopiccircuitryis
time will still allow for gate operations to occur due to
then discussed. The initial and final states of the read-
the faster operation time (estimated to be of order 10
out process in both types of SSQC is then described in
ps16).
detail. In this paper we extend our previous descrip-
From the above description of both SSQC types it is tion of SET sensitivity32 to calculate both the charge
apparentthatthefinalread-outsignalforbothSSQCsis
inducedonaSETisland,andthecurrentvariationsthat
dependent onthe locationofa single electrononsubsur-
can be expected through the device due to charge trans-
facePdonors. HencebyusingSETstomonitorthelocal
fer on nearby quantum dots (QD) (i.e. the nuclear or
electrostatic environment we can perform SSQC read-
charge qubits). We then present a technique for calcu-
out. Though our investigation is motivated by this spe-
lating these currents in the steady state, based on a rate
cificapplicationofSETs,theanalysistechniquewefollow
equation and geometric considerations of the SET. The
is general enough to be useful to any field intending to
experimental setup which motivated our research, and
use the SET as an electrometer.
the techniques we use to determine the capacitance ma-
Ourresearchismotivatedbyexperimentsinwhichsin- trixofoursystemarethenpresented. Finallywepresent
glechargetransferbetweenclustersofP-donorswasmon- the results of simulations in which we varied the SET
itored by two SETs30,31 (Fig.4). We use the structures geometry in order to optimize its sensitivity as an elec-
of the experimental SETs as a starting point for our in- trometer.
vestigation into SET geometry optimization. In keeping
with the experimental architecture,our simulations con-
II. SINGLE ELECTRON TRANSISTORS
As electronic devices approach the nanometer scale,
the sea of continuous electronic energy levels normally
presentinmetalsbecomesdiscreteandpartsofthedevice
begin to behave as low dimensional quantum systems.
Whenthisoccurs,thepassageofsingleelectronsthrough
atunnelbarriercanbecontrolled. Byjoiningtwotunnel
barriersin series (effectively creating an isolatedisland),
it is possible to create a transistor, the current though
which is determined by single electron tunneling events.
FIG.3: Initialandfinalstatesoftheread-outprocessforthe The controlledtunneling of single electrons comes about
Kane and charge-qubit SSQCs. due to the Coulomb blockade present on the SET island
4
the transistor gate, the energies for having n or n+ 1
excess electrons on the island become degenerate. Elec-
trons will now able to tunnel sequentially from sourceto
drain, creating a measurable current through the SET.
The behavior of the SET can be characterized by
sweeping the gate voltage for a variety of source-drain
(transport) voltages. Mapping the current through the
SET for each point in this gate-transport voltage plane
produces regions of zero current in which the tunneling
is suppressed, and the current is essentially zero. These
regions correspond to Fig.5 (a), and have a characteris-
tic diamond shape. Each successive Coulomb diamond
(in the gate voltage direction) represents a region of the
parameter-spaceinwhichthe chargingenergyofthesys-
temisminimizedbyaddingasingleelectrontotheisland.
Byincreasingthe transportvoltage,higher chargestates
become available in the tunneling process, reducing the
efficacy of the Coulomb blockade, as shown in Fig.6.
FIG. 5: Discrete energy levels on a SET island. For sin-
gle electron tunneling to be observable, the energy spacing
between levels must be greater than thermal fluctuations
(∆E > kBT). (a) First unoccupied energy level on the is-
land isgreaterthan theFermienergy(E ) ofthesourcepre-
f
venting tunneling through the potential barrier U. (b) The
energy levels of the island are shifted by the application of
a nearby potential, allowing an electron to tunnel from the
source. The field effect of theexcess electron on thedot pro-
duces a Coulomb blockade. This prevents further electrons
FIG.6: Normalizedcurrentmappedacrossthegate-transport
from tunneling onto tothe island.
(source-drain) voltage plane (also normalized). The charac-
teristic Coulomb diamonds show stable regions of zero cur-
rent. VaryingthegatevoltagemovestheSETislandthrough
successive states of excess electrons, while increasing the
(Fig.5). Anyelectrontunnelingontotheislandfacesfind-
transport voltage reduces the Coulomb blockade, by intro-
ing a particular energy level to inhabit, rather than the
ducinghigher order charge states.
conduction band continuum normally occurring in met-
als. Tunneling phenomena of single electrons may be
observed and controlled by varying the height of the is-
land’s energy levels through the application of a nearby
electric potential (e.g. the gate of the transistor). Ther- A. The SET as an Electrometer
mal fluctuations in the device must be small enough not
to override the resolution of the discrete energy levels, As described above, the variation of the potential ap-
i.e. ∆E >kBT, where ∆E is the energy level spacing of plied to the SET gate varies the energy level on the is-
the island. land,changingthe strengthofthe Coulombblockade. In
TheCoulombicrepulsionofelectronspresentontheis- an analogous fashion, any charge on a nearby QD will
land prevents any further electrons from tunneling onto vary the island’s potential. We can associate a capaci-
it. Though the field produced by a single electron is tivecouplingbetweentheislandandthenearbyQD.The
quite small, the effect of this field on the device is in- potential variation on the island is a continuous variable
versely proportional to the size of the island. Hence in based on the island-QD capacitive coupling. From the
the nanometer regime the field effects can become quite definition of capacitance (C Q/V) we can associate
≡
large. This prevention of further electrons moving onto thispotentialvariationwithaninducedchargeδq,which
the island is referred to as the Coulomb blockade. Only may be a fraction of an electron charge.
when an electron tunneling to the island has an energy The greater this δq, the stronger the variation in the
equaltoorgreaterthanoneoftheunfilleddiscreteenergy energylevelsonthe island. Assuch,thisδq iscommonly
levels of the island is it able to overcome the Coulomb usedtodescribe the sensitivityofthe SET tothe system
blockade. By tuning the energy level on the island via being measured. The actual signal measured however is
5
C C C C
1i 1s 1d 1g
nq = + + + nq . (2)
e e
(cid:18)C C C C (cid:19)
1Σ 1Σ 1Σ 1Σ
Inspection of Equation (2) tells us immediately that the
charge induced on the island due to the QD is simply
C
1i
δq = nq . (3)
e
C
1Σ
2. The Electrostatic Energy Variation of the System
FIG. 7: Circuit diagram of the SET-QD system. Isolated
regions with discrete energy levels are represented by solid
boxes. The dotted outline indicates the region considered as This second method for determining δq, though more
theSET island. complicated than the previous, has the advantage of al-
lowing for calculation of energy variations of the system
due to tunneling events, which is essential for determin-
a currentthroughthe SET which we calculate in section ing the current through a SET (see section III).
III. TheelectrostaticenergyofacapacitorisgivenbyE =
q2/(2C) where q is the magnitude of the charge on one
of the plates of the capacitor. The electrostatic energy
B. The Induced Island Charge of the SET-QD system under investigation is simply the
matrix equation extension of this, accounting for all the
capacitancesinthe system. Anyelectrodesconnectedby
We presenttwomethods ofdetermining the chargein-
wires to regions outside the system have their potentials
ducedonanislandforthesimplestsystemofaSETcou-
controlled externally, and so we need only consider the
pled to a single QD. We extend the technique to a two
chargesonanyfloatingelectrodes(inthis casetheisland
QD system for our analysis, though present the single
and a single QD). The charging energy for the system is
QD case here for clarity. The motivation for developing
given by
two approaches is that the first is a common method in
the literature, and is very simple, while the second al-
1
lows us to determine the energy variations of the entire E = QTC−1Q. (4)
2 E
system due to tunneling events. These energy variations
can then be used to determine current variations in the Q is a vector containing the charge on the island and
SET. Comparison of the two techniques indicated that the QD and we refer to C as the ‘energy’ capacitance
E
both gave equivalent results for our simulations. matrix,definedbelow. The chargeonthe islandandQD
The circuitdiagramforthe systembeing consideredis is a combination of actual electrons, and induced charge
shown in Fig.7. due to voltage differences,
Q=Q˜+nq , (5)
e
1. Ratio of QD’s Total Induced Charge
where n represents the number of electrons on each low
dimensionalquantumsystem(inthis casethe islandand
SupposewehaveaQDchargedwithnexcesselectrons the subsurface QD), while Q˜ represents the continuous
(q = nq ) coupled to a SET. Let the self capacitance
− e chargeinducedontheseobjects. Inmatrixformwehave,
of the QD be C . This self capacitance is simply the
1Σ
summation of the QD’s capacitive coupling to each of
the objects in its environment. In our model we assume Q˜ n
Q= i + i . (6)
thecapacitivecouplingtodefectsinthelattice(dangling (cid:18)Q˜1 (cid:19) (cid:18)n1 (cid:19)
bonds,localimpuritiesetc.) isnegligiblesothatwehave,
Considering only the island for a moment, we can write
the induced charge from the source, drain and gate elec-
C1Σ =C1i+C1s+C1d+C1g , (1) trodes as
where the roman subscripts refer to the SET’s island, Q˜ =C V +C V +C V , (7)
i is s id d ig g
source, drain and gate. Conservation of charge necessi-
tates that the chargeonthe QD mustinduce a chargeof where we are referencing all voltages to the island (V =
s
nq in its environment. Hence we can write the manner V assuming the tunnel barriers in our SET are identi-
e d
inwhichthis chargeis sharedacrossthe environmentas, c−al). For our system then, we can write Q˜ = C V. We
c
6
Todeterminethe shiftinthesystem’selectrostaticen-
ergyprofilewe firstdetermine the twolowestenergylev-
elsofthesystemforthecaseoftheQDbeingneutral. We
set the source and drain voltages to +V /2 and V /2
sd sd
−
(in our simulations V = 0.1 mV) and step through a
sd
voltagerangeonthe biasgate V . Foreachapplied
gmax
±
biasvoltage(V )wecyclethrough nexcesselectronson
g
±
theislandanduseEquation(4)todeterminetheelectro-
static energy of the system. We then keep the difference
betweenthe twolowestenergiesandplotthis energydif-
ferencewithrespecttothebiasgate’svoltage(seeFig.8).
The zeros of this plot are voltages at which the island
having n, or n + 1 excess electrons is degenerate with
respect to the energy of the system, and correspond to
maximum currents through the SET. Similarly, the pe-
riodoftheplotcorrespondstoaddingoneelectrontothe
island.
To determine δq we then perform the entire process
again, this time placing one electron on the QD. This
FIG. 8: Plot of difference between two lowest energy levels
will give a similar energy difference plot, but shifted due
on the island. The blue line is for the quantum dot being
measured neutral, the red line for a charge of −qe on the to the dot’s influence on the island. Since we know the
periodoftheplotcorrespondstoanelectronbeingadded
quantumdot. Peak topeakdistancecorresponds toacharge
of qe, enabling δq to be determined by the shift between the to the island, we can thus deduce what fraction of an
red and blueplots. electron corresponds to the shift, giving δq.
refer to C as the ‘correlation’ capacitance matrix, III. CURRENT THROUGH A SET
c
V Though we have focused on the induced island charge
C C C s
Cc =(cid:18)C1iss C1idd C1igg (cid:19) , V =VVd  . (8) (mδeqa)suasrinchgaoruarctQerDizisnygsttehme,stehnesiaticvtiutaylosfigonuarl dSeEteTctwedheinn
g
 
experiments is a variation of current. Having demon-
where the subscript 1 again refers to the QD. This can stratedhowto determine the chargingenergyofoursys-
easily be extended to an arbitrary number of QDs. We tem in section IIB2, we now show how to extend the
can now write the total charge on the QD and the is- methodtodetermineactualcurrentsthroughthedevice.
landas a functionof the electrode voltagesand elements OurtechniquecloselyfollowsGrabertandDevoret33 and
of the system’s capacitance matrix, with the number of gives currents in the nA regime, which is in agreement
electronsoneachasafreeparameterwewillvaryinorder with the measured results for SETs in general1,34.
to find the minimum charging energy of the system. To determine the current through the SET we need
The C matrix describes the crosscapacitancesof the to consider the tunneling rates on and off the island.
E
QD and the island and is given by, The rate is determined by the energy difference of the
system between different configurations of electrons on
C C the island. We make the assumptions that co-tunneling
CE =(cid:18) CiΣ1i −C1Σi1 (cid:19) , (9) events can be ignored, and assume that only the island
− has quantised energy levels.
where C is defined in Equation (1). The island’s self The rateatwhichelectronstunnel fromthe electrodes
ıΣ
capacitance (C ) has an equivalent form. to the island is dependent on the number of excess elec-
iΣ
trons on the island. Giventhe island is in a state with n
While not necessary to the theory, it is worth noting
excess electrons, and at a temperature T we denote the
that this matrix must be symmetric, which can reduce
rate at which electrons tunnel from the island (i) to the
computationtime formorecomplicatedsystems. Weare
electrode χ (either the source or drain) as,
now in a position to calculate the electrostatic energy of
thesystemforarbitrarychargeconfigurationsontheQD
andislandby simplyvaryingthe integersconstituting n. 1 ∆En
From this the δq induced on the island by the presence Γn = χi . (10)
χi q2R ∆En
ofachargeontheQDwillbedetermined. Wedothisfor e texp χi 1
(cid:16) kBT (cid:17)−
the case of a single electron being added to the QD, and
determine δq by the shift in the system’s electrostatic We used 4K for our simulations to avoid low temper-
energy profile. ature convergence issues in determining the capacitive
7
coupling between the objects. In practice the opera- Equation (14) simply states that the rate of change of
tional temperature of a SSQC would be of order mK, occupation number must simply be the rate at which
though this won’t effect the qualitative results of our electrons are entering the state, less the rate at which
analysis. The parameters of interest are the change in they are leaving.
energy (∆Eχni) and the barrier tunnel resistance (Rt). In order to solve the problem numerically we must
We determine ∆Eχni by using Equation (4) for the truncatethesummation. Welettheislandchargeconfig-
charging energy of the system, and considering the work urationrangeover N (withN =10),andEquation(14)
done (either on or by the system) in pushing an elec- becomesasystemo±f2N+1equations. WefoundN =10
tron through the tunnel barrier. The energy variation tobe sufficientforconvergence,withN =5,N =10and
for Equation (10) is given by: N =20allgivingthesameresultsforEquation(13). We
write these 2N +1 coupled equations in a single matrix
∆En =E(n 1) E(n)+V q , (11)
χi − − χ e equation. For clarity we make the substitutions
For the tunnel resitance we use the expression35 Γn−1+Γn−1 =A ,
is id n
~3exp(2W√2m φ)
~ e
R = , (12)
t 2πm∗q2E A
e e F
where W is the width of the tunnel barrier and A is its Γnis+Γnsi+Γndi+Γnid =Bn, (15)
surfacearea. Thevariableφistheheightofthepotential
barrier (taken to be a typical value of 2eV36 and E is
F
the Fermi energy of the island (taken to be 11.65eV at Γn+1+Γn+1 =C .
si di n
4K37). The effective mass of an electron in aluminium
oxide is taken to be m∗e =0.35me38. NoticethatA−N containstheprobabilityforthe N 1
To obtain the current through the SET we must con- state, and C for the N +1 state. We must th−eref−ore
N
sider the contribution from all possible processes. Since makethefurtherapproximationthatA =0andC =
−N N
weareignoringco-tunnelingprocessesweneedonlycon- 0. The problem can be simplified further by recognizing
sider events that change the number of electrons on the thatweareafterthe currentinthe steadystate,andthe
island by 1. The current is then given by p are probabilities, hence
n
∞
I =qe pn(Γnχi−Γniχ), (13) p˙n =0, (16)
n=X−∞χX=s,d
and
where p is the probability that the island is in a state
n
withnexcesselectrons. Todeterminetheseprobabilities
p =1. (17)
n
we consider the master equation
Xn
p˙ = (Γn−1+Γn−1)p (14)
n is id n−1 The matrixequationcanthereforebe writtenas(replac-
−(Γnis+Γnsi+Γndi+Γnid)pn ing the final equation with the normalization condition
+(Γn+1+Γn+1)p . of Equation (17))
si di n+1
B C 0 0 p
0 − −N −N ··· ··· ··· −N
A B C 0 p
0  −N+1 − −N+1 −N+1 ··· ··· ···  −N+1 
  . . . . . . . .
. . . . . . . .
0 . . . . . . . .
  
0 = 0 A B C 0  p  . (18)
   n n n  n 
0   ... ·...·· ... −... ... ·...·· ...  ... 
0    
   0 A B C  p 
1   1 ·1·· ··· ··· N−1 − 1N−1 N1−1  pN−1 
 ··· ··· ···  N 
Equation (18) is a simple matrix equation of the form (13) we thus obtain the current through the SET. Fig.9
X=YP, which we solve for P, giving all the occupation shows an example of the currents calculated using this
probabilities. Inserting these probabilities into Equation method. ItclearlydisplaystheexpectedCoulombblock-
8
FIG. 10: Geometry of experimental motivation for SET ge-
ometries under investigation. Control and barrier electrodes
induce charge motion between the clusters. SET gates com-
pensatetheeffectofthecontrolandbarrierelectrodesonthe
SETs’ currents, allowing the detection of charge motion be-
tween the clusters.
FIG. 9: Example of currents calculated using the above this to calculate read-out signals using the above argu-
method. Blue plot is for measured QD being neutral, red ments. To determine the capacitance matrix of such a
is for the QD having −qe excess charge. The charge on the complicated geometry, it is necessary to use the numeri-
QD induces a shift in the current (δI), due to the charge in-
caltechniqueoffiniteelementmodeling(FEM).InFEM,
ducedontheislandbytheQD(δq)behavinginananalagous
the physics of a system being investigated is calculated
fashion to a voltage variation on thegate.
onameshwhichdiscretisesthegeometryofthesituation.
To reduce computing requirements careful consideration
needs to be given the mesh construction. Finer meshes
ade regions. The two plots are the currents for the QD
arerequiredinregionswheretheelectricfieldisexpected
neutral,andhavingchargeof q . The shift inthe plots
e
− to vary rapidly. Fig.11 shows an example of this for the
isdue tothechargeinducedontheislanddue totheQD
single SET single QD system discussed in section IIB2.
charge. When performing measurements of a SSQC, the
WeuseISE-MESH39 andin-housecomputationalgeome-
signalwillbethevariationinthecurrent,duetotheshift
tryalgorithmsto generatethe meshautomaticallybased
brought about by the charge on the quantum dot. Our
onourinputgeometery. Theexperimentsmotivatingour
resultsbelowarethusdisplayedasthedifferencebetween
researchemployedtwinclustersof 600Pdonorsasatest
theneutralandchargedplots. Wecanmaximizeoursig-
forsingleelectrontransfer. WethusmodeledourQDsas
nalbymakingmeasurementswhentheSETcurrentisat
metalic cubes (60 60 60nm3) with a surface area of
a maximium rate of change with respect to voltage vari-
× ×
approximatelythesamesizeasthespheroidsdetermined
ation on the gate (i.e. at maximum transconductance).
bySRIM42 calculationsoftheionimplantationprocess40
used to make the clusters. In section VD we investigate
the change in the measured signal through the SET due
to measuring single donors rather than clusters.
IV. DETERMINING CAPACITANCES
A. Experimental Motivation
B. Capacitance Calculation
WebaseourSETgeometryinvestigationaroundrecent
Once the mesh is determined, capacitances are deter-
experiments30. The experiment involves two Al SETs
mined by performing an AC analysis through the device
(with oxide tunnel barriers) atop a 5nm SiO layer de-
2 using ISE-DESSIS39. This constitutes placing a known
posited on high resistivity Si. The SETs are coupled
alternatingcurrentbetweenthe objects wewishto know
to a double QD system embedded in the Si substrate.
the capacitance of and solving
Control and barrier gates are used to mediate the elec-
tron transfer process between the QDs. The gates of I =A V +j ω C V , (19)
the SETs compensate the effect of the control and bar- · · · ·
rier electrodes on the SETs’ islands (See Fig.10). This I isthevectorcontainingthecurrentateachofthenodes
ensures that only uncompensated effects due to electron (determined by applying Kirchhoff’s current law), V is
motion between the QDs being measured give a varia- the vector containing the known potential at each node,
tion in the currents through the SETs. Once the full A is the admittance matrix containing the resistancebe-
capacitance matrix of the system is determined, we use tween each node, ω is the frequency of the AC. We use
9
FIG.11: Exampleofmesh variationbased ongeometry. The
colours represent the electrostatic potential due to an excess
electron placed on a subsurface quantum dot. Finer meshes
near metallic objects are generated automatically to capture
therapidly spatially varyingelectric field.
a frequency of 1MHz, but the model is not sensitive to
thisparametersincewehaveapureSisubstrate. Finally
C is the capacitance matrix containing the capacitances FIG. 12: Geometry of tunnel barrier. The vertical and hori-
in the device. It is worth noting that some properties zontal areas contribution to the SET current calculated sep-
of the capacitance matrix can serve as a check for the arately and added.
correct behavior of the simulation. Firstly the elements
of the capacitance matrix (C ) denote the capacitance
αβ
between the αth and βth objects hence C =C . Di- the islandtowardsthe QDto confirmthatthisimproved
αβ βα
agonal terms (C ) denote the total self capacitance of thesignalfortheexperimentaldevice. Wethenrepeated
αα
the αth object. Since the charge on the αth object in- this setup with the gate placed between the island and
duces a charge of opposite sign on all other objects: the drain, in order to shield the island’s antenna from
the gate. An investigation into the signal variation for
increasing the size of the island was then performed, as
C =C . (20)
αβ αα
− thiswillvaryitscapacitivecouplingtotherestoftheen-
αX6=β
vironment. Finally we investigated varying the overlap
of the source and drain with the island as this varies the
tunnel barrierarea, implying an increase in the detected
V. GEOMETRIES INVESTIGATED
current variation. The geometry variation can be seen
inFig.13, Fig.14, andFig.15. For the geometriesinvesti-
TheeffectsofmisalignmentoftheQDstothetwinSET gated the barrier wraps over two faces of the island. We
architecturehasbeenconsideredinpreviousstudies41us- treat this by calculating the resistance, width and area
ing the FASTCAP package, though the geometry of the of the vertical and horizontal parts individually, adding
SETs was not investigated. Due to the computational the results for the current (see Fig.12).
requirementsofFEM,wewerenotabletoperformsimu-
lations of the entire twinSET device that motivated our
investigations (see section IVA). We instead chose to
A. Results of Growing Antenna
model a single SET based on the experimental set-up,
coupled to two QDs. While this does not change the
qualitative analysis of our results, we do expect the sig- Fig.13(a) shows the series of geometries investigated
nalsforthefulltwinSETsystemtobesmallerduetothe for growing an antenna from the island to the two QDs
largernumberofmetallicelementsthe chargewewishto representing the measured qubit. Increments of the an-
measure can couple to. When modeling the Kane qubit tenna length for each simulation was 10nm. The proce-
the SET is coupled to two QDs. Initially the currentare durewascarriedoutforthegateplacedbesidethedrain,
determined with both QDs neutral. The final configura- and for the gate placed between the source and drain.
tionis then with oneQDhaving qe andthe otherhaving Fig.13(b)and(c)showthevariationincurrentthrough
qe. ForachargequbittheSETisalsocoupledtoatwo the SET with respect to the gate voltage. The variation
−
QDs, one with the charge of a hole, which changes its is for the initial and final qubit read-out states for the
location (see Fig.3). charge and Kane qubits respectively, for each antenna
For both qubit types we investigated four aspects of length shown in Fig.13(a). Note that for the purpose
the SET geometry. Firstly, we grew the antenna from of comparison we have kept the spacing between QDs
10
constant for all simulations, although donor spacing for
the Kane and charge SSQC will most likely be different.
The resultsclearlyshowanimprovementin the detected
signal for the read-out event as the antenna is extended
towardsthesystembeingmeasured. Thesignalsforboth
the charge and Kane qubit read-out events are of the
same size, with the different qubit types simply giving
a shift in the required gate voltage for maximizing the
detected currentvariation. This trendwasconsistentfor
allthesimulationsperformedandhenceforthonlyresults
for the charge qubit will be discussed.
Fig.13(d) showsthe effectofplacingthe gatebetween
the sourceanddrain. The size ofthe currentvariationis
the same for both cases (charge qubit data displayed in
figure),buttheperiodicityisgreatlyincreasedforthebe-
tweencase(note the changeinthe voltagescalebetween
Fig.13 (c) and Fig.13 (d)). This is due to the self capac-
itance of the island increasing when the gate is placed
between the source and drain. Though the signal size is
notaffected,theincreaseinperiodicityisnotdesirableas
it would make the current in the SET more susceptible
to voltage fluctuations in the gate.
B. Results of Increasing Island Size
Fig.14(a) showsthe geometries usedin a series of sim-
ulations investigating the effect of increasing the size of
theisland. Thewidthoftheislandwasincreasedby5nm
for each simulation. Though the width of the island was
more than doubled, no significant change in the current
variation of the SET due to a read-out event was shown
(Fig.14(b)). Aslightincreaseintheperiodicityofthesig-
naldidoccur. Ofcoursetheislandmustbesmallenough
sothattheenergylevelseparationduetochargingevents
are greater than the average thermal energy of the elec-
trons in the system. Our results show that this occurs
for a wide range of island sizes. As such the size of the
island is not a significant parameter for consideration in
SET design. It should be noted however that our sim-
ulations did not include any stray capacitances to other
objects in the environment near the SET. In reality we
expect randomimpurities in the siliconandsiliconoxide FIG. 13: (a) Geometries for a series of simulations investi-
regions, to which the island will capacitively couple, re- gating the effect of an antenna on the SET island. Each in-
ducingthesignalsize,implyingasmallerislandsizemay crement of the antenna length is 10nm. Simulations were
be desirable. run with the gate beside the drain, and then between the
source and drain. (b) Current variation for growing antenna
with SET gate beside drain when measuring a charge qubit
read-out event. (c) Current variation for growing antenna
C. Results of Varying Tunnel Barrier
withgatebesidedrainwhenmeasuringakanequbitread-out
event. (d) Current variation for growing antenna with gate
Fig.15 (a) shows the geometry for a series of simula- betweenthesourceanddrainwhenmeasuringachargequbit
tions investigating the effect of increasing the overlap of read-out event.
the source and drain with the island. The overlap was
increased by 5nm for each simulation. We expect to see
an increase in the current through the SET as the over- read-out event for the geometries in Fig.15 (a). Al-
lap is increased, as more area is available for electrons though an increase in current is desirable as it gives a
to tunnel through. This is clearly shown in Fig.15 (b) larger detectable signal, note that for large overlaps,the
which displays the current variation for a charge qubit Coulomb blockade no longer completely suppresses the