toward a silicon-based nuclear-spin quantum computernatural silicon contains a mixture of three...

One of the major challenges inquantum computing is to iden-tify a system that can be

scaled up to the number of qubitsneeded to execute nontrivial quantumalgorithms. Peter Shor’s algorithm forfinding the prime factors of numbersused in public-encryption systems(numbers that typically consist of morethan a hundred digits) would likelyrequire a quantum computer with sev-eral thousand qubits. Depending on theerror correction scheme appropriate tothe particular computer, the requirednumber could be much larger.Although ion-trap or nuclear-magnetic-resonance (NMR) quantum “computers” containing a few (lessthan 10) qubits have been successfully

demonstrated, it is not clear that thosesystems can be scaled up.

Solid-state quantum computersmay be more likely candidates. As aresult, a number of solid-stateschemes have been proposed, most ofwhich fall into two categories: Thephysical qubits are spin states of indi-vidual electrons or nuclei, or they arecharge or phase states of supercon-ducting structures.

A particularly promising scheme isthe silicon-based nuclear-spin com-puter, proposed a few years ago byBruce Kane (1998), then of theUniversity of New South Wales inSydney, Australia, and now of theUniversity of Maryland in CollegePark, Maryland. Shown in Figure 1,

the Kane computer architecture con-sists of a linear array of phosphorusatoms embedded beneath the surfaceof a pure silicon wafer. Each phospho-rous atom serves as a qubit, and thelinear array forms the computer’squantum register.1 The entire wafer isplaced in a strong, static magneticfield B0 at sub-Kelvin temperatures,and alignment of the phosphorousatom’s nuclear spin as parallel orantiparallel to B0 corresponds to the|�⟩ and |�⟩ states of the qubit, respec-tively. (Throughout this article, wewill follow the convention of Kane

284 Los Alamos Science Number 27 2002

Toward a Silicon-Based Nuclear-Spin Quantum Computer

Developing the technology for a scalable, solid-state quantum computer

1 Two-dimensional qubit arrays are also possi-ble but require complex electrode geometriesand additional insulators.

Robert G. Clark, P. Chris Hammel, Andrew Dzurak, Alexander Hamilton,Lloyd Hollenberg, David Jamieson, and Christopher Pakes

as told to Jay Schecker

and use the notation |�⟩, |�⟩ to desig-nate both qubit and nuclear spinstates. We will use arrows, |↓⟩ or |↑⟩,to designate electron spin states.)

In order to execute a quantum algo-rithm, we need to rotate individualqubits in Hilbert space and couple twoqubits together. We accomplish bothoperations with an array of gate elec-trodes2 that lies on top of the wafer butis isolated from the pure silicon by athin insulating layer of silicon dioxide(SiO2). Referring to Figure 1, noticethat each A-gate sits precisely above aphosphorous atom and each J-gate liesbetween adjacent atoms. As discussedlater, a small positive voltage applied tothe A-gate gives independent control ofthe qubit directly under the gate, whilevoltage applied to the J-gate allowscoupling two qubits together through anelectron-mediated interaction.

The Center for Quantum ComputerTechnology (CQCT), headquartered inSydney, Australia, and Los AlamosNational Laboratory are trying to fabri-cate Kane’s silicon-based quantumcomputer. Although we can call uponthe technology, techniques, and collec-tive experience of the huge siliconsemiconductor industry, building thecomputer is still a daunting enterprise.We need to manipulate individual phos-phorus atoms and place them preciselywithin a defect-free, isotopically puresilicon matrix. We need to create metalgates on the nanoscale, lay them withina few atoms of each other, and thenensure that each gate is aligned properlyover the buried qubits. Simply put,the ability to do this level of nanofabri-cation does not exist at this time.

Employing a “see-what-works-best”strategy, we have initiated parallelresearch approaches for nearly everyfabrication stage. If one approach fails,we still have a backup. Our currentfocus is on developing a prototype

two-qubit device. By stretching manytechnologies beyond their heretofore-assumed limits, we have come tantaliz-ingly close to achieving that goal. In the sections that follow, we describethe computer and some critical technologies in greater detail, and wealso outline our progress in building a prototype.

Design Features of theSilicon-Based Computer

In our solid-state architecture,individual phosphorus atoms areembedded in a sea of silicon. These

two elements were chosen for severalreasons, the first and foremost beingthat phosphorous is the standarddopant for conventional silicon-basedsemiconductor devices and there istremendous working knowledge ofphosphorus and silicon from the con-ventional computing industry.

The second reason stems from theneed to control the spin environment.We require our qubits to have nuclearspin I = 1/2, but we also want the surrounding environment to be spinfree. Otherwise, unwanted spin-spininteractions between the qubit and anynearby nuclear spins would compro-mise the coherent states needed for

Number 27 2002 Los Alamos Science 285

31P31P

31P

Siliconsubstrate

~20 nm

~20 nm

SiO2 barrierV = 0

V > 0

B0

J-gateJ-gate

SET

A-gateA-gate

J-gateA-gate

+

++

+

|�⟩

|�⟩

Figure 1. Schematic of the Silicon-Based Quantum ComputerThe architecture of Kane’s solid-state quantum computer has a linear array of phos-phorous donor atoms buried into a pure silicon wafer, with an interdonor spacing ofabout 20 nm. In the presence of a large magnetic field and at sub-Kelvin tempera-tures, the nuclear spins of the donor atoms can be aligned either parallel or antiparallel with the field, corresponding to the |��⟩ and |��⟩ qubit states, respectively.An array of metal gates lies on the surface of the wafer, isolated from the silicon bya barrier layer of SiO2. The A-gates lie directly above the donor atoms and enableindividual control of single qubits. The J-gates lie between the donors and regulatean electron-mediated coupling between adjacent nuclear spins, thus allowing fortwo-qubit operations. Readout of the qubit is performed with either a single electrontransistor, as shown, or with a magnetic-resonance force microscope (MRFM, notshown). The electron clouds shown here are schematic—the actual situation is more complicated.

2 In this context, a “gate,” like a transistor gate, is adevice used for controlling charge motion. It doesnot refer to a logical operation such as a cnot gate.

quantum computation. The only stablephosphorous isotope, phosphorous-31,is a spin-1/2 nucleus, so nature hasautomatically satisfied our qubit crite-rion. Creating a spin-free environment,however, is a little more difficult.Natural silicon contains a mixture ofthree isotopes: silicon-28, -29, and -30. Whereas the even-numbered iso-topes are spin free, silicon-29 has aspin of I = 1/2. As a result, we esti-mate that to do quantum computation,we will need to reduce the silicon-29content in our silicon wafer to aboutone part in 105. Those stringent isotopic purity levels can be reachedwith current technology.

Finally, the nuclear spin of a phos-phorous atom in a silicon host is sta-ble. One way to infer the stability is toexamine the spin-lattice relaxationtime T1, which characterizes the timeit takes for a spin system with a netalignment (a net magnetization) toreturn to its thermally equilibratedmagnetization. At temperatures ofabout 1 kelvin, the nuclear-spin relaxation time T1,n (where the subscript “n” is for the nucleus) forphosphorus in silicon is expected tobe longer than 10 hours (Feher 1959).

The nuclear spin qubits, however,interact with the environment throughtheir donor electrons; as a result, theelectron spin stability is also impor-tant, particularly for qubit readout (seelater discussion). The electron-spinrelaxation time T1,e (where the subscript “e” is for the electron) isapproximately 1 hour at about1 kelvin (Honig and Stupp 1960). The phase decoherence time of theelectron spin, as measured by thespin-spin relaxation time T2,e, isshorter still. Although never measuredfor an isolated electron system such asour qubit scheme, the T2,e for anensemble of electrons was measuredto be approximately 0.5 millisecond(Gordon and Bowers 1958). A recenttheoretical study, appropriate for asingle phosphorus donor atom in sili-

con, indicates a T2,e of the order of1 second (Mozyrsky et al. 2002). This value for T2,e should be longenough for us to perform a quantumcomputation and read out the result.

The Spin Hamiltonian andSingle-Qubit Operations. To under-stand the physics underlying the oper-ation of the silicon-based computer,recall that phosphorus has one moreelectron in its outer electron shell thansilicon. When it is placed into a sili-con crystal lattice, phosphorus fulfillsits role as a surrogate silicon atom andstill has one electron left over. At verylow temperatures, that “donor” elec-tron remains bound—albeit ratherloosely—to the phosphorus nucleus.The electron “talks” to the nucleusprimarily through the charge(Coulomb) interaction and to a lesserdegree through the hyperfine interac-tion, which is between the electronspin and the nuclear spin.

We exploit the hyperfine interac-tion to individually address singlequbits. The effective low-energy, low-temperature Hamiltonian describingthe spin interactions for the electronspin and the nuclear spin of an atomin the presence of a static magneticfield B0 is given by

H = µB B0 σze – gn µn B0 σz

n

+ Aσe•σn , (1)

where σze and σz

n are Pauli spinmatrices, µB and µn are the Bohr andnuclear magnetons, respectively, andgn is the nuclear g-factor. The hyper-fine interaction is described by theterm Aσe•σn.

For large values of B0, theHamiltonian in Equation (1) leads to aset of energy levels that correspond tothe four hyperfine levels, |�↓⟩, |�↓⟩, |�↑⟩,and |�↑⟩. At the sub-Kelvin operatingtemperature of the computer, however,the electron spins are completely spin-polarized in the lower-energy state |↓⟩.

Thus, for one-qubit operations, we mayignore the electron spin polarization to agood approximation and consider onlythe two nuclear states |�⟩ and |�⟩ (Goanand Milburn 2000). The energy differ-ence between those states is

∆E0 = hω0≅ 2gn µn B0 + 2A + 2A2/µB B0 ,

(2)

where ω0 is called the nuclear reso-nance frequency. The resonance fre-quency, which is typically in theradio-frequency (rf) range, is equal tothe Larmor frequency, or the rate atwhich the nuclear spins precess aboutthe magnetic-field lines.

Suppose B0 is aligned along the z-axis, and the nuclear spin is initiallyin the |�⟩ state. If we subject the spinsto a secondary magnetic field that isoscillating in the x-direction at thenuclear resonance frequency, that is, afield B1 = B1 cos(ω0t) x, then thenuclear spins will begin to rotatetoward the (–z)-axis, or from the par-allel to the antiparallel alignment (seethe box “Spin Manipulation withMagnetic Resonance” on page 288 ).The spin rotation is equivalent torotating a qubit in Hilbert space fromthe |�⟩ state to some superposition ofthe |�⟩ and |�⟩ states.

As described, the B1 field willaffect all spins simultaneously. Toaddress a particular spin, we use theA-gate directly above it and modifythat donor atom’s hyperfine coupling.The parameter A in Equation (1) isproportional to the magnitude of theelectron probability density at the siteof the nucleus, Ψe(0):

A = 8/3π µB gn µn Ψe(0)2 . (3)

As seen in Figure 2, placing a positivevoltage on the A-gate above the phos-phorous atom attracts the atom’s elec-tron cloud toward the surface and awayfrom the nucleus, thereby reducing the magnitude of Ψe(0). The hyperfine



energy levels of that one atom changeslightly, and the resonance frequencyneeded to rotate the nuclear spin isreduced from, say, ω0 to ω–. If the fre-quency of the B1 field is set to ω–, thatis, B1 = B1 cos(ω–t) x, then only thespin directly under the A-gate will bein resonance and will begin to rotate.Removing the voltage on the A-gatehalts the rotation.

A one-qubit gate operation is there-fore implemented if the silicon waferis subjected to a transverse oscillatingB-field of frequency ω– and if the A-gate above a qubit is pulsed for adefined period. Throughout the dura-tion of the pulse, the qubit is in reso-nance with the secondary magneticfield and rotates through some anglein Hilbert space. When the voltage isremoved at the end of the pulse, thequbit is left in the desired superposi-tion of the |�⟩ and |�⟩ states.

Two-Qubit Operations. To selectadjacent pairs of qubits for two-qubitoperations, we apply a positive voltageto the J-gate between them. As seen inFigure 3(a), this procedure causes theelectron wave functions of the twodonor atoms to overlap, and the elec-tron spins couple together through theelectron-spin exchange interaction.Because each electron is coupled to itsnucleus through the hyperfine interac-tion, turning on the electron-spinexchange interaction also couples thenuclear spins together.

The coupled nuclear-electron spinsystem is fairly complex, but we cangain insight into it by looking at theeffective Hamiltonian for the system:

Hcoupled = H1 + H2 + Jσ1e• σ2e . (4)

This Hamiltonian is valid at energyscales that are small compared with the electron-binding energies of the donor atoms. The first twoterms correspond to the hyperfineHamiltonian—Equation (1)—of eachdonor, respectively, and the last termaccounts for the spin exchange



Figure 2. A-Gate Control of One Qubit We use magnetic resonance techniques to rotate nuclear spins and place them inarbitrary superpositions of |��⟩ and |��⟩ qubit states. (a) In this cartoon, a small volt-age is applied to the A-gate directly above a qubit. The donor electron moves awayfrom the 31P nucleus. (a′) This plot of the electron probability density surrounding adonor atom with no voltage on the A-gate was obtained by solving the Schrödingerequation nonperturbatively in an isotropic effective-mass hydrogenic basis. The plotis a cross section through the nucleus, with the color variations on a logarithmicscale. The atom is buried 20 nm below the Si/SiO2 interface. (b) The graph showsthe variation of the nuclear transition frequency as a function of A-gate voltage.(b′) The color plot shows the electron probability density. A positive voltage on the A-gate pulls the electron away from the nucleus, thus reducing the hyperfinecoupling—the parameter A in Equation (1) in the text. In a B-field of about 2 T, theresonance frequency of a phosphorous nucleus q1 is ν0 = ω0/2π ≈ 93 MHz. With a gatevoltage of 1 V, the resonance frequency of q1 reduces to about ν– = ω–/2π ≈ 90 MHz,while a neighboring nucleus q2 is in resonance at about 87 MHz. (The proximity ofthe oxide barrier has a fairly large effect on the qubits, and the positive gate voltageaffects q2 more than q1.) Subjecting the silicon wafer to a transverse, oscillatorymagnetic field of frequency ν– would cause only q1 to respond. (c)–(c′) Initial calculations indicate that the electron probability density is more responsive to a negative gate bias, which results in better frequency discrimination between adjacent qubits.

95

90

85

80

750 0.5

A-gate potential (V)

Nuc

lear

res

onan

ce

freq

uenc

y (M

Hz)

1.0 1.5

q1

q2

A

q2q1

J A

SiO2 barrier

Silicon substrate

Electroncloud

+++++A-gate, 0 V

0

–5

A-gate, +1 V

0

–5

A-gate, –1 V

0

–50 –0.2 –0.4A-gate potential (V)

Nuc

lear

res

onan

ce

freq

uenc

y (M

Hz)

–0.6 –0.8

q2

q1

–1.0

100

80

60

40

Continued on page 290

(a) (a′)

(b)

(c)

(b′)

(c′)

Spin Manipulation with Magnetic Resonance

Magnetic resonance is the traditional technique for detecting and manipulating any particle, such as an electron, atom, or nucleus, that has a magnetic moment µ.The manipulation is controlled by a combination of static and oscillating magnetic fields. Classically, a particle’s magnetic moment is proportional to itsangular momentum J through the relation

µ = q /2m J , (1)

where q is the charge of the particle and m is its mass. Remarkably, a similar relation holds true in quantum mechanics, although we must also take into accountthe particle’s intrinsic spin angular momentum. In general, we can write

µ = γ J , (2)

where the parameter γ is known as the gyromagnetic ratio. It is related to the constants in equation (1) by a dimensionless constant known as the g-factor,

γ = g (q/2m) . (3)

The magnitude and sign of the g-factor depend on the specific atom or nucleus,but are always approximately 1.

In the classical picture of a randomly oriented moment in a magnetic field B0 = B0 z, the moment would like to lower its energy by aligning itself parallel tothe applied field. But the magnetic field can only produce a torque on the moment,� = µ × B0. Because the torque is directed perpendicular to the plane defined bythe field, the moment does not align with the field, but like a spinning gyroscopethat resists the force of gravity, precesses around the magnetic field line. By usingthe fact that the torque is equal to the rate of change of the angular momentum,we can derive the angular precession frequency of the moment (see Figure A):

ωL = γ B0 , (4)

where ωL is called the Larmor frequency and is measured in radians per second.Equation (4) is the single most important equation of magnetic resonance. It saysthat the frequency of precession about a magnetic-field line is proportional to boththe magnitude of the magnetic field and the gyromagnetic ratio. Interestingly, asderived in the equations accompanying the figure, the frequency is independent ofthe angle θ that specifies the orientation of the magnetic moment. The Larmor fre-quency enables us to identify the particle because the gyromagnetic ratio is distinctfor electrons and different nuclei. The Larmor frequency (ω0/2π) for an electron isabout 28 gigahertz per tesla (MHz/T) and for a proton, roughly 45 MHz/T.

The moments precessing in the applied field can be manipulated in several ways.One common method is pulsed magnetic resonance. For a short period, we apply anoscillating magnetic field along the x-axis, B1 = B1 cos(ωosct) x, where B1 << B0.The moment will begin to precess around a time-dependent total magnetic fieldconsisting of B0 plus B1. This complicated motion can be better understood byexamining the moment in a reference frame that rotates around the z-axis with frequency ωosc, the same frequency as B1. In the rotating frame, B1 becomes a staticfield and the precession frequency about the z-axis is reduced: ωL → ωL – ωosc.

∆µµ

sin

µ′ µ

µ

B0

= L∆t

θ

θ

φ

µ

|

| |||

|

×

µ∆


ωeff

x

B0 – ωosc

B1

Beff

–y

z

γ

µ

Figure A. Larmor Precession Magnetic moments precess aroundmagnetic field lines at the Larmor precession frequency ωL, which isderived above.

Figure B. Effective MagneticField in the Rotating Frame The motion of the moment about Beffis easier to describe in the framerotating about the z-axis at the same

frequency ωosc as the oscillating fieldB1, since then B1 is static. B0 isreduced by the amount ωosc/γ.


Phenomenologically speaking, in the rotating frame the magnetic moment “sees”an effective field of magnitude

(5)

which is illustrated in Figure B. Equation (5) tells us that, when the frequency ofthe B1 equals the Larmor frequency, namely, at the resonance condition ωL = ωosc,the effective field has no z-component. Only the B1 field remains, and the momentwill precess around the x-axis at an angular frequency ω1 set by the magnitude ofB1, namely, ω1 = γB1. Thus in the laboratory, we can rotate a moment about the x-axis by setting the frequency of B1 to the Larmor frequency. We control the rateof rotation by adjusting the field strength and the amount of rotation by restrictingthe length of time that the B1 field is applied.

Pulsed magnetic resonance can be used to manipulate a qubit. Suppose a qubit stateis defined by the nuclear spin orientation such that the spin aligned parallel to B0represents the state |�⟩ whereas the spin aligned antiparallel to the field representsthe state |�⟩. We send a current pulse through an inductive coil to create the field B1.If the pulse is timed to last for one-half of a precession period, or t = π/ω1, then the spins will rotate around the x-axis for π radians, or by 180°. If the qubit was initially in the |�⟩ state, it would now be in the |�⟩ state. Similarly, we can pulse thecurrent for a time t = π/(2ω1)—a so-called π/2 pulse—and rotate the qubit into anequal superposition of the |�⟩ and |�⟩ states, namely the state 1/√2 (|�⟩ + |�⟩). (See Figure C.)

We can also make moments rotate continuously about the x-axis. In a processknown as cyclic adiabatic inversion, we sweep ωosc through a range that includesthe Larmor frequency. When we start the sweep, ωosc << ωL. According toEquations (5), there is little cancellation of the static field B0, and Beff will liesubstantially along the z-axis. As the frequency approaches ωL, there is more cancellation, and Beff begins to rotate toward the x-axis. When ωosc = ωL, Beffpoints along the x-axis. Continuing to sweep the frequency to ωosc >> ωL willeventually cause Beff to point along the (–z)-axis. If ωosc is swept slowly enough(the adiabatic condition), the moments will continue to precess around Beff andwill follow its rotation in the x-z plane from +z to –z (See Figure D). Reversingthe sweep will cause Beff to rotate backwards. By continuously sweeping ωoscback and forth through the resonance frequency, we effectively make the spinsrotate continuously around the y-axis.

Cyclic adiabatic inversion provides one of the mechanisms by which we detect elec-tron moments with a magnetic resonance force microscope (MRFM). A small numberof moments are in resonance with B0, B1, and the gradient field produced by the magnetic tip at the end of the MRFM cantilever. We use cyclic adiabatic inversion toselectively rotate those moments, thus producing a tiny oscillating magnetizationwithin the sample that in turn produces an oscillating force on the MRFM cantilever.By adjusting the rate at which we sweep ωosc, we can match the forcing frequency tothe cantilever’s resonant frequency, and even a small number of moments can drivethe cantilever into a detectable oscillation.

Beff =

+

= −( ) +

B z B x

z B x

0 1

11

– ˆ ˆ

ˆ ˆ ,

ωγ

γω ω

osc

L osc


B1x

–y

z

µf

µi

(|�⟩ + |�⟩)1√2

|�⟩

µ

z

x

Beff = B1 for ωosc = ωL

Path of precessing moment as ωosc is swept

Beff for ωosc << ωL

Beff for ωosc >> ωL

Figure D. Cyclic AdiabaticInversionBeff rotates about the y-axis when ωoscis swept through the resonance frequency ωL. If ωosc changes slowly,the moment continues to precessabout Beff and we can rotate themoment about the y-axis.

Figure C. Pulsed MagneticResonanceWhen ωosc is made equal to ωL, amoment will begin to rotate about the x-axis. We place a qubit into an equalsuperposition of logical states by rotat-ing the moment through 90° with a π/2pulse, in which B1 is turned on for atime t = π/(2ω1), where ω1 = γ B1.


interaction. The exchange couplingcoefficient J is proportional to the over-lap between the wave functions of thetwo donor electrons, so its strength is afunction of the J-gate voltage.

We first examine the coupled electron-spin states by ignoring (for a moment) the contribution of thenuclear spins to H1 and H2 inEquation (4). The effect of the spinexchange interaction is to create coupled electron-spin states, threewith total spin S = 1 and one withtotal spin S = 0. The respective wavefunctions are

S = 1 |↑↑⟩ ,1/√2 |↑↓ + ↓↑⟩ , and|↓↓⟩ ,

S = 0 |S⟩ = 1/√2 |↑↓ – ↓↑⟩ .

In the absence of a magnetic field, theenergy difference between the stateswith S = 1 and S = 0 is 4J, an amountknown as the exchange energy. In the

presence of the magnetic field B0 thatpermeates the quantum computer, the|↑↑⟩ and |↓↓⟩ states are Zeeman-splitaround zero by an amount ±2µBB0,and the energies of the coupled elec-tron-spin states vary as a function ofJ, as seen in Figure 3(b). Notice thatthe lowest-energy S = 1 state and theS = 0 state cross when the exchangeenergy becomes equal to the Zeemansplitting, that is, when 4J = 2µBB0.We exploit that crossing in a qubitreadout scheme discussed later.

We now consider the nuclear spinstates. Conceptually, for every cou-pled electron-spin state, there are fourpossible orientations of the twonuclear spins, corresponding to theuncoupled (J = 0) nuclear states |��⟩,|��⟩, |��⟩, and |��⟩. Thus, there are six-teen nuclear spin states in all.Formally, we must use Equation (4)to find the energies and eigenfunc-tions of all sixteen.3 If we focus onlyon those states associated with theelectron ground state |↓↓⟩ and assume

4J< 2µBB0, then in order of decreas-ing energy, the coupled nuclear-spinstates are the following:

|��⟩ ,|Φ+⟩ = 1/√2 |�� + ��⟩ ,|Φ–⟩ = 1/√2 |�� – ��⟩ , and|��⟩ . (5)

The electron-spin exchange inter-action shifts the energy of the |Φ–⟩state below that of |Φ+⟩ by an amount

(6)

where ωJ is the nuclear exchange fre-quency. For B0 = 2T and 4J = 0.124milli-electron-volt (meV), ωJ has avalue of about (2π)75 kilohertz, a fre-quency that approximates the rate atwhich binary operations can be per-formed on the computer.

The spin exchange interaction caus-es the wave functions of Equation (5)to evolve and rotate between spinstates. One possible result is that thenuclear spins undergo a coordinatedswapping of states: |q1q2⟩ → |q2q1⟩(see the box “The Swap” on the facingpage). Thus, the spin exchange interac-tion should automatically implementthe logical two-qubit swap gate.

Of more interest is the cnot gate,which along with single-qubit opera-tions, forms a universal set of gatesfrom which any quantum algorithmcan be executed. In the Kane system,the cnot corresponds to the conditionalrotation of a target spin by 180°, pro-vided the control spin is in the state|�⟩. In principle, it can be realized bysubjecting the wafer to a transverse

∆E

AB J B

J J=

=−

−

hω

,21

2

12

0 0µ µB B



+++++ +++++ +++++A1 J A2

SiO2 barrier

Silicon substrate

(a)

Figure 3. Coupling Two Qubits with a J-Gate(a) When the gates A1 and A2 are appropriately biased, application of a small posi-tive voltage to the J-gate lowers the potential barrier between adjacent donor sitesand turns on an electron-spin exchange interaction between qubits, as seen in thiscartoon. The electrons interact with the nuclei through the hyperfine interaction;hence, the two nuclear spins become coupled to each other. (b) The graph showsenergy levels of the coupled electron-spin system as a function of the electron-spinexchange coefficient J, which can be modified by voltage applied to the J-gate. Theelectrons couple their spins to form three states with S = 1 (shown in blue) and onewith S = 0 (shown in red). For J/µBB0 < 0.5, the electrons occupy the lowest energyS = 1 state |↓↓⟩. Two-qubit operations are performed in this low J regime.

4

2

0

–2

–4

0 0.5 1.0

|↑↑⟩

|↓↓⟩

|↑↓ + ↑↓⟩1√2

|↑↓ – ↑↓⟩1√2

J/µBB0

E/µ

BB

0

4J

2µBB0

(b)

3 The energy differences between the fournuclear states associated with each electronstate are very small. A graph of the nuclear-electron energy levels would look identical to Figure 3(b), except that under high magnifica-tion one would see that each line consists offour closely spaced lines.

Continued from page 287

magnetic field B1 and applying voltages to the A- and J-gates (Goanand Milburn 2000).

Suppose that the two electron spinsare initially uncoupled (J = 0) and thatthe hyperfine coupling constants A1and A2 of the two donor atoms areequal (A1 = A2). In that case, biasingthe A-gates such that A1 > A2 distin-guishes the control qubit from the tar-get. We then turn on the spin exchangeinteraction (J > 0) and slowly make A1 equal to A2. The result would bethat the uncoupled qubit state |��⟩evolves adiabatically into the state|Φ+⟩ = 1/√2 |�� + ��⟩, and |��⟩ evolvesinto |Φ–⟩ = 1/√2 |�� – ��⟩. When A1 equals A2, the energy splittingbetween the two states is given byEquation (6). The states |��⟩ and |��⟩are unaffected by the procedure.

We next apply a linearly polarizedoscillating field B1 with frequencythat is resonant with the |��⟩ – |Φ+⟩energy difference. The field is left onuntil the |��⟩ state has rotated into the|Φ+⟩ state and vice versa. By execut-ing a reverse of the sequence of stepsperformed at the beginning of theoperation, we adiabatically transformthe |Φ+⟩ and |Φ–⟩ states back into |��⟩and |��⟩, respectively. We effect thechange

|��⟩ → |��⟩ ,|��⟩ → |��⟩ ,|��⟩ → |��⟩ , and |��⟩ → |��⟩ .

That is, the only qubits that areflipped are those in which the controlqubit is in the state |�⟩. Thus,we expect to be able to perform thecnot operation.

Approaches to Readout

One can evaluate the result of aquantum computation only by readingthe final state, |�⟩ or |�⟩, of a qubit.Likewise, the ability to determine the

state of a given qubit is critical to ini-tializing the quantum register. Ideally,we would read the qubit state directlyby measuring the donor atom’s nuclear-spin state. But direct detection of a sin-gle nuclear spin is currently impossibleand may forever be out of our grasp.(The strength of the coupling between amagnetic field and the nuclear spin isset by the magnitude of the nuclearmagneton µn, which is very small.) Weare therefore forced to find some othermeans of reading out the qubit state.

The potential solution is to correlatethe nuclear spin states of a target atomwith the electron spin and to find someway of determining the electron spinstate. We are currently pursuing twodistinct detection schemes, one involv-ing a single electron transistor (SET)and the other, a magnetic resonanceforce microscope (MRFM). Bothapproaches require that we push the respective technologies beyond thecurrent state of the art.

Single-Charge MeasurementUsing SETs. The idea behind thistechnique, first described by Kane(1998), is to couple the target qubit qtto a readout qubit qr by a J-gate, andthen infer the state of qt by monitor-ing the donor electrons of the coupledsystem. If qt is in the state |�⟩, we cancause both electrons to become local-ized around the readout atom (theywould occupy the so-called D– state).If qt is in the state |�⟩, each donorelectron would remain bound to itsrespective atom. An SET would beused as an ultrasensitive electrometerto determine whether one or two elec-trons were bound to the readout atom.

The procedure can be understoodwith reference to Figure 4(a), whichshows the coupled nuclear-spin statesin the vicinity of the electron spincrossing. As discussed in the previoussection, for J/µBB0 < 0.5, the lowest-energy electron spin state is the S = 1state |T⟩ = |↓↓⟩, but for J/µBB0 > 0.5,



The Swap

Before the J-gate is turned on, the two nuclear spins are uncoupled, and eachis described by the following energy eigenstates: |Ψ1⟩ = |��⟩, |Ψ2⟩ = |��⟩,|Ψ3⟩ = |��⟩, and |Ψ4⟩ = |��⟩. Once the J-gate is turned on, the coupled eigen-states are |��⟩, |Φ–⟩ = 1/√2 |�� – ��⟩, |Φ+⟩ = 1/√2 |�� + ��⟩, and |��⟩.

Suppose the uncoupled nuclear spins were originally in the state |Ψ2⟩ = |��⟩,and then voltage was applied quickly to the J-gate. In terms of the eigenstatesof the coupled system, the system finds itself in the state

|Ψ2⟩ = 1/√2 (|Φ+⟩ – |Φ–⟩) . (1)

The time evolution of this wave function (up to an overall phase) is given by

|Ψ2(t)⟩ = 1/√2 (|Φ+⟩ – e–iωJt |Φ–⟩) , (2)

where ωJ is the nuclear exchange frequency. After a time t = π/ωJ, the wavefunction will evolve into

|Ψ2(π/ωJ)⟩ = 1/√2 (|Φ+⟩ + |Φ–⟩) = |Ψ3⟩ . (3)

That is, the system will have evolved from the state |��⟩ to the state |��⟩. The spins will have swapped. If we quickly remove the voltage from the J-gate, the two-spin system will remain in the state |��⟩.

the S = 0 state |S⟩ = 1/√2 |↑↓ – ↓↑⟩assumes the lower energy.

Figure 4(a) shows what happens tothe eight lowest-energy nuclear-spinstates as the electron-spin states cross.Focusing on the four states initiallyassociated with |T⟩, we see that afterthe crossing, the two higher-energynuclear states |��⟩ and |Φ+⟩ remaincoupled to |T⟩, while the two lower-energy states |��⟩ and |Φ–⟩ get cou-pled to |S⟩. In other words, as weincrease J, we can adiabaticallyevolve both the nuclear- and electron-spin systems. If the target qubit wasoriginally in the state |�⟩, then regard-less of the state of the readout qubit,the electrons will evolve into the S = 0spin state. If qt is originally in thestate |�⟩, the electrons will remain inthe lowest energy S = 1 spin state. The sequence of steps, similar tothose used to implement the cnotgate, is outlined in Figure 4(b).

We next use the fact that the onlytwo-electron bound state of a phospho-rous atom in silicon is the D– statewith total spin S = 0. As seen inFigure 4(c), we would bias the A- andJ-gates to create an electric fieldbetween the two donor atoms. If theelectrons are in the S = 0 state, the tar-get electron can transfer to the readoutatom, and we would know that the tar-get atom was initially in the state |�⟩.

An SET would be used to detectthe presence of the second donor elec-tron about the readout atom. In manyways, an SET is like an ordinary tran-sistor, in that a gate electrode moder-ates the current flowing between asource and drain electrodes. The dif-ference is that between the SET’ssource and drain lies an extremelysmall metallic island, which is isolatedfrom each electrode by small patchesof insulating material. The insulatoracts as a tunnel junction. For currentto flow, electrons must tunnel from the source to the island and then fromthe island to the drain. The tunnelingcurrent is greatly affected by the



|T ⟩

|T ⟩

|��⟩

|��⟩

|��⟩

|��⟩

|��⟩

|��⟩

0.4965

–1.492

–1.497

–1.502

–1.507

–1.5120.4975 0.4985 0.4995 0.5005 0.5015 0.5025

|��⟩

|��⟩

| –⟩

| –⟩

|S⟩

|S⟩

| +⟩

| +⟩

|T ⟩ |��⟩

|T ⟩ |��⟩

|T ⟩ |��⟩

|T ⟩ |��⟩

|T ⟩ |��⟩

|T ⟩ | +⟩

|T ⟩ | –⟩

|T ⟩ |��⟩

|T ⟩ |��⟩

|T ⟩ | +⟩

|S⟩|��⟩

|S⟩| –⟩

|S⟩ = 1/√2 |↑↓ – ↓↑⟩ |T ⟩ = |↓↓⟩

Electron States

S = 1

S = 0

Nuclear States

(b) States Adiabatically Evolved through the Crossing

(a)

| –⟩ = 1/√2 |�� – ��⟩ | +⟩ = 1/√2 |�� + ��⟩

J/µBB0

E/µ

BB

0

qt initially in state |�⟩

qt initially in state |�⟩

J/µBB0 < 0.5, At – Ar >> hωJ

J/µBB0 < 0.5, At = Ar

J/µBB0 > 0.5, At = Ar

ΦΦ

ΦΦ

| –⟩

| –⟩

| +⟩

| +⟩ΦΦ

Φ

Φ

Φ

Φ

Φ

Φ

Φ

Φ

SETSiO2 barrier

SET

Ar ArJJ J

qrqt = |�⟩ qt = |�⟩qr

JAt At

SiO2 barrier

(d) No Tunneling (qt = |�⟩) (c) Tunneling (qt = |�⟩)

+ ++ ++++ ++ +++

qr

E-field E-field

Figure 4. Single-Electron Transistor (SET) Readout Scheme(a) The graph shows the eight lowest-energy nuclear-spin states for the coupledtarget and readout qubits |qt, qr⟩ in the region where the S = 0 and the lowestenergy S = 1 electron-spin states cross. (b) We can adiabatically evolve thenuclear-electron states by biasing the J- and A-gates, as seen in this (partial)sequence of steps. The electrons are initially in the S = 1 state |T ⟩. If qt was ini-tially in the |��⟩ state, then the electrons will remain in |T ⟩ regardless of the stateof qr. If initially qt = |��⟩, then at the end of the sequence, the electrons will be inthe S = 0 state |SS⟩. (c) Only the two electrons in the |SS⟩ state can bind to a singlephosphorous atom in silicon. Given a suitable biasing of the gate electrodes, wecan try to induce an electron to tunnel to a readout qubit qr. If the tunneling issuccessful, the electrons were in the |SS⟩ state, and qt = |��⟩. The tunneling currentwould be detected by an SET located near qr. (d) If no tunneling occurs, the twoelectrons were in the |T ⟩ state, and hence qt = |��⟩.

capacitive coupling between the gateand the island. This means that forparticular voltage biases on the gate,source, and drain, current flowthrough the SET becomes exquisitelysensitive to minute changes in thecharge distribution of the local envi-ronment. The presence of a singleadditional electron is readilydetectable as a change in the SET’ssource/drain conductance.

We have developed several readoutsimulation devices to test the proper-ties of our SETs built in house. In thedevice seen in Figure 5, two thinmetal bars, isolated from each otherby a tunnel junction, substitute for thephosphorous atoms. Control gates areused to electrostatically “push” singleelectrons from one bar to the next.The two SETs are then used to detectthe change in the charge distribution

due to the discrete, single-electrontunneling events. Those events causethe output of both SETs to changeabruptly. In contrast, signals due tounwanted charge noise (reproduciblefluctuations in the conductance versusvoltage curve) tend not to affect bothSETs simultaneously. By correlatingthe outputs of the two SETs, we areable to clearly identify the single-charge transfer events and reject



Figure 5. Twin SET Device forReadout Simulation(a) The figure shows one half of a twinSET/ double-bar test device. The SETconsists of a small metal island con-nected to source and drain leads bytunnel junctions and a gate electrodethat is capacitively coupled to theisland. Electrons can pass from sourceto drain only by tunneling through bothjunctions. The SET is located next to ametal bar B2, which is isolated from theother bar B1 by a tunnel junction.(b) The tunneling current Isd in the SETis strongly influenced by a change inthe local charge distribution. If the gatevoltage is originally biased at V0, (bluedot), then a change in the local chargedistribution effectively modifies it to V0– δ, and the source-drain current willchange dramatically (red dot). (c) Thisis an image of the twin-SET test deviceobtained with a scanning electronmicroscope. The image to the right is amagnified version of the central region.The twin-SET device is fabricated by adouble-angle evaporation process,which replicates each of the features.Unequal voltage on A1 and A2 causesan electron to tunnel from one bar tothe next. (d) The movement of chargeis detected as a change in thesource/drain conductance in bothSETs simultaneously. The two signalscan be correlated to discriminate thecharge transfer signal from repro-ducible charge noise or from randomnoise events.

Source

SET Double bar

Drain

Gate A2

B2

Island

Tunnel junction

40

20

00 0.02 0.04

Gate voltage (V)I s

d (p

A)

(a) (b)

0.015

0.005

0.010

0 0.025

G′left SET × G′right SET

0.05Gate voltage (V)

0.075 0.10

Con

duct

ance

, G (

e2 /h)

Cor

rela

tion

(a. u

.)

Left SET Charge noise

Right SET

(c)

(d)

spurious signals that would interferewith the readout.

Other factors, however, also needto be considered before we use anSET in a qubit readout scheme.Suppose the target qubit qt is initiallyin the |�⟩ state. Then, for high valuesof J, the coupled electrons will remainin the higher-energy state |T⟩ (refer toFigure 4). This means that the cou-pled atomic system could lower itsenergy if one of the polarized elec-trons “relaxed” and flipped to formthe state |S⟩. The electron would thentransfer to the readout qubit, and wewould erroneously deduce that qt wasinitially in the |�⟩ state! Recent resultssuggest that the spin relaxation time isof the order of milliseconds. We musttherefore pull information out of theSET on an even shorter time scale.We must be able to determine that achange occurred in the SET conduc-tance at a time t0, rather than a fewmilliseconds after t0.

Unfortunately, that is difficult to dowith an ordinary SET. The measure-ments are made at liquid helium tem-peratures, and the SET, sitting in acryostat, must somehow be connectedto the outside world. The capacitanceof the connecting cables is fairlylarge, and when combined with theintrinsic resistance of the SET,produces a resistance-capacitance (or R-C) time constant for the devicethat is longer than the spin relaxationtime. Information about the SET con-ductance takes too long to propagateto the outside world.

The solution to this problem is todevelop a fast readout SET(Schoelkopf et al. 1998). Known as anrf SET, it is basically an ordinary SETcoupled to an impedance-matchingcircuit that negates the effects of theexternal capacitance. We have recentlydeveloped a very sensitive reflection-mode rf SET that operates at430 megahertz. It can detect themovement of a single electron in thedevice shown in Figure 5 in about

1 microsecond. For a system contain-ing discrete phosphorous atoms, thereadout time would likely increase toabout 100 microseconds, but that isstill sufficient for the readoutapproach discussed in this section.

The MRFM. The second approachto readout is to perform a direct meas-urement of the spin state of the elec-tron surrounding the qubit and therebyinfer the qubit state. To do so, we aredeveloping an MRFM, which com-bines the versatility and chemicalspecificity of magnetic resonance withthe high-resolution and ultrahigh sen-sitivity of an atomic force microscope(AFM). The key feature of the MRFMis that only spins contained within adefined area in the sample—the so-called sensitive slice—contribute to the detected signal. Because thelocation and size of that slice can be controlled, there is selective sensi-tivity to deeply buried structures.

Our MRFM, developed at LosAlamos in collaboration with MichaelRoukes of Caltech, is illustrated inFigure 6. The microscopic, sharp-pointed magnetic tip is bonded to theend of a tiny cantilever. As in an ordi-nary AFM, the tip is scanned over asample, and signals are recorded atevery point. In our instrument, howev-er, the magnetic field from the tip B(r)interacts with all the electron spins inthe substrate through the magnetic gra-dient force, F(r) = (m•∇)B(r), wherem is the net magnetization of thespins. Depending on the spin orienta-tion, the force on the tip is eitherrepulsive or attractive. The net orienta-tion of the electron spins in the sam-ple, therefore, causes a tiny deflectionof the cantilever.

We interact with only a subset ofthe spins through magnetic resonance.The sample is immersed in a staticmagnetic field B0 = B0z, so the pre-cession frequency of the spins aroundthe magnetic-field lines is proportion-al to B0 plus the z-component of B(r),

that is, the total magnetic field in thez-direction. Magnetic resonancecomes into play when we subject thespins to an oscillating magnetic fieldB1 that is aligned in the x-direction.Because the magnitude of B(r)decreases rapidly with distance, onlythose spins that are at the correct dis-tance from the tip are in resonancewith the B1 field. Spins that are tooclose to the tip will have a higher res-onant frequency; those that are fartheraway, a lower frequency. Thus, for agiven field gradient and a fixed can-tilever position, a resonance frequencybecomes correlated with positionsinside the sample. With reference toFigure 6, all spins that lie within asmall, hemispherical shell beneath thetip (the sensitive slice) have the sameresonance frequency.

To detect those spins, we use thetechnique of cyclic adiabatic inver-sion, discussed in the box “SpinManipulation with MagneticResonance” on page 288. In essence,we continuously rotate the selectedspins at the resonant frequency of thecantilever. The continuous up anddown reorientation of the spins createsan oscillating force on the tip thatamplifies the cantilever’s natural up-down motion. The situation isanalogous to pushing a child’s swingat its natural frequency of oscillation:with each push, the amplitude of themotion becomes larger. After thou-sands of spin rotations, the amplitudeof the cantilever’s up-down motionhas increased by about an angstrom,which is large enough to be detectedwith a fiberoptic interferometer. The fiber sits slightly above the backof the cantilever, and laser light sentdown the fiber interferes with itself as it reflects from both the cantileverand the fiber’s end. By monitoringchanges in the interference pattern,we can detect the oscillations.

The orientation of the nuclear spinscan be inferred from the frequency atwhich the electron spin resonance



occurs. Because of the hyperfineinteraction, the resonance frequencyof an electron spin flip depends on thenuclear-spin state. Considering thehyperfine states of a single qubit,the |�↓⟩ to |�↑⟩ transition has a differ-ent energy than the |�↓⟩ to |�↑⟩ transi-tion, and thus there are two resonancefrequencies for an electron spin transition. Measurement of, say, thehigher resonance frequency wouldcorrespond to the nuclei in the samplebeing aligned with the B0 field.

The discussion so far has centeredon detecting many nuclear spins, butto read out the result of a quantumcomputation, we need to measure asingle nuclear spin. That such meas-urement is at all possible is due to theexceedingly high spatial resolution ofthe MRFM, which is determined bythe thickness ∆z of the hemisphericalshell. The thickness is inversely pro-portional to the magnitude of the fieldgradient:

(7)

where γ is the gyromagnetic ratio and∆ωr is the linewidth of the resonantelectron-spin transition that is beingdriven by the MRFM. For phosphorusatoms in silicon, ∆ωr/γ is on the orderof 1 milligauss. A field gradient ofabout 105 tesla per meter (T/m) willthen produce a thickness that is muchless than 1 angstrom, even when thehemispherical shell extends severalhundred angstroms beneath the sub-strate surface. In that case, the sensi-tive slice would be so thin that only asingle donor electron would be in res-onance with the MRFM probe.

We have conducted numerousexperiments to measure the field gra-dient of our specialized magnetic tips.With the tip about 2 micrometers fromthe surface, we have measured a fieldgradient approaching 104 T/m (seeFigure 7). From this value, we esti-mated ∆z and the volume of our hemi-

spherical shell. Then, knowing thespin density of the sample, we esti-mated the number of spins that con-tribute to the signal. For the datashown in Figure 7, the number isbetween one thousand and ten thou-sand electron spins.

Because the field gradient increases nearer to the tip, sensitivityshould be greater if the tip is closerto the surface. But mechanical andthermal noise also deflect the tip andcantilever. As we begin to interactwith fewer spins, the “signal” forcedue to spins eventually becomes less than the “noise” force due to

unwanted sources. By equatingexpressions for the signal force to thenoise force, we can derive an expres-sion for the minimum detectablemagnetic moment, mmin, needed togive a signal to noise of 1:

(8)

In Equation (8), kB is the Boltzmannconstant; T, the temperature; and ∆ν,the detection bandwidth. The otherparameters describe the cantilever: itsforce constant k, resonant frequency f,and quality factor Q. The three key

mkk T

Qfmin( )

.=∇

1 2

B rB

c

∆νπz

∆∆

z ,r

z

≅∇

ωγ B r( )

295Los Alamos Science Number 27 2002

F

B1cos( 1t ) Resonant frequency 1 = γ [B0 + B(r)]ω

ω

Sensitive slice

Magnetic tip, B(r)

B0

r

Figure 6. The Magnetic ResonanceForce Microscope (MRFM)(a) The MRFM can, in principle, deduce thestate (up or down) of a single nuclear spin.The entire device sits at approximately liquidhelium temperatures in a static magnetic fieldB0. The tip at the end of the cantilever is coated with a magnetic material that generatesa magnetic field B(r) that changes rapidly with the distance r. The interactionbetween the electron spins in the sample and the magnetic field gradient due to B(r)produces a force that deflects the cantilever. We interact with only a small subset ofspins, located within a hemispherical shell of radius r1, by subjecting the sample toan oscillating magnetic field B1cos(ω1t), where ω1 = γ[B0+B(r1)]. By using the tech-nique of cyclic adiabatic inversion, we can cause the spins to oscillate between theup and down states at the cantilever resonance frequency, thus driving the can-tilever into measurable oscillation. We detect the oscillation with an optical device.The electron-resonance frequency can then be correlated with a nuclear spin orien-tation. (b) The MRFM tip assembly and sample mount are shown in this photo.The vertical tube is a piezo scanning tube, which moves the tip over the sample,while the circular feature is the induction coil that produces B1. The white box high-lights the magnetically coated tip, shown under high magnification in (c).


(a) (b)

(c)

10 µm

parameters that we can optimize arethe field gradient, temperature, andquality factor.

We believe that the sensitivity ofthe MRFM is currently limited bysurface contamination on the sample.As the tip approaches the surface, thecontamination acts like a viscousforce that damps the oscillatorymotion—that is, it lowers the Q inEquation (8). To solve this problem,we are upgrading the equipment sothat the sample be transferred from asurface preparation chamber into themicroscope without leaving the ultra-high vacuum environment. The systemwill also be cooled to temperaturesbetween 250 and 300 millikelvins in a helium-3 dilution refrigerator, atechnique that is compatible withmaintaining the sample under ultrahigh vacuum.

Detection of a single electronmoment requires that

mmin = 1 µB ≅ 10–23 joule/tesla . (9)

Given a field gradient of 105 T/m,the signal force on the cantilever is

approximately 10–18 newton (theweight of approximately two millionphosphorus atoms). We believe that an upgraded, low-temperaturemicroscope will allow us to observethe magnetic resonance signal of asingle electron spin.

SSQC Fabrication Progress

Implementing our quantum-computing scheme requires that weproduce a very regular array of phos-phorus atoms in pure silicon, inwhich each donor is located preciselybeneath a metal A-gate on the sur-face. The spacing between adjacentphosphorus donors is chosen toensure that the electron-spin exchangeinteraction is minimal when there isno voltage on the J-gate lyingbetween the donors. We want the twoelectron wave functions to overlap,but only slightly. Calculations (Goanand Milburn 2000) indicate that aseparation of 10 to 20 nanometersbetween donors is required.

A nominal donor spacing of

20 nanometers translates into gatestructures that are less than10 nanometers in width. Fabricating ahighly regular metal array on thatscale, even with state-of-the-art techniques, is at the limits of the electron-beam techniques used inmaking conventional electronics. That problem, however, pales whencompared with the difficulties we facein making a precisely aligned array of phosphorous donors that is buriedunder layers of silicon. The difficultieshave led us to pursue two differentfabrication strategies, known as thetop-down and bottom-up approaches.

In the top-down approach, phos-phorous atoms will be implanted byion bombardment into specific siteson the silicon wafer. Because the ionscatters as it slows down in the sili-con, we will not know the exact loca-tion of the donors, only that they willlie within close range of the definedimplantation area. The top-downapproach provides a rapid means todemonstrate proof of principle andallows us to fabricate a two-donordevice that can be used to test readout


d = 2.1µm

2.1µm

3380 3420B0 (G)

3460

Sig

nal (

a. u

.)

~104µB

3.2 µm

5.5 µm7.0 µm

8.7 µm

12.3 µm

13.9 µm

(a)

100

10

10

1

0.1

0.011 10

Distance (µm)

Probe fieldSlice width

Fie

ld (

G)

Slic

e w

idth

(µ

m)

(b)

(a) If the magnetic tip is kept at a fixed distance d from thesample, then lowering the value of the magnetic field corre-sponds to sweeping the sensitive slice upwards, toward thesurface. At some point, the slice leaves the sample, and theresonance condition changes dramatically. That change isseen in the derivative of the MRFM signal as a dip, indicated

by the arrows. (b) By following the dip as a function of tipheight, we can measure the tip’s magnetic field. From the fieldgradient, we then calculate the width of the sensitive sliceusing Equation (7) in the text. Knowing the spin density withinthe sample, we use the slice width to deduce how many spinsproduced the signal and thereby infer the MRFM sensitivity.

Figure 7. Sensitivity of the MRFM

strategies and, possibly, quantumoperations. Scaling this approach tolarge numbers of qubits will be challenging, because of the irregularspacing of the donor array.

In contrast, the bottom-upapproach will use a scanning tunnel-ing microscope (STM) with which toplace phosphorous atoms on a cleansilicon surface in a precisely arrangedarray. The array will then be over-grown with silicon, and the gate struc-tures will be laid down by electronbeam lithography (EBL). Thisapproach, although more difficult toimplement, could in principle allow usto build a Kane-type computer withthe required precision. The bottom-upapproach is not discussed here but isdescribed in detail in the article“Fabricating a Qubit Array with aScanning Tunneling Microscope”on page 302.

Top-Down Approach forCreating a Two-Donor Device. Ahost of issues surrounds the operationand readout of the nuclear-spin quan-tum computer. A key concern involvesthe transfer of the electron from thetarget to the readout atom during read-out. The two electrons in the D– stateare not bound very strongly to thephosphorous atom, and the electronmay be lost during the transfer. Theinitial phosphorus-phosphorus (P-P)system would transform into aP-P+ system. If that is the case, wemay need to use extra electrodes inorder to create a deeper potential thatwill confine the electron or to employa different readout atom (such as tel-lurium) that has a more stronglybound two-electron state.

Our current goal for the top-downapproach is to produce a device thatcan be used to study the controlledelectron transfer between two donors.We intend to ionize one of the twophosphorus atoms and then study thecoherent transfer of the remainingelectron between the two donor atoms

in the P-P+ system. For this purpose,we have relaxed the stringent con-straints of the Kane computer archi-tecture and designed the simple deviceshown in Figure 8. It consists of twoA-gates separated by a single J-gate.Two phosphorous atoms will be

implanted between the A- and J-gates,an arrangement that is sufficient forcharge transfer experiments.

Device fabrication starts with awafer that is already topped with abarrier layer of SiO2. To depositmetal A- and J-gates on the surface



Figure 8. Fabrication of a Two-Donor Device(a) For proof-of-principle experiments involving the transfer of an electron betweentwo phosphorus donors, it is not necessary to configure A-gates above the donors.Instead, we are configuring the three-gate device shown in side view in this illustra-tion. (b) The top view schematic shows that the donors will be implanted betweenthe A- and J-gates (red circles). The single horizontal line at the top of the gates rep-resents an opening in a polymer resist layer. Ions can enter the substrate onlywhere the line crosses the gates. The schematic also shows representative SETdevices located near the implantation sites. (c) AFM image of an actual device priorto implantation. The narrow horizontal line near the end of the gates is a 20-nm-wideopening in the resist. (d) An SEM image of a fabricated metal-gate array is shown.The large metal structures on either side of the gates are aluminum electrodes usedto detect the impact of ions during implantation. (e) This magnified view shows thecentral A-, J-, and A-gates. We have fabricated gate arrays with J-gate widths of lessthan 15 nm and gate separations down to 30 nm. The image in (f) shows a J-gatemade from a titanium/gold alloy that is only 12 nm wide.

(a) Two-Donor Test Device

1 µm

12 nm

J A

(d) Fabricated Test Device

(b) Top View Schematic

(c) Two-Donor Test Device

(e) Magnified View

Site 1 Site 2

(f ) J-Gate

before ion implantation, we use EBLtechniques. A new layer of resist iscreated that covers the entire surface,including the gates. A second EBLexposure then patterns a thin lineacross the gates. This pattern isdeveloped so that two tiny channels,each approximately 15 × 30 nanome-ters, are created on each side of theJ-gate. The channels extend down toclean SiO2 and define the implanta-tion sites.

Next, we bombard the wafer withphosphorous ions. Although most ofthe ions are stopped in the mask,some go through the channels, strikethe wafer, and get implanted about10 nanometers below the Si/SiO2interface. After implantation, thedevice is heated to between 900°Cand 950°C to anneal any damage tothe silicon lattice. As a final step, welay down the SETs. Creating the SETs

after the anneal (instead of makingthem in the same step as the controlgates) protects their fragile tunnel bar-riers, which would likely be degradedshould they be submitted to tempera-tures above 900°C.

Because we want only one phos-phorus atom per implantation site, thekey to this entire process is the abilityto detect a single ion after it hasstruck the silicon. And it is the proper-ties of the silicon itself that help usfulfill this task. The energetic phos-phorous ion produces a cascade ofelectron-hole pairs as it slows downand comes to a stop in the siliconmatrix. Those charge species can beseparated by an applied electric field,accelerated, and detected as a currentpulse in an external circuit (seeFigure 9). Voltage applied to surfaceelectrodes straddling the implantationsites produces the field and transmits

the pulses. The intrinsic silicon substrate makes this in situ particle-detection system highly efficientbecause the accelerating electric field extends fully between the twoelectrodes. We have demonstrateddetection efficiencies of over 99 per-cent. Unfortunately, we cannot tellwhere the ion falls, and as there aretwo holes, there is only a 50 percentchance of creating a two-donor devicewith one donor in each hole. Althoughthere are ways to improve those odds(by masking all but one hole with aspecialized AFM cantilever, for exam-ple), in the short term, a 50 percentsuccess rate is acceptable.

So far, no one has come close toseeing the transfer of a single electronbetween two precisely located,nanofabricated donor atoms. We hopeto do so with a device similar to theone described above by September orOctober of 2002. We would then be ina position to study the coherent trans-fer of the electron between two donoratoms and possibly obtain informationon decoherence mechanisms of rele-vance to spin readout.

Unlike the simple test deviceshown in Figure 8, an ideally config-ured device would have the A-gatesdirectly above the phosphorousdonors. We have designed a process to fabricate such a device (McKinnonet al. 2001). A multilayer electron-beam-sensitive resist is deposited ontop of a SiO2-coated silicon wafer,the resist is partially developed, and alinear array of ion-implantation chan-nels is patterned in the resist with EBLtechniques. The wafer is bombardedwith phosphorous ions. The resist isthen fully developed, and triple-anglemetal evaporation is used to depositthe metal gate array and the SETs onthe SiO2 surface. Because only onemask is used to define the location ofboth the ion implantation sites and thegates, the A-gates are automaticallyregistered over the qubits.

Figure 10 shows a six-donor test



Phosphorus ions

Detectorelectronics

Single-ionstrike

Silica

– +

A

E

A AlAl J

Figure 9. Detection of a Single-Ion ImpactWe can detect in situ the impact of a single phosphorous ion in the silicon wafer.Aluminum electrodes are deposited on either side of the implantation site. An elec-tric field applied between the two electrodes is used to separate electron-hole pairsthat are created by ion impact. Each type of charge carrier migrates to its respectiveelectrode and produces a current pulse that is detected by an external circuit.The implantation is halted after two such pulses have been detected.

device made by triple-angle evapora-tion. No ions were implanted into thisdevice, and as can be seen in the fig-ure, the triple-angle process does notyet result in gates that are sufficientlynarrow to allow implementing theKane scheme. We also have to addressthe problem of maintaining theintegrity of the SET through the high-temperature anneal. However, the fundamental idea is robust.

The precision of the donor arraysproduced by the top-down approachwill be limited by straggling, which isinherent to ion implantation, and bythe diffusion of dopants during theannealing step. Recent calculationsindicate that small irregularities in theion array could impair the operation

of the quantum computer. That is whywe are also pursuing the bottom-upfabrication approach, which mightlead to a device with a very regular,well-characterized donor array.

Concluding Remarks

Phosphorous in silicon is a veryclean, well-understood solid-state sys-tem. In its turn, NMR is a very wellunderstood nuclear-spin manipulationtechnique. Performing NMR on a sili-con chip implanted with phosphoruscan therefore make for a very power-ful quantum computer.

But the creation of a silicon-basedsolid-state computer presents such an

enormous technical challenge that wemust explore several strategies forbuilding and implementing almostevery aspect of the device. Hence, weinvestigate both SETs and magneticresonance force microscopy as ameans to read out the qubit state.Similarly, we have pursued two com-plementary fabrication strategies: thetop-down process, which uses indus-trial production techniques, such asion implantation and EBL, to producea few-qubit device, and the bottom-upprocess, which involves advancedSTM techniques and conventionalmolecular-beam epitaxy. Although thebottom-up approach is less suited tohigh-throughput production, it has the potential of leading to large, high-ly regular qubit arrays. We have madesignificant progress along all theseparallel development paths.

Currently, scaling up a solid-statecomputer to over a million qubits is agoal that appears so distant as to benearly out of sight. Yet less than fiftyyears ago, computer companiesattempting to reduce the size of theirmachines were just becoming awareof a new strategy known as integratedcircuits. Those early chips were crudeand contained but a few transistors,but from them, evolved the modern,densely packed integrated circuits oftoday. Like those early chips, thequantum devices developed so far arerudimentary. No doubt, the challengeswe face in building a real silicon-based quantum computer are signifi-cant, but our initial results offer hopethat large-scale quantum computingmay one day be realized. �

Acknowledgments

Building a silicon-based solid-statequantum computer is a difficult andchallenging project that cannot beachieved without the efforts of a largeresearch team. At the University ofNew South Wales, Tilo Buehler,



Figure 10. Creating a Multidonor DeviceWe have developed a process for creating gates directly over implanted ions.(a) In separate steps, patterns 1 and 2 of this EBL mask are partially developed in atrilayer resist that coats the substrate. The resist then sustains a series of lines withtiny channels that extend down to the substrate, where patterns 1 and 2 intersect.(b) A cross section of the resist after partial development shows the channels.The wafer is now bombarded with phosphorus ions. Some ions make it to the sili-con surface and are implanted 5–10 nm below the Si/SiO2 interface. (c) The resist isfully developed and material is removed, leaving behind a large cavity between theSiO2 surface and the self-supporting top layer. Triple-angle shadow evaporation isthen used to lay down an array of metal gates on the SiO2 surface. (d) This photoshows a potential six-donor device with two readout SETs. No ions were implantedinto this device. (e) A schematic side view of the device reveals the metalization pattern that results from the triple-angle shadow evaporation process.

(a) EBL Mask Pattern 2Pattern 1(b) Ion Implantation 31P+ ions

(c) Multiangle Shadow Evaporation (d) Multiangle Shadow Evaporation

SETSET

X

AJAJAJAJAA J

X

(e) Metalization Pattern along the Line XY

Rolf Brenner, David Reilly,Bob Starrett, and Dave Barber have worked in the development ofSET and rf SET readout schemes.Similarly, important contributions tothe top-down fabrication approachhave been made by Fay Stanley,Eric Gauja, Nancy Lumpkin,Rita McKinnon, Mladen Mitic,Linda Macks, and Victor Chan. At the University of Melbourne,Cameron Wellard has supported thedevice modeling; Changyi Yang,Paul Spizzirri, and Robert Short haveworked on single-ion implantationtechnologies; and Jeff McCallum andSteven Prawer have contributed in arange of areas. The MagneticResonance Force Microscope researchwould not have been possible withoutthe contributions of Denis Pelekhov,Michael Roukes, Andreas Suter, andZhenyong Zhang.

Further Reading

Feher, G. 1959. Electron Spin ResonanceExperiments on Donors in Silicon. Phys. Rev. 114 (5): 1219.

Goan, H., and G. Millburn. 2000 (February).Silicon-Based Electron-Mediated NuclearSpin Quantum Computer. CQCT, Universityof Queensland internal report.

Gordon, J. P., and K. D., Bowers. 1958.Microwave Spin Echoes from DonorElectrons in Silicon. Phys. Rev. Lett. 1: 368.

Honig, A., and E. Stupp. 1960. Electron Spin-Lattice Relaxation in Phosphorus-DopedSilicon. Phys. Rev. 117 (1): 69.

Kane, B. E. 1998. A Silicon-Based NuclearSpin Quantum Computer. Nature 393: 133.

McKinnon, R. P., F. E. Stanley, T. M. Buehler,E. Gauja, K. Peceros, L. D. Macks,M. Mitic et al. 2002. NanofabricationProcesses for Single-Ion Implantation ofSilicon Quantum Computer Devices. (To bepublished in Smart Mater. Structures.)

Mozyrsky, D, Sh. Kogan, and G. P. Berman.2001. Time Scales of Phonon InducedDecoherence of Semiconductor Spin Qubits.[Online]: http://eprints.lanl.gov (cond-mat/0112135).

Schoelkopf, R. J., P. Wahlgren, A. A.Kozhevnikov, P. Delsing, and D. E. Prober,1998. The Radio-Frequency Single-ElectronTransistor (RF-SET): A Fast and Ultra-sensitive Electrometer. Science, 280: 1238.



P. Chris Hammel received his B.A. from the University ofCalifornia at San Diego and his Ph.D. from Cornell University(1984). Following a postdoctoral appointment at the MassachusettsInstitute of Technology, Chris came to Los Alamos NationalLaboratory on a J. Robert Oppenheimer Fellowship. He became astaff member at Los Alamos in 1989. He is a Fellow of Los AlamosNational Laboratory and was awarded the Los Alamos Fellows Prizein 1995. He is also a Fellow of the American Physical Society. In June, 2002, Chris joined the Physics Department at the Ohio StateUniversity as an Ohio Eminent Scholar. In addition to his currentresearch, which is focused on the development of an ultrasensitivemagnetic resonance force microscope, he is actively engaged in magnetic resonance studies of correlated electron systems including high-temperature cuprates andheavy fermions.

Robert G. Clark received Bachelor of Science and Ph.D. degrees inphysics from the University of New South Wales in Sydney,Australia, in 1973 and 1983, respectively. Between 1984 and 1990,Bob was a lecturer in physics at the University of Oxford, England,and research group head at Clarendon Laboratory at Oxford. Duringthis period, he received a number of awards and distinctions, includ-ing the Wolfson Award for prestigious research (1988). In 1990, Bobbecame professor of experimental physics at the University of NewSouth Wales. He is now heading the Center for Quantum ComputerTechnology in Sydney, Australia, and is leading the Australian effortfor developing a silicon-based solid-state quantum computer.



Andrew Dzurak is a program manager at the Centre for QuantumComputer Technology and Associate Professor within the School ofElectrical Engineering at the University of New South Wales, Sydney.He has collaborated with researchers at Los Alamos since 1996.Andrew has 15 years of experience in the fabrication and cryogenicelectrical measurement of quantum effects in semiconductor nanostruc-tures, beginning with a Ph.D. at the University of Cambridge (theUnited Kingdom). Andrew moved to UNSW to take up an AustralianPostdoctoral Fellowship and has worked closely with Robert Clark toestablish the Semiconductor Nanofabrication Facility, of which he hasbeen Deputy Director since 1995. Within the Centre, Andrew managesresearch in integrated devices for the control and readout of phospho-rous qubits in silicon.

Alex Hamilton received his Ph.D. from the University ofCambridge, the United Kingdom, in 1993. He is a senior lecturer inphysics in the School of Physics at the University of New SouthWales, as well as manager for the Quantum Measurement Program atthe Centre for Quantum Computer Technology. He is interested inthe study of the fundamental properties of submicron electronicdevices, with particular interest in the regime in which strong inter-actions between adjacent devices influence the behavior of the com-plete system.

Llloyd Hollenberg is a senior lecturer at the University ofMelbourne, and since 2001 has been manager of the DeviceModeling Program at the Centre for Quantum Computer Technology.He received his Ph.D. from the University of Melbourne in 1989. His interests include the general aspects of quantum many body systems, and in particular field theoretic and spin systems. At pres-ent, his work focuses on the physics of the Kane quantum computerand on quantum algorithms.

David N. Jamieson is an associate professor and reader at theUniversity of Melbourne, Australia. He received his Ph.D. from theUniversity of Melbourne, and has been Director of theMicroanalytical Research Centre (MARC) of the School of Physics,University of Melbourne since 1996. In 2000, he joined the Centrefor Quantum Computer Technology as the manager of the Ion BeamProgram. He aims to adapt the technology of single ion detectionand control to the construction of phosphorous arrays in silicon.

Christopher I. Pakes received his Ph.D. in physics in 1999 from the College of Physics, Birmingham, the United Kingdom. He has studied metrological applications of nanotechnology andspin-polarized properties of oxide functional materials, using ultrahigh vacuum, low temperature STM and dc SQUIDs. In 2000,Chris joined the Center for Quantum Computer Technology and wasappointed manager of the Atomic Level Manipulation and ImagingProgram in 2001.

toward a silicon-based nuclear-spin quantum computernatural silicon contains a mixture of three...

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