Letter pubs.acs.org/NanoLett
Probing the Spin States of a Single Acceptor Atom Joost van der Heijden,*,†,‡ Joe Salfi,†,‡ Jan A. Mol,†,‡ Jan Verduijn,†,‡ Giuseppe C. Tettamanzi,†,‡ Alex R. Hamilton,‡ Nadine Collaert,§ and Sven Rogge†,‡ †
Centre for Quantum Computation and Communication Technology and ‡School of Physics, University of New South Wales, Sydney NSW 2052, Australia § Inter-University Microelectronics Center, Kapeldreef 75, 3001 Leuven, Belgium S Supporting Information *
ABSTRACT: We demonstrate a single-hole transistor using an individual acceptor dopant embedded in a silicon channel. Magneto-transport spectroscopy reveals that the ground state splits as a function of magnetic field into four states, which is unique for a single hole bound to an acceptor in a bulk semiconductor. The two lowest spin states are heavy (|mj| = 3/2) and light (|mj| = 1/2) hole-like, a two-level system that can be electrically driven and is characterized by a magnetic field dependent and long relaxation time, which are properties of interest for qubits. Although the bulklike spin splitting of a boron atom is preserved in our nanotransistor, the measured Landé gfactors, |ghh| = 0.81 ± 0.06 and |glh| = 0.85 ± 0.21 for heavy and light holes respectively, are lower than the bulk value. KEYWORDS: Single acceptor, silicon MOSFET, single atom transistor, boron atom, magneto-transport spectroscopy, hole spin states
C
such that a heavy and light hole state form the lowest-energy spin states of the acceptor. A qubit composed of these two levels is characterized by a magnetic field dependent Larmor frequency and relaxation rate, which have suitable values for qubit operations (e.g., for a boron atom a Larmor frequency of 4 GHz with a relaxation rate of 6 μs at ∼0.3 T).22 In this Letter, we experimentally demonstrate a single atom transistor where single hole tunneling through an acceptor dopant is controlled by a gate electrode. Remarkably, we find that the silicon nanotransistor environment for the acceptor, including the gate required to manipulate its charge state, preserves the full symmetry of the acceptor spin states. The nanotransistor used in this work provides the necessary tunnel coupling between the dopant and the leads to identify a single atom using direct current measurements.26−30 The measured charging energy (47 ± 1 meV) and the energy splitting between the ground and first excited state (27 ± 1 meV) are close to the values found for ensembles of boron acceptors in bulk silicon. Magnetic field-dependent hole transport spectroscopy shows that the ground state splits into four well-defined spin states. Previously only observed in ensemble measurements on acceptors in unstrained silicon,23,31−33 the availability of this 4-fold degenerate ground state for a single acceptor in a nanotransistor is promising to form a single qubit that can be electrically manipulated via gates. The observation of this 4-fold
oupling of spin degrees of freedom to oscillating electric fields has the potential to boost the prospects of scalability for spin qubits. In particular, electric field control enables fast, local single-qubit operations,1,2 and enhances the coupling of spin qubits to photons in microwave resonators, which is of interest for long-range coupling of qubits via circuit quantum electrodynamics.3,4 Coupling of spins to electric fields has recently been realized via spin−orbit coupling in III−V based, gate-defined electron2,5 and hole6 quantum dots. Meanwhile, single dopants have emerged as promising candidates for the realization of robust spin qubits7−11 with fabrication of devices9 and control of electron10 and nuclear spin11 having been accomplished at the single-donor level, using oscillating magnetic fields. Yet, the coupling of the spin of a single electron bound to a donor atom in silicon to an oscillating electric field is extremely weak, as it experiences a negligible spin−orbit coupling.12 As an alternative to gate-defined quantum dots in both III− V6,13−16 and group IV17−20 material systems, the spin states of acceptors also couple to oscillating electric fields, forming favorable two-level systems for electrically controllable qubits.21,22 In contrast to donor bound electron spins in silicon, ensemble acceptor measurements have demonstrated that a large (∼1 D) dipole coupling exists between the states of heavy (mj = ± 3/2) and light (mj = ± 1/2) holes bound to acceptor atoms.23 Moreover, because of the symmetric confinement of a hole by the Coulomb potential of an acceptor in an unstrained crystal the light hole and heavy hole manifolds are degenerate.24,25 A finite magnetic field lifts this degeneracy, © 2014 American Chemical Society
Received: December 18, 2013 Revised: February 14, 2014 Published: February 26, 2014 1492
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and electric fields.27 This large splitting between the heavy and light hole states makes a qubit formed by a heavy and a light hole unsuitable to drive electrically, because of a large Larmor frequency and a short relaxation time. As illustrated in Figure 1b, the silicon fin field effect transistor (FinFET) used in this work has two undoped fins connecting p-type (boron doped) source and drain regions. All the measurements shown in this work were performed in a dilution refrigerator with a FinFET with fin heights of 40 nm and widths of 25 nm. The width of the gate, which defines the channel length, is 40 nm and the two fins are 200 nm apart from each other. In similar devices with n-type leads, experimental indications of dopant diffusion from the leads into the channel have been observed.8,26,43,44 This mechanism has recently been confirmed by atom probe tomography.45 When a boron atom is present in the channel, two distinctive and well separated current peaks are measured (see Figure 1d) as a result of single hole tunneling through the neutral (A0) and positively charged (A+) acceptor states. To detect the acceptor atom, a small voltage is applied to the source terminal (bias voltage), while the current is measured between the drain terminal and ground as a function of the applied gate voltage (see Figure 1c). In all measured FinFETs, several current peaks are detected at low temperature, related to sequential hole tunneling through gate induced quantum dots, before a conductive channel opens. The current peaks of the acceptor states show up before these quantum dot peaks. By using multichannel devices we increase the chance of finding a single acceptor. A stability diagram was recorded by measuring the current as a function of the source-drain bias and gate voltage and plotting the numerical differential conductance dI/dVbias (see Figure 2a). The two highest visible zero-bias resonances occur at ∼−76 and ∼−210 mV gate voltage, respectively. Between these resonances the current is Coulomb blocked up to an observed charging energy of 47 ± 1 meV. This is in good agreement with the charging energy of a boron atom in bulk silicon (44 meV).35,46 Using a simplified model of a metal sphere, a selfcapacitance leading to such a charging energy is found for a radius smaller than 3 nm,8 which shows that the confining potential leading to these resonances is very localized. In this batch of samples, we often observe signatures of additional charging events, periodically disturbing the shape of the observed Coulomb diamonds. This has been observed in similar devices and interpreted as the charging of an island, close to one of the leads47 and is further discussed in the Supporting Information. The conventional measurement of the charging energy, which is directly related to the bias voltage at the top of the Coulomb diamond (see figure 2a), is not disturbed by these additional charging events. At this intersection, where both states are in the bias window, they are affected in the same way by the charge state of the island. The observation of the same disturbances in the finite current regions of both the first and the second resonance, confirms that these resonances originate from the same entity and thereby validates the interpretation that they are separated by a charging energy. The origins of the positive differential conductance lines within the current regions of the stability diagram are discussed later. Next we extract the hole temperature from the experiments. In Figure 2c, part of the bias trace at −75.90 mV gate voltage (highlighted with a blue line in Figure 2b) is fit to a convolution of the Fermi-Dirac and Lorentzian lineshapes (red dashed
degeneracy in a single-hole transistor is nontrivial, as previous single acceptor measurements have shown a splitting between the heavy and light hole Kramers doublets due to local strain and electric fields.27,34 In bulk silicon, the top of the valence band consists of a heavy and a light hole band, which are degenerate at the gamma point (|k| = 0), and a lower lying split-off band separated by 44 meV due to the spin−orbit coupling (see Figure 1a).
Figure 1. (a) Energy states of a neutral boron acceptor in bulk silicon. The ground state has a binding energy Eb of 46 meV35 and the first excited state lies 23 meV lower.36 The degeneracies of the acceptor states (blue numbers) resemble the degeneracies of the valence band of silicon at the gamma point (|k| = 0), which consists of a heavy and light hole band and a 44 meV lower lying split-off band. (b) SEM image of a multichannel p-type FinFET, similar to the one used in this work. For more fabrication details, see the Supporting Information and reference 37. (c) Schematic view of the direct current measurement setup and the potential profile through the channel of the transistor with a single acceptor located in the middle of the channel. Voltages are applied to the source and gate leads and the current is measured between the drain lead and ground. (d) IV traces of the single boron device at room temperature (black) and 5 K (red) at 5 and 1 mV source-drain voltage respectively. Two isolated current resonance peaks are the signatures of the transitions from the negatively charged acceptor (A−) to the neutral acceptor (A0) and from the neutral acceptor to the positively charged acceptor (A+) by binding one and two holes to the boron atom, respectively.
Accordingly, the spin states of confined holes in nanostructures are predominantly built up from the heavy and light holes in a way that is strongly dependent on the symmetry of the confinement potential. The s-like ground state confined by the Coulomb potential of an acceptor retains the 4-fold degeneracy, as the confinement does not break the cubic symmetry of the silicon crystal.24,25 In contrast, in a two-dimensional hole gas the heavy holes form the lowest energy states,38,39 while in an ideal one-dimensional structure with cylindrical confinement these consist of light holes.40 Furthermore, the confinement of self-assembled or gate-confined quantum dots also leads to a considerable splitting between the heavy and light holes, making the heavy holes the lower lying states.14,41 Similarly, the bulk 4-fold degenerate acceptor ground state can be split in two Kramers doublets by an electric field or strain.24,25 Split Kramers doublets have been observed in ensembles of boron dopants in intentionally strained silicon42 and in a single acceptor in a silicon Schottky barrier subject to random strains 1493
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Figure 2. (a) Measurement of the differential conductance (dI/dVbias) as a function of the applied bias and gate voltages. A charging energy of 47 ± 1 meV is measured between the first and second hole transport through the acceptor states. The Coulomb diamonds are distorted by charging events of a nearby island, which is capacitively coupled to the boron atom. White lines indicate the different slopes caused by the different capacitive couplings to the drain terminal of the boron atom and the nearby island. Arrows indicate lines of positive differential conductance in the first hole transport region. Green arrows indicate the effect of excitations in or near the drain reservoir. The blue arrow indicates where an excited state enters the bias window. (b) A zoom-in of (a) at the zero bias crossing of the A− to A0 transition, which is not disturbed by the nearby defect and is used to determine the relative capacitive couplings to the three terminals. (c) A part of the bias trace at a top gate voltage of −75.9 mV, indicated by a blue line in panel (b), which is used to determine the hole temperature by fitting it to a convolution of a thermal and lifetime broadened line shape (red) and compared to the thermal Fermi-Dirac line shape (blue).
line).48 To convert the voltage scale to an energy scale, the capacitive coupling to the three terminals was determined, by using an undisturbed section of the stability diagram (see Figure 2b). The conversion factors for the gate and bias voltage are αg = e(Cg/Ct) = 0.42 ± 0.02 meV/mV and αs = e(Cs/Ct) = 0.37 ± 0.01 meV/mV respectively, where Ct = Cg + Cs + Cd. A hole temperature of 55 ± 8 mK and a small lifetime broadening of hΓ/kT ∼ 0.2 was found, by fitting the data over a range of gate voltages to the thermally broadened Lorentzian (see Supporting Information). This hole temperature determines the differential conduction peak shape and is later used in the analysis of the Landé g-factors. To study the degeneracy of the ground state, we investigated its behavior in a magnetic field. A bias trace at −75 mV gate voltage is measured as a function of magnetic field, which is applied parallel to the top plane of the silicon chip and with a 19 ± 5° angle to the axis perpendicular to the direction of the current flow (the ⟨110⟩ direction). As the ground state is expected to shift toward a higher gate voltage when a magnetic field is applied, a gate voltage higher than the zero-field zerobias crossing is chosen. At the positive bias side, the ground state clearly splits in four states (see Figure 3a). Between 0.8 and 0.9 T a charge change in a nearby defect causes a small shift in the data of figure 3a. At higher magnetic fields, the current steps of the excited states decrease in amplitude, which might be related to increasing relaxation rates. The 4-fold degeneracy of the ground state, indicates that its cubic symmetry is not broken by the confining potential, preserving the degeneracy of the heavy-hole and light-hole like states. This excludes the confinement potentials of gate-defined and disordered quantum dots as the source of these states.
Combined with the measured charging energy, we conclude that the observed resonances are the transitions to the A0 and A+ states of a boron acceptor. The Zeeman-type interaction is approximated as a linear function EZ = gmjμBB for each of the four states, where μB is the Bohr magneton, B the magnetic field, and mj the projection of the total angular momentum in the direction of the magnetic field. The effects of electric field and strain are not taken into account as they are, although not precisely known, small compared to the Zeeman interaction as evidenced by the 4-fold degenerate acceptor ground state. Thermally broadened peaks, using the earlier determined hole temperature, are used to approximate the conductance peaks as shown in the trace at 0.7 T in Figure 3b. As the tails of the conductance peaks overlap, it can not be excluded that some of these peaks have extra broadening due to a finite lifetime of the states but for finding the relative positions of the peaks the thermal fit is sufficient. A fit to the linear Zeeman model for the data up to 0.8 T is shown in Figure 3c with 95% confidence intervals. We find a ΔEZ of 0.141 ± 0.010 meV/T between the ground state and third excited state (red fits in Figure 3b,c) and a ΔEZ of 0.049 ± 0.012 meV/T between the first and second excited states (green fits in Figure 3b,c), again using the 1 − αs to convert the voltage scale to energy. To determine the gfactors and compare them to the values found in EPR measurements,23 we follow the assumption made in ref 23 that the four boron states are built up from solely light or heavy holes, having mj = +3/2, +1/2, −1/2, and −3/2, respectively. The g-factors for the Δmj = 3 (red) and Δmj = 1 (green) case are |gΔmj=3| = 0.81 ± 0.06 and |gΔmj=1| = 0.85 ± 0.21. The g-factors found in our nanostructure are significantly smaller compared to the g-factors found for boron in bulk 1494
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Figure 3. (a) Magnetic field dependence of the A0 state at a gate voltage of −75 mV. The shown numerical differential conductance is determined after the data is corrected for a density of states background (see Supporting Information). Between the 0.8 and 0.9 T a clear shift is observed, which is caused by a change of the charge state of a nearby defect. The conduction peaks between 0 and 0.8 T are fit to four thermally broadened peaks with a linear dependence on the magnetic field and constant peak heights. The trace at 0.7 T is compared to this fit in (b). In (c), the fit results with 95% intervals are shown on top of the differential conductance data. The spitting between the heavy (red) and light (green) hole-like doublets are used to calculate the g-factors.
for an excited state and observed for the line at −27 mV. As the broad lines run parallel to the drain edge of the Coulomb diamond for positive and negative bias voltages, they are expected to be a result of excitations near or in the drain lead. The observation of similar signatures in the finite current region of the second resonance, again confirms that this resonance is due to tunnelling through the same entity as the first resonance. For a boron atom spin−orbit qubit, silicon offers a very suitable platform. First, silicon can be isotopically purified with zero nuclear spin isotopes, reducing hyperfine mediated decoherence effects. In addition, compared to electrons, hyperfine processes are expected to be suppressed for holes. An upper bound of ∼10 MHz for the hyperfine coupling to the boron nucleus has been obtained from EPR spectra of borondoped silicon,50 which is ten times smaller than for phosphorus donors in silicon.12 And second, the silicon platform is compatible with the successful CMOS technology. In conclusion, we have measured a single hole transistor, where the hole is confined by the Coulomb potential of a single boron atom. Even though this atom resides in a gated silicon nanostructure, the unique bulk 4-fold degenerate ground state of an acceptor is retained. The measured charging energy of 47 meV, the distinct excited state spectrum and the magnetic field splitting of the ground state in four well-defined spin states identify the boron atom. The measured values of the Landé gfactors, |gΔmj=3| = 0.81 ± 0.06 and |gΔmj=1| = 0.85 ± 0.21 for the heavy and light holes, respectively, are significantly smaller than the bulk Landé g-factor. Our results demonstrate that electrostatic control of a single acceptor can be obtained in a nanoscale transistor without losing the unique signature of the Coulomb confinement, the 4-fold degenerate ground state. These results pave the way toward a gate electric-field controlled single acceptor qubit.
silicon, whose absolute value ranges from 1.08 to 1.18,23 depending on the direction of the magnetic field with respect to the crystal axes. The lower g-factor could be associated with a change in the shape of the orbital, which influences the Zeeman splitting due to the spin−orbit interaction. This is evident from the deviation of the g-factor found for boron in bulk silicon from the free electron g-factor. Remarkably, we find that this distortion has not appreciably broken the cubic symmetry, leading to an undetectable splitting between the heavy and light hole doublets but does lead to a measurable change in the gfactor. In this nanoscale transistor with a channel of 25 × 40 × 40 nm, we believe that the effect of the nearby interfaces between the silicon and oxide layer (at most ∼12.5 nm away) is the dominant contribution in the observed reduction of the gfactors. In the same batch of devices, acceptors with a 2-fold degenerate ground state with a nearby first excited state and a second excited state with a lower energy splitting than the bulk value of 23 meV, have been measured, which is in agreement with acceptor atoms positioned very close to the interface.49 In addition to the charging energy and the 4-fold degenerate ground state, the distinct excited state spectrum of a boron acceptor was observed. The presence of an excited state shows up as a sharp positive differential conductance line at the negative bias voltage side, intersecting with the Coulomb diamond at −27 ± 1 mV bias voltage (see the blue arrow in Figure 2a and more details in the Supporting Information). At this intersection, the ground and excited state are exactly leveled with the source and drain reservoir energies, respectively. This bias voltage can therefore be directly translated to an energy splitting of 27 ± 1 meV, which is close to the expected bulk value for the first excited state (23 meV) of a boron acceptor.36 Broader lines that show up on both the positive and negative bias side (see the green arrows in Figure 2a), represent a bump in the current measurement, after which the current returns to its original value (see Supporting Information). This is in contrast to a step in current, expected 1495
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ASSOCIATED CONTENT
S Supporting Information *
Further details about the device geometry and the measurements of the spin states of the single acceptor. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank A. Srinivasan for his assistance with running the dilution refrigerator and J. Cochrane for technical support. S.R. acknowledges support from the ARC FT scheme (FT100100589) and the ARC DP scheme (DP120101825). A.R.H. acknowledges support from the ARC DP and DORA schemes. G.C.T. acknowledges financial support from the ARC DECRA scheme (DE120100702) and from the UNSW under the GOLDSTAR Award 2013 scheme.
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