Quantum non-demolition measurements of single donor spins in semiconductors

0.85 MB·English

by  Mohan Sarovar

Quantum non-demolition measurements of single donor spins in semiconductors
Mohan Sarovar1, Kevin C. Young1,2, Thomas Schenkel3, and K. Birgitta Whaley1
Berkeley Center for Quantum Information and Computation,
Departments of Chemistry1 and Physics2, University of California, Berkeley, California 94720
3Accelerator and Fusion Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720
We propose a technique for measuring the state of a single donor electron spin using a field-
effect transistor inducedtwo-dimensional electron gas and electrically detected magnetic resonance
techniques. The scheme is facilitated by hyperfine coupling to the donor nucleus. We analyze the
9 potential sensitivity and outline experimental requirements. Our measurement provides a single-
0 shot, projective, and quantumnon-demolition measurement of an electron-encoded qubit state.
0
2
PACSnumbers: 73.23.-b,03.67.Lx,76.30.-v,84.37.+q

n
a
J Semiconductorimplementationsofquantumcomputa- paratusandthetechniques of2DEGmediatedspinreso-

5 tion have become a vibrant subject of study in the past nancespectroscopy. InsectionIIwepresentourproposal
decadebecauseofthepromisequantumcomputers(QCs) for spin state measurementin detail, and then in section

] hold for radically altering our understanding of efficient IIIweanalyzethesensitivityofthemeasurementscheme
l
l computation,andtheappealofbootstrappingthewealth and establish the key factors that determine signal-to-
a
h of engineering experience that the semiconductor indus- noise. Then section IV concludes with a discussion.
- tryhasaccumulated. Apromisingavenueforimplement-
s
ingquantumcomputinginsiliconwasproposedbyKane
e
m [1], suggesting the use of phosphorous nuclei to encode I. THE PHYSICAL SETTING
quantum information. However, while the long coher-
.
t encetimesofthenucleiareadvantageousforinformation
a
m storagetasks,theirweakmagneticmomentalsoresultsin
longgateoperationtimes. Incontrast,donorelectrons in
-
d Sicouplestronglytomicrowaveradiationandpermitthe
n fast execution of gates; and while electron spin decoher-
o ence times are shorter than their nuclear counterparts,
c the tradeoff of decreased robustness to noise for faster
[
operation times could be appropriate to implementing a

3 fault-tolerant QC. This has led several authors to sug-
v gest the use of electron spin qubits as a variant on the FIG. 1: A cross section of the field-effect transistor (FET)
3
original Kane proposal (e.g. [2, 3, 4]), and we focus on used to create the 2DEG. In order to reduce qubit decoher-
4
such a modified Kane architecture here. ence, it is beneficial to implant into isotopically purified sili-
3
con.
2 An integral part of any quantum computation archi-
.
1 tecture is the capacity for high-fidelity qubit readout.
The use of electrical conductivity properties of semi-
1 Whilesmallensemblesofdonorspinshavebeendetected
conductorstoinvestigatespinpropertiesof(bulk-doped)
7 [5, 6] and single spin measurements have been demon-
impurities has a long history [12, 13], including studies
0 strated (e.g. [7, 8]), detection of spin states of single
: of donor polarization using a 2DEG probe [9]. Figure
v donorelectronsandnucleiinsiliconhasremainedelusive.
1 shows a cross section of a 2DEG spin readout device
i In this paper we analyze spin dependent scattering be-
X with a single implanted donor. Prior studies have used
tweenconductionelectronsandneutraldonors[9,10]asa
r spin-to-charge-transportconversiontechnique, and show similar devices with bulk-doping [9, 14] or a large num-
a berofimplanteddonors(106)[10]inthe2DEGchannel.
that quantum non-demolition (QND) measurements of
The 2DEG is operated in accumulation mode and thus
single electron spin-encoded qubit states are realistically
conduction electrons scatter off the electron(s) bound to
achievable when mediated via nuclear spin states. Such
the shallow donor(s). The basic principle exploited in
a measurement will also be of value to the developing
these studies is the role of the exchange interaction in
field of spintronics [11] where the electrical detection of
electron-electron scattering. At a scattering event be-
spin states is valuable. Our readout takes advantage of
tween a conduction electron and a loosely bound donor
two features: i)the ability to performelectronspinreso-
impurity electron, the Pauli principle demands that the
nance spectroscopyusing a two-dimensionalelectron gas
combinedwavefunctionofthe twoelectronsbe antisym-
(2DEG), and ii) the hyperfine shift induced on dopant
metric with respect to coordinate exchange. This con-
electron Zeeman energies by the dopant nuclear spin
straint, together with the fact that the combined spin
state.
state can be symmetric (triplet) or antisymmetric (sin-
In the next section we describe the experimental ap- glet), imposes a correlationbetweenthe spatialand spin
2
partsofthewavefunctionandresultsinaneffectivespin state ofthe target(donorelectron)spinwith someprob-
dependenceofthescatteringmatrix,leadingtoaspinde- ability), it will only produce a faithful measurement if
pendent conductance. Application of a static magnetic the time over which it acts is extremely short. We will
fieldwillpartiallypolarizeconductionandimpurityelec- now show that this is indeed the case and that direct
trons leading to excess triplet scattering. A microwave measurement of the electron spin via the 2DEG current
drive will alter these equilibrium polarizations when on is consequently not possible within experimentally real-
resonance with impurity (or conduction) electron Zee- izable times. This negative result will motivate our sub-
man energies and hence alter the ratio of singlet versus sequent presentation in Section IIB of a more complex
triplet scattering events, registering as a change in the scheme for measurementthat is both faithful and exper-
2DEG current. Thus, the spin dependent 2DEG current imentally realizable.
can be used as a detector of spin resonance and accord-
ingly this technique is commonly known as electrically
detected magnetic resonance (EDMR). Ghosh and Sils- A. Direct measurement of electron spin
bee, and later Willems van Beveren et. al., employed
EDMRinbulkdopednaturalsilicontoresolveresonance Toinvestigatethe ability ofthe spin-dependent 2DEG
peaks correspondingto donor electronspins that are hy- currentto measurethe electronic spinstate,we shalluse
perfine split by donor P nuclei [9, 14]. Recently, Lo et. a minimal model of the scattering process. Since we are
al. have used this technique to investigate spin depen- primarily concerned with the spin state of the particles
dent transport with micron-scale transistors on isotopi- involvedinthescattering,weexaminethetransformation
cally enriched 28Si implanted with 121Sb donors [10]. that a singlescattering eventinduces onthe spinor com-
ponents of the conduction and impurity electrons. We
write this transformation as
II. THE PROPOSAL: EDMR BASED SINGLE
SPIN MEASUREMENT ρ (k,k′)= Tk,k′ρinTk†,k′ , (2)
out tr(Tk,k′ρinTk†,k′)
From hereon we will consider the experimental setup
described above in the particular situation where there where ρin/out are the density operatorsfor the spin state
is a single donor P nucleus present, the electron spin of of the combined two-electron system, and: k,k′ =
which encodes the quantum information that we wish Fd(k,k′) + Fx(k,k′)σc σi [17]. Here Fd (FxT) is the
·
to measure. A crucial question in the context of quan- amplitudeforun-exchanged(exchanged)conductionand
tumcomputingiswhetherthespin-dependent2DEGcur- impurity electron scattering [34]. Note that the spatial
rent can be used to measure the state of an electron- aspects of the problem only enter into the amplitudes.
spin qubit, as spin-dependent tunneling processes have Wewillassumeelasticscatteringwiththedonorelectron
been employed [7, 8]. The fundamental concern here is remaining bound, and no scattering of conduction elec-
whether the spin exchange scattering interaction at the tronsoutsidethe2DEG.Thereforetheamplitudescanbe
core of the spin-dependent 2DEG current allows for a parameterizedbytwoparameters: Fd/x Fd/x(θ,k),the
≡
quantum state measurement of a single donor impurity scattering angle within the 2DEG, θ; and the incoming
electron spin. momentum magnitude k (determined by the Fermi en-
ergy of the 2DEG electrons). These amplitudes are free
A spin (1/2) state measurement couples the micro-
parameters in our model and we explore a wide range
scopicstateofthespin,giveningeneralbya(normalized)
of values for them in the simulations below. The direct
density matrix,
andexchangescatteringamplitudesaresimplyrelatedto
the more familiar singlet (f ) and triplet (f ) scattering
a c s t
ρi =(cid:18)c∗ b (cid:19) (1) amplitudes as:
1
(in the measurement basis, with a+b =1), to a macro- Fd(θ,k) = (fs+3ft)
4
scopic meter variable I, the 2DEG current in our case.
1
The meter variable can take one of two values, and at Fx(θ,k) = (ft fs) (3)
4 −
the conclusion of the measurement, a faithful measur-
ing device would register each meter variable with the Now, assume an initial state ρ = (p +(1
in |↑ich↑| −
correct statistics, i.e., I↑ with probability a and I↓ with p)|↓ich↓|)⊗ρi,wherethefirstterminthetensorproduct
probability b. A QND measurement device will have the is the state of the conduction electron (the conduction
additional property that once a meter variable has been bandisassumedtobe polarizedto thedegreeP0 =2p
c −
registered,the measured spin remains in the state corre- 1, 0 p 1), and the second term is the general state
≤ ≤
sponding to the value registered so that a second mea- of the donor electron given above. After applying the
surement gives the same result [15, 16]. scatteringtransformationandtracingouttheconduction
One might expect that because the exchange interac- electron (because we have no access to its spin after the
tion is destructive (in the sense that it will change the scattering event in this experimental scheme) we get a
3
map that represents the transformation of the impurity exchange amplitude F and for any polarization, P0.
| x| c
electron state due to one scattering event: Figure 2 shows worst-case fidelity decay as a function
of scattering amplitude parameters for various values of
ρi →ρ′i(θ,k) 2DEG polarization Pc0. These simulations clearly show
(1 p) thattherelaxationofageneralelectronspinstateisrapid
= − (F +F σ )ρ (F∗+F∗σ )+4F 2σ ρ σ
d x z i d x z | x| − i + acrossvirtually allreasonableparameterranges. In fact,
pN (cid:2) (cid:3) forrealistic2DEGpolarizationvalues w typicallydrops
+ (Fd−Fxσz)ρi(Fd∗−Fx∗σz)+4|Fx|2σ+ρiσ− to near zero already after 103 104Fsncattering events.
N (cid:2) (cid:3) ∼ −
(4) Thus the measurement induced population mixing time
is T 1 10ns, which is drastically smaller than τ .
mix m
whereN isanormalizationconstanttoensuretr(ρ′i)=1, These s∼imul−ations thus show conclusively that whatever
and we have not explicitly written the (θ,k) dependence the precisevaluesofthe scatteringamplitudes,under re-
of the scattering amplitudes for brevity. alistic experimental conditions the electron spin relax-
In order for the measurement to be faithful, the diag- ation induced by the scattering interaction makes the
onal elements of the impurity spin state (the population 2DEG current an ineffective measurement of the elec-
probabilities) must be preserved under the interaction – tron spin state. This makes it impossible to faithfully
thatis,themeasurementinteractionmayinducedephas- map the electronspin state onto the meter variable,and
ing (in the measurement basis), but no other decoher- hence impossible to perform a single electron spin state
ence. However,thetermsproportionalto Fx 2 inEq.(4) measurement using the 2DEG current directly.
| |
suggest that there will be population mixing. Using this For completeness we note that since we only have ac-
equation, we can write the transformation of the diago- cesstothetotal2DEGcurrentandnoangle-resolvingde-
nal elements (which are uncoupled from the off diagonal tectors,theactualimpurityelectrondensitymatrixmust
elements by the transformation Eq. (4)) as: alsoinvolveanaverageoverθandkoverthe2DEGFermi
surface in Eq. (4). However, as we have shown that the
a a′(θ,k)
→ direct measurement will not work for any value of θ and
= (1−2Pc0Λcosχ+(2Pc0−1)Λ2)a+2(1−Pc0)Λ2 k, the averageddynamics will only resultin a worseper-
4P0Λ(Λ cosχ)a+1+3Λ2+2P0Λcosχ 2P0Λ2 formance analysis.
c − c − c
However, as we will now show, it is possible to make
where Λ(θ,k) Fx(θ,k)/Fd(θ,k), and χ(θ,k) use of the nuclear spin degree of freedom in order to
≡ | | | | ≡
argFx(θ,k) argFd(θ,k),andwehaveusedthefactthat utilize EDMR for projective and QND measurement of
−
a+b=1 to normalize the transformation. single spin states. The key is that the state of the nu-
We can iterate this recursion to simulate the effects clearspinaffectstheZeemansplittingoftheelectronspin
of the repeated scattering events that contribute to the (and thus its resonant frequency) via the mutual hyper-
current. An appropriate quantification of measurement fine coupling. Therefore our strategy is to transfer the
quality is the measurement fidelity [18]: qubit state from the electron to the nucleus and then to
perform an EDMR readout.
=2( a(n) a(0)+ b(n) b(0))2 0.5, (5)
n
F | − |
p p p p
where a(n) and b(n) are the diagonal elements of ρ after
i B. Nuclear spin mediated electron spin state
n scattering events. An ideal measurement has = 1,
n measurement
F
while = 0 indicates a measurement that yields no
n
F
information – i.e. no correlation between the original
Thelow-energy,low-temperatureHamiltoniandescrib-
qubit state and the meter variables. Since the measure-
ing the electron and nuclear spins of a phosphorous
ment should work for all initial states, we consider the
dopant in a static magnetic field, B=Bzˆ is
worst-case measurement fidelity: w =min .
Fn a(0),b(0)Fn
In order to calculate this fidelity we need to deter- 1
mine how many scattering events will take place within H = 2[geµBBσze−gnµnBσzn]+Aσe·σn (6)
the time required to do the measurement. Given the
current state of the art, and factoring in improvements whereµ andµ aretheBohrandnuclearmagnetons,g
B n e
in 2DEG mobility [19] and conduction electronpolariza- (g ) is the electron (nuclear) g-factor, and A character-
n
tion, we estimate a shot-noiselimited measurementtime izesthestrengthofthehyperfineinteractionbetweenthe
of τ 10−3 s (this calculation is given below, in sec- two spins [1] (we set ~ = 1 throughout the paper). For
m
tion II∼I). Within this time, there will be 109 scatter- moderate and large values of B, the σ terms dominate
z
∼
ing events (see the Appendix for details on calculating andwecanmakethesecular approximation, toarriveat:
the number of scattering events per second). Although H 1/2[g µ Bσe g µ Bσn]+Aσeσn. The energy
≈ e B z − n n z z z
we do not assume specific values of the scattering am- levels and eigenstates of this Hamiltonian are shown in
plitudes, we find from iterating the above recursion for Fig. 3. Note that we have ignored the coupling of both
a broad range of values F /F that after 109 scat- spins to uncontrolled degrees of freedom such as para-
x d
tering events w is 1 for any non-zero v∼alue of the magnetic defects and phonons (coupling to lattice spins
Fn ≪
4
n=10 n=100 n=10000
1 1 1
P0=0.01
0.5 0.5 0.5 c
0 0 0
0 0 0
0.5 0 0.5 0 0.5 0
0.5 0.5 0.5
|F |/|F | 1 1 χ/2π 1 1 1 1
x d
1 1 1
P0=0.1
0.5 0.5 0.5 c
0 0 0
0 0 0
0.5 0 0.5 0 0.5 0
0.5 0.5 0.5
1 1 1 1 1 1
1 1 1
P0=1
0.5 0.5 0.5 c
0 0 0
0 0 0
0.5 0 0.5 0 0.5 0
0.5 0.5 0.5
1 1 1 1 1 1
FIG. 2: Evolution of worst-case measurement fidelity, Fw, during 2DEG scattering dynamics as a function of the number of
n
scattering events n, for a range of values of the ratio of direct and exchange scattering amplitudes and of 2DEG equilibrium
polarization. The independent (base plane) axes on the plots parametrize the complex scattering amplitude ratio Fx/Fd ≡
|Fx|/|Fd|eiχ: one axis is the magnitude, Λ ≡ |Fx|/|Fd| (shown for 0 < |Fx|/|Fd| < 1; the plots are restricted to this range
because the worst-case fidelity is negligibly small outside it), and the other is the phase, χ (shown for 0 < χ/2π < 1). The
numberofscattering events,n,variesacross thecolumns,with valuesn=10,102 and104 shown here. The2DEGequilibrium
polarization, P0,variesacrosstherows,withvaluesP0 =0.01,0.1and1shownhere. ForP0 ≥0.1,weseethattherearefairly
c c c
large regions in the Fx/Fd parameter space for which the worst-case measurement fidelity is non-zero: however, (i) Fnw still
decaysrapidlywithnumberofscatteringevents,andisrarely>0.9(thefidelitiesdesirableforhigh-qualitymeasurement),and
(ii) Fnw is highly sensitive to theprecise value of Fx/Fd and Pc0 in these regions.
can be mitigated by the use of a 28Si substrate). These will simply assume that this results in some effective re-
environmental couplings will contribute to decoherence laxationanddephasingoftheelectronandnuclearspins.
ofthenuclearandelectronspinstates(e.g. [20]),andwe
We see that the resonance frequency (Zeeman energy) (CNOT) gates [21] SWAP = CNOT CNOT CNOT ,
n e n
of the electron is a function of the nuclear spin state. where the subscript indicates which of the two qubits
Therefore,ourstrategywillbetotransferthequbitstate isactingasthecontrol. However,the completeexchange
from the electron to the nucleus and then use EDMR to of electron-nuclear states is unnecessary, since the spin
measure the nuclear spin. This is in effect a spin-to- state of the impurity electron is lost to the environment
resonance-to-chargeconversionmeasurement. by the application of resonant pulses and elastic scat-
To perform the state transfer, we appeal to the qubit tering with conduction electrons in the 2DEG. There-
SWAPgate: SWAP[ρ τ ]SWAP† =τ ρ . SWAPcan fore, the final operator in the sequence can be neglected
e n e n
be decomposed into th⊗e sequence of thre⊗e controlled-not since it only alters the state of the electron. This leads
5
Afterperformingthestatetransferonthecombinedstate
andtracingovertheelectrondegreesoffreedom(because
itislosttotheenvironment),weareleftwiththereduced
density matrix describing the nucleus,
tr TRANS [ρ τ ]TRANS† = (8)
e(cid:16) e e⊗ n e(cid:17)
α2 αβ∗(w+w∗)
(cid:18)α∗β(|w|+w∗) β 2 (cid:19)
FIG. 3: Four-level system of electron-nuclear spin degrees of | |
freedom. Theenergyeigenstatesinthesecularapproximation
are the eigenstates of σe and σn. The transitions indicated Becauseofthehyperfinecoupling(Fig. 3),electronreso-
z z
by arrows are required for the state transfer described in the nance will occur at the lower frequency with probability
text. α2 and at the higher frequency with probability β 2.
| | | |
Theelectricaldetectionofthisshiftfromthefreeelectron
resonancefrequency by EDMR constitutes a single-shot,
to the definition of the electron-to-nucleus transfer gate, projective measurement in the σ basis of the original
z
TRANSe = CNOTeCNOTn. To apply these CNOT electron state (and therefore, qubit state) with the cor-
gates we use resonant pulses: CNOTe interchanges the rect statistics.
states and and so can be implemented
|↑ie|⇑in |↑ie|⇓in In detail, the single qubit spin readout can proceed
byapplicationofaresonantπ-pulseatfrequencyω (see
n as follows. Following state transfer to the nuclear spin,
Fig. 3),anRFtransition;similarly,CNOT interchanges
n one of the two hyperfine split electron spin resonance
and and is implemented by a resonant
|↑ie|⇑in |↓ie|⇑in lines that corresponds to a given nuclear spin projection
π-pulse at ω , a microwave transition. Each of these
e is addressed by dialing in the corresponding microwave
transitionsisdipole-allowed,ensuringthatgatetimesare
frequency for resonant excitation of electron spin tran-
sufficiently fast. The ability to apply pulses faster than
sitions. At the same time, the transistor is turned on
relevant decoherence times is required for successful im-
andthe channelcurrentis monitored. Now,assume that
plementation of the state transfer. We note that after
the magnetic fields have been tuned to address the
this work was completed, a state swap scheme very sim- nuclear state projection. Then with probability α2|↑thine
ilar to the state transfer scheme outlined above was suc- | |
transistorcurrentwilldifferfromtheoff-resonantcurrent
cessfully performed in experiments on bulk-doped Si:P
valueandwithprobability β 2 itwillbejustequaltothe
samples [22]. | |
off-resonant channel current. In either case, monitoring
Supposetheelectronisinaninitial(pure)state, ψ =
| ie the current at one hyperfine resonance for the τm mea-
α +β , while the nucleus is in a general mixed
|↑ie |↓ie surement duration constitutes a readout of the nuclear
state,
spin. And due to the prior state transfer, it effectively
measures the spin state of the original donor electron
u w
τ = . (7) spin.
n (cid:18)w∗ v (cid:19)
FIG. 4: Illustration of single spin readout. In experiments with large ensembles of donor spin qubits, lines from all nuclear
spin projections are present in EDMR measurements (left). In measurements with single donors (right), only single lines are
present for measurement times shorter then thenuclear spin relaxation time. Monitoring the current at a given resonant field
measures thespin state of thedonor nucleuswith the correct statistics.
6
III. MEASUREMENT SENSITIVITY AND s 1. The final term in Eq. (10) represents the ratio
≈
MEASUREMENT-INDUCED DECOHERENCE between impurity scattering (1/τ ) and total scattering
n
(1/τ ) rates. We assume 1/τ =1/τ +1/τ , where 1/τ
t t 0 n 0
Two critical practical issues need to be addressed for is the scattering rate due to all other processes (such as
realizationofthis protocolformeasurementofthe donor surface roughness scattering and Coulomb scattering by
electronspinquantumstate. Theseare: i)thesensitivity charged defects).
of the EDMR measurement needs to be sufficiently high To estimate the expected magnitude of this current
to allow single donor spins to be detected, and ii) the differential, we begin by considering present state of the
measurement time τ must be small compared to the art 2DEG mediated EDMR experiments where this cur-
m
lifetime of the nuclear spin. rent differential is 10−7 (with T 5K,B 0.3T, a
We first address the issue of the sensitivity of the dif- 2DEG channel area∼of 160 20µm2,∼probe cur∼rent 1µA,
ferentialEDMRcurrentin the limitofsingle donorscat- anda donordensity of 2 ×1011donors/cm2) [10]. We as-
tering. As detailed above, the spin state of the donor sume that α′ will be sim×ilar for the single donor device
electronis continually changingdue to the scattering in- as in current experiments. Then in scaling down to a
teraction, and hence is time-dependent. However, ignor- singledonor,the firstaspectto consideristhe scattering
ing the transient, we can approximate it with a time in- rate ratio: ̺ 1/τn. To first order this ratio can be
≡ 1/τt
dependentvaluegivenbythesteadystatesolutionofthe kept constant if we scale the 2DEG area concomitantly
recursion relation, Eq. (4). This approximation can be withthe donornumber. Fromthe channelareaandden-
thought of as taking the equilibrium spin value, where sityofcurrentexperiments,weextrapolatethata 2DEG
the “spin temperature” of the impurity has equilibrated area of 30 30nm2 – well within the realm of current
with that of conduction electrons via the scattering in- technolo∼gy[2×4]–wouldkeep̺unchanged. Optimization
teraction. The explicit simulations of the scattering re- of donor depth might relax this size requirement [23]. A
cursion Eq. (5) shown in Section IIA indicate that this higherorderanalysiswouldrequiredetailedinvestigation
equilibrationhappenswithin104 scatteringeventsforall ofthedevicespecificinterfaceandintrinsiccontributions
possible values of scattering amplitudes. Thus the time to the other scattering processes,and hence to the chan-
scale for this equilibration is 10ns, much faster than nelmobilityandτ . Relatedtothisconcern,themobility
∼ 0
the observable times scales of the measurement, justify- ofthe2DEGchannelcanbeimproved–e.g.,byusinghy-
ing our use of the steady-state solution of the recursion drogen passivation to mitigate surface roughness at the
(see the Appendix for details on calculating the number oxide interface [19] – to increase ̺. We conservatively
of scattering events per second). Solving for the steady- estimate a factor of 10 increase in ∆I/I from such im-
0
state (ρ(n) =ρ(n−1) ρss), gives us a time-independent, provements. The saturation parameter s 1 for large
i i ≡ i ∼
non-resonant single donor “polarization” equal to enough microwave powers in the recent measurements
[10]andsodoesnotpresentanareaforimprovement. Fi-
σ ss tr(σ ρss)= Λ− (Pc0)2cos2χ+Λ2(1−(Pc0)2) nally,anavenueforsignificantimprovementinsignalisto
h zii ≡ z i p P0(Λ cosχ) increase the conduction electron polarization, P0, which
c − c
(9) iscurrently 0.1 1%. Thispolarizationisroughlypro-
∼ −
It should be noted that this expression for the steady portionalto the applied static magnetic field, and there-
state “polarization” is not valid when P0 = 0 or Λ = 0, fore a factor of 10 improvement is possible by operating
c
but neither of these limits is relevant to spin measure- at B = 3T. Additionally, spin injection techniques can
ment. Now, we can follow the analysis of Ref. [9], using be employed to achieve P0 > 10% (e.g. [25, 26]), re-
c
donor “polarization” σ ss, to estimate the on-resonant sulting in a 100-fold improvement in ∆I/I . Hence, by
h zii 0
(I)andoff-resonant(I )currentdifferential(normalized) improvementsindevicescalingandchannelmobility,and
0
as: by incorporating spin injection, we estimate a realistic,
improved current differential of ∆I/I 10−4. Given
∆I I −I0 α′ s σ ssP01/τn. (10) this ∆I/I and a probe current of I 0 ∼1µA, to achieve
I ≡ I ≈− h zii c 1/τ 0 0 ∼
0 0 t ansignal-to-noise(SNR)of10throughshot-noiselimited
Hereα′ ≡hΣs−Σti|z=zi/hΣs+3Σti|z=zi,Σs andΣtare detection we require a τm satisfying:
singlet and triplet scattering cross sections, respectively,
andh·i|z=zi denotesanaverageoverthescatteringregion ∆I I0τm >10 I0τm (11)
with the donor location in z (see Fig. 1) held fixed [23]. (cid:18) I (cid:19) e r e
0
s = 1 (1 s )(1 s ), and s and s (both between 0
i c i c
− − −
and1)aresaturationparameterswhichcharacterizehow wherethelefthandsideisthe signal,therighthandside
much of the microwave power is absorbed by the impu- istheaccumulatedshot-noisemultipliedbytheSNR,and
rity and conduction electrons, respectively [9]. s is a e is the fundamental unit of electric charge. Solving this
i
function of the broadening at the single donor electron yields a measurement integration time of τ 10−3 s.
m
∼
resonance frequency: if we work in a regime where this Inordertocompletethemeasurementanalysisweneed
broadeningis minimal (as requiredto performthe quan- to address the second issue identified above and confirm
tum state transfer described above), s 1 and thus that the state of the nuclear spin does not flip within
i
≈
7
the measurement time – i.e. the measurement time τ can be measured. One concern is that at the required
m
has to be shorter then the nuclear spin flip time T . transistor size of 30 30nm2, the MOSFET device
1
∼ ×
Once the 2DEG current is switched on (the 2DEG cur- will no longer act as a 2DEG but rather more like a
rent is off during the electron-nucleus state transfer) the quantum dot. However, while the proposed MOSFET
dynamics of the donor electron due to scattering and is small, there are no physical tunnel barriers between
microwave driving will contribute to the decoherence of the transistor island and the source and drain leads (no
the nuclear spin. Donor nuclear spin relaxation is not intentional confinement). Important Coulomb blockade
well characterized under these conditions but we expect effectshaveindeedbeenobservedinsuchMOSFETs,but
that in large magnetic fields the donor electron dynam- onlyinabiasingregimewithverylowsource-drainbiases
icscontributesprimarilyonlytodephasingofthenuclear [24,30,31]andinsub-thresholdcurrentsatverylowgate
state. This can be made precise by performing pertur- biases[32]. Inthesestudies,confinementwasfoundtobe
bation theory on Eq. (6) in the parameter A/∆, where induced by impurity potentials [30, 31]. It has also been
∆ ω ω =B(g µ g µ ). In the detuned regime shownthatonecaneasilytuneinandoutoftheCoulomb
e n e B n n
≡ − −
whereA/∆ 1,theeffectiveHamiltoniandescribingthe blockade regime by suitable gate and source-drain bias-
≪
coupled systems is ing [30] (and see also Ref. [31] for related results with
tunable tunnel barriers). In the experiments we envi-
H H = 1ω σe 1ω σn+Aσeσn+A2(σe σn) (12) sion, which are closely related to and inspired by recent
≈ eff 2 e z−2 n z z z ∆ z− z demonstrations of spin dependent transport in micron-
scale devices [9, 10], the source-drain as well as the gate
This effective Hamiltonian is of course also the justifica-
biascanbetunedoveralargerangeofvoltages,fromthe
tion for the secular approximation made earlier (see be-
very low values needed to study interesting and impor-
lowEq.(6)). Thereforewesee thatto firstorderinA/∆
tant Coulomb blockade effects in a low current regime
thedonorelectroncanonlydephasethenuclearspinand
(< 50 nA) to higher bias values desirable for electri-
that direct contributions to nuclear spin T through the
1 cal detection of magnetic resonance through scattering
hyperfine interaction are small. Secondary mechanisms
of conduction electrons off neutral donors [9, 10] with
suchasphonon-assistedcrossrelaxationcan,inprinciple,
much higher channel currents ( 1 µA). Such MOSFET
alsocontributetothe nuclearT1 duringelectrondriving. devices can be operated as 2D∼EGs and away from the
However,thesecontributionswereshowntobeverysmall
quantum dot regime, by imposing large enough source-
by Feher and Gere who demonstrated that the electron-
drain and gate biases.
nuclear cross relaxation time, T , under electron driving
x
Finally, we note that the fact thatthe measurementis
conditions is on the order of hours [27]. Given this ex-
facilitated by the nucleus of the donor atom intimates a
tremely long cross-relaxation time and the equivalently
hybriddonorqubitwherequantumoperationsarecarried
long nuclear T in a static electron environment at low
1
outontheelectronspinandthestateistransferedtothe
temperatures[27,28],weconcludethatthenuclearT in
1
nucleus for measurement and storage(advantageous due
thepresenceofelectrondrivingwillbecomfortablylarger
than τ 10−3 s. Indeed, this has very recently been tothelongerrelaxationtimes). Althoughtheaboveanal-
m
∼ ysis was done with the example of a phosphorous donor,
confirmed by explicit measurements of nuclear T under
1
it applies equally well to other donors,such as antimony
the conditions of electron driving [29]. This analysis im-
[4, 10], and some paramagnetic centers [33]. One merely
plies thatonce the measurementcollapsesontoa nuclear
hastoisolatetwo(dipole-transitionallowed)nuclearspin
basis state, the nuclear spin state does indeed effectively
levelsto serveas qubitbasis states andtransferthe elec-
remainthereandthereforethe EDMRmeasurementsat-
tronstate to these nuclearstateswith resonantpulses as
isfies the QND requirement on the qubit state.
outlined here.
IV. CONCLUSION
V. APPENDIX
Byutilizingresonantpulsegatesand2DEG-mediated-
EDMR readout, we have proposed a realistic scheme for Here we detail the procedure used in the main text
measuring the spin state of a single donor electron in for calculating the amount of time taken for n scatter-
silicon. By making use of the hyperfine coupled donor ing events. We assume a MOSFET channel of width
nuclear spin, the readout scheme provides a single shot W = 30nm probed with a current of 1µA. This means
measurementthatisbothprojectiveandQND.TheQND that there are I/e 1.6 1013 electrons per second
≈ ×
aspect also makes this technique an effective method for crossing the channel. However, to gain a more accu-
initializing the state of the nuclear spin. rate estimate of the number of electrons interrogating
Wehaveanalyzedthemeasurementprocedure,thefac- the donor electron per second we scale this total num-
tors which influence the signal-to-noise ratio, and the ber by a ratio of the scattering length to the width of
experimental apparatus to arrive at realistic modifica- the channel: √σ/W. A crude estimate of the scattering
tions/improvementsthatcanbe made to current2DEG- cross section, σ, can be obtained using the singlet and
based EDMR apparati [10] so that a single nuclear spin triplet scattering lengths givenin Ref. [12]: a =6.167˚A
s
8
and a = 2.33˚A. These scattering lengths are for three Usingthis estimateofthe averagescatteringcrosssec-
t
dimensional electron-hydrogen scattering in bulk-doped tion,thenumberofelectronsinterrogatingthedonorper
systems, however we consider them sufficient for an or- second is given by
derofmagnitudeestimateofthenumberofinterrogating
electronspersecond. Intermsofthesescatteringlengths,
18.5 10−10
thescatteringcrosssectionforthe2DEGinteractingwith n 1.6 1013 × 1 1012 (15)
e ≈ × · 30 10−9 ≈ ×
an ensemble of donors is [9]:
×
σ =2π (a2+3a2) (a2 a2)P0P0 (13)
s t − s− t c i Therefore we estimate that the time taken for n scat-
(cid:2) (cid:3) tering events is n 10−12 seconds.
where P0 and P0 are the conduction band and impu- ×
c i
rity polarization. Since P0 1 we will approximate the
c ≪
above expression as:
σ 2π(a2+3a2)=341.3 ˚A2 (14)
≈ s t Acknowledgments
For a single donor, Eq. (13) should be modified to take
into account the time-dependent “polarization” of the We thank NSA (Grant MOD713106A) for financial
single donor spin (see Sec. III). However, since this support. MS and KBW were also supported by NSF
quantity drops out in the final approximate expression (Grant EIA-0205641) and TS by DOE (Contract DE-
for the cross section, Eq. (14), we will not be concerned AC02-05CH11231). We are grateful to S. Lyon, A. M.
with this modification. Tyryshkin,C.C.LoandJ.Bokorforhelpfuldiscussions.
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