Letter pubs.acs.org/NanoLett
Wave Function Control over a Single Donor Atom J. Verduijn,*,†,‡ G. C. Tettamanzi,† and S. Rogge†,‡ †
Centre for Quantum Computation and Communication Technology, University of New South Wales, Sydney NSW 2052, Australia Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
‡
S Supporting Information *
ABSTRACT: Single donor atoms in semiconductor nanostructures are attractive basic components for quantum device applications. In this work, we demonstrate the ability to manipulate the wave function of a single donor electron with an electric field. The deformation of the wave function is probed by the tunnel current which, furthermore, allows for the determination of the location of the atom in the device. This experiment demonstrates the control necessary for the utilization of single donors in quantum electronics. KEYWORDS: Single dopant, wave function control, quantum transport, phosphorus, MOSFET, single electron transport
T
method we use here provides direct access to the spatial distribution a single donor wave function and has the advantage that it does not rely on the spin-coupling of the electron to the environment. Furthermore, the presented results show that it is in principle possible to control the hyperfine coupling of the electron spin to a single donor nucleus by reducing the charge density at the nucleus.19 Nanowire field-effect transistors (FETs) with a small number randomly placed phosphorus donors in the channel (i.e., SATs) are used as a platform to study single dopants;8,7,20 see Figure 1. The device used here consists of a nanowire etched out a of a 20 nm thick silicon film of a silicon on insulator wafer (SOI). The gate, which is isolated from the channel by a 5 nm thick silicon dioxide layer, protects the channel region during the high dose source/drain implantation; see Figure 1a and b. To obtain single dopant transport in a controllable way, the silicon film has been preimplanted to obtain 1 × 10 17 cm −3 phosphorus after an activation anneal. For the device studied here, with channel region of (40 × 60 × 20) nm3, where the dimensions are the channel length, width, and height, respectively, this results in on average only 3−7 donors in the channel. The position of the donors is random, and thus also the tunnel coupling to the source and drain contacts is different for each donor. Therefore we measure many devices and select a device with a dopant in a favorable location in the channel. For additional electrostatic control of the channel, the substrate wafer below the 145 nm thick buried oxide is used as a gate.21,22 This degree of tunability allows us to control the wave function of a single donor, as shown in Figure 1c and d. We find a charging energy of 20.1 meV close to flatband for the donor studied here. This value is extracted from a measurement
hanks to their unique properties, single isolated dopants hold great promise for quantum device applications. A famous example of a prototypical application is the dopantbased quantum computer.1−3 For this proposal long electronspin coherence times are important. Donors in silicon seem to be able to meet these prerequisites because extremely long coherence times can be achieved by drastically reducing the number of silicon-29 nuclear spins in the lattice.4 Even though, by now, many experiments demonstrate access to single dopants (see, for example, refs 5−9), control over the charge distribution and adiabatic manipulation of the donor-bound electrons are relatively unexplored aspects. Further, though quite extensively studied theoretically,10−13 the influence of the conduction-band valley structure on the shuttling of electrons between the donor and the interface, as required for some applications,1,14,15 has not been investigated experimentally to date. Here we present data of a single atom transistor (SAT) that demonstrates for the first time a smooth transition of the single electron wave function from a bulk-like dopant state to an interface-well state, passing via a molecular dopant-interface state. By monitoring the amplitude of the tunnel current as a function of the applied electric field we probe the electron wave function directly. In the process of shuttling the electron, the valley population of the state changes in a nontrivial but controllable way. Previously, Lansbergen et al. demonstrated tuning of the orbitals of a single donor by changing the virtual occupation in Coulomb blockade with a gate.16 More recently, Hoehne et al. employed an electrically detected spin−echo technique to map the charge distribution of an ensemble of interface defects.17 Our experimental technique achieves a similar goal but allows for single dopant sensitivity. Another approach was proposed by Park et al.: by using the hyperfine interactions with silicon-29 nuclear spins it should be possible to map out the wave function of a donor.18 The experimental © 2013 American Chemical Society
Received: December 7, 2012 Revised: March 19, 2013 Published: March 20, 2013 1476
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the voltages on the topgate, Vtg, and backgate, Vbg, a current will flow between source to drain when a small source/drain voltage Vsd = 0.2 mV is applied. At low gate voltages the current is blocked; see Figure 2a. As the gate voltages are increased, a localized state in the channel moves into resonance with the source and drain chemical potential, μs,d, and current can flow; see Figure 2b.5 Note that we keep the source/drain bias small such that (μs,d ≈ eVs ≈ eVd) < kBT at all times (kB is the Boltzmann constant) and thus only the ground state is probed. When the localized state is situated below μs,d, the current is blocked again due to a Coulomb blockade.25 This mechanism allows us to measure a clean resonant tunnel current signal, probing only a single donor in the channel of the device, denoted by green circles in Figure 2a. In the region labeled B the current flow is mediated by a bulk-like wave function localized at the donor nucleus as will be shown later; see Figure 2b. In regions labeled A and C the current is mediated by a wave function localized in an interface-well that is induced by the electric field; see Figure 2c. Please note that in the interfacewell regime the splitting between ground state first excited state splitting may become comparable to the bias voltage or thermal energy, but this would at most double the current, whereas, as we will show below, the changes in current that we observe are more then 2 orders of magnitude in size. The black arrow at Vbg ≈ 2.5 V denotes the position where the conduction band is flat on average, and thus the electric field at the donor location is approximately zero. This flatband point is signaled by a change of the slope of the resonant tunneling lines and is further discussed in the Supporting Information. Thus, by changing the frontgate and backgate voltage, the local electric field can be tuned, while maintaining the donor in resonance with the source/drain chemical potential. Hereby we realize field tuning of a single donor which is one of the key elements of Kane quantum computers.1,10,14,26 Silicon is an indirect band gap material. This renders the conduction band minimum 6-fold degenerate with states at a finite momentum, k0, in each ±x-, ±y-, and ±z-direction in
Figure 1. Device structure of the single donor device. (a) Scanning electron micrograph of a device similar to the one studied in this Letter. The channel (blue) is a nanowire etched from a silicon film on a silicon dioxide layer. The gate (yellow) is wrapped around the channel on three sides. (b) A schematic cross-section of the device across the dashed line in a. Spacers (gray) on both sides of the channel cause the structure to be slightly underlapping.24 The source and drain contacts are doped with a high concentration of arsenic donors to make them metallic. The topgate and backgate can be used to control the electric field in the channel. (c and d) Simulated wave function of an electron bound to a single phosphorus donor in the center of the channel in a schematic representation of the device. By applying appropriate gate voltages, and thereby inducing an electric field in the z-direction, Fz, the one-electron electron wave function is deformed in a controlled way. In the shown case, the electron is delocalized along the interface of the channel.
at large Vsd and is consistent with the expected charging energy of a phosphorus donor in a nanowire FET;23 see Supporting Information. At low temperature (T = 4.2 K) we perform single donor current spectroscopy on the SAT; see Figure 2. Depending on
Figure 2. Low-temperature transport data and wave function manipulation in real space and reciprocal space. (a) The drain current as a function of backgate and topgate voltage at a source/drain bias of 0.2 meV at 4.2 K. The white dashed line indicates the intrinsic device threshold voltage, and the green circles a single donor resonance. In region B, between the black vertical dashed lines, the resonance is induced by a bulk-like donor state and in the regions A and C an interface-well-like one. Panels b and c schematically visualize how the real space electron wave function is modified by an electric field. A 1-D cut of the wave function and the conduction band structure is shown as a function of position between the topgate and backgate. On the resonance the donor level aligns with the source/drain chemical potential, μs,d. Panels d and e show a schematic reciprocal space representation corresponding to b and c. An electric field lifts the 6-fold degeneracy; see inset of panel e. Therefore low energy wave functions localized at the interface are mainly built up from the two z-valleys. The inset of d shows a detail of the +z-valley, centered at k0, with the transversal, mt, and lateral, ml, effective mass indicated. 1477
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Figure 3. Tunneling spectroscopy of a single donor and its valley composition. (a) The magnitude of the current at the resonance denoted by the green circles in Figure 2a as a function of the backgate voltage. A light blue area is used to show the size of the 98% confidence bounds of the resonance height from the fit. Whenever this area is not visible, the confidence region is smaller than the markers. The gray band at the bottom denotes the region in which the electron is located on the donor, region B, or in the interface-well, regions A and C. Even though the band is divided in discrete sections, the transition from donor-like to interface-like is smooth. The horizontal lines at the bottom of the graph indicate which data has been used to match the model curves i and ii, black and green, respectively. (b) A fit allows to extract the valley composition of the ground state (see main text). (c) The gradual delocalization of the electron as a function electric field can be detected in the tunnel current, because states in different valleys have a different spatial extent. (d) When the donor state crosses the interface-well state, a hybridized state is formed. The dashed lines are the positions of the uncoupled states, i.e., with Δ = 0; see Supporting Information.
reciprocal space, with k0 ≈ 0.85 × 2π/a = 2π/0.64 nm−1; see the familiar reciprocal space representation in Figure 2d. In addition, the effective mass in each of these minima is anisotropic, in the lateral direction the mass is 0.914me, and in the two transversal directions 0.191me,27 where me is the free electron mass. When confined in one direction by an fieldinduced interface-well, this results in the splitting of the conduction band states in a two-fold degenerate band at low energy and 4-fold degenerate band at higher energy, see the inset in Figure 2e. As a consequence, the ground state populates only the two valleys in the confined direction; see Figure 2e. For spherical confinement by a donor potential, on the other hand, this results in a nondegenerate singlet ground state and triplet and doublet excited states.28 For the groundstate singlet the valley population is equal for all valleys, as shown in Figure 2d. As the applied electric field in our device induces a well at the interface and pulls the electron wave function away from the donor, the splitting of states varies dramatically, and the ground-state population of the valley states evolves from donor-like to interface-well-like.12,13,23 This also changes the wave functions’ spatial extent and thus the tunnel coupling with the source and drain contacts. Therefore, these modifications to the wave function can be probed by measuring the tunnel current. The resonant tunnel current into the donor state at small source/drain bias is proportional to the tunnel coupling of the contacts to the ground state. As explained above, the backgate is used to tune the electric field at the position of the donor.
Figure 3a shows the measured drain current as a function of the applied backgate voltage extracted from a fit of a thermally broadened resonance to the peak denoted by the green circles in Figure 2a; see also the Supporting Information. We assume a linear relation between the backgate voltage and the electric field. Note that this assumption is only justified in the absence of space charge in the channel region but is confirmed by the analysis presented below. This relation between the field and backgate voltage allows us to compare the measured tunnel current to a model describing the field dependence of the wave function. Due to the anisotropy in the effective mass, states in different valleys have a different spatial extent in the transport direction, which we have chosen as the x-direction; see Figures 1 and 2. This means that the zero-field tunnel coupling is larger for the y-valleys and z-valleys than for the x-valleys, because the xvalley states have a large effective mass in the x-direction and are therefore more confined. If an electric field is applied, however, the spatial extent of the wave function as well as the population of the x-valleys changes. Initially, at a small field, the effect on the donor wave function is to increase the spatial extent while the electron stays localized at the donor. This is due to the conventional Stark effect.19 At some critical field however the electron is pulled to the interface and delocalizes further. Figure 3c schematically illustrates this effect for a donor at ∼5 Bohr radii from the interface (the Bohr radius for phosphorus in silicon is ∼2 nm). Since the tunnel coupling increases with an increase of the spatial extent of the wave 1478
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effect on the valleys parallel to the field (z-valleys) then on the perpendicular ones. It should be noted here that, even though the exact shape of the obtained curve does depend on the details the fitting procedure, the general trend is that the valley population of the z-valleys increase when the magnitude of the electric field is made larger. These results demonstrate, for the first time, control over the valley population of a single gated donor. At the backgate voltage where the interface-well state aligns with the donor state, these states hybridize with each other and shift in energy. Since the energy shift can be detected in the position of the Coulomb blockade resonance as a function of the topgate and backgate voltage, this provides an independent way to validate the donor-interface transition of the localized electron.29 Figure 3d shows the peak position in topgate voltage as a function of backgate voltage in the vicinity of the hybridization point. The shift of the state manifests itself clearly as one branch of an anticrossing. Fitting a two-level dispersion to this crossing results in a hybridization gap of (1.4 ± 0.2) meV; see the Supporting Information for more details. Using the values published by Calderón et al.,26 we find that this corresponds to a donor at ∼12 nm from the interface, approximately in the center of the channel, halfway between the top and back sides of the 20 nm thick channel. This result agrees well with the estimate from the fitting of the tunneling current amplitude and is consistent with the electrostatic analysis presented in the Supporting Information. The prototypical donor-based quantum computer as proposed by Kane1 requires the tunability of the electron charge distribution. Very recently, Pla et al.30 demonstrated the ability to perform spin rotations on a single gated phosphorus donor. In their experiment the strength of the hyperfine interaction could not be controlled. Our experiment shows that this control should in principle be possible by reducing the charge density at the donor nucleus.10,19,31−34 There are two mechanisms by which this can be achieved: by removing the electron from the nucleus or by changing the valley composition of the donor state, which is known to change the hyperfine interaction.31,35 Another aspect that is closely related to the results presented here is the tuning of the exchange interaction between neighboring donors.36−38 The demonstrated control provides encouraging prospects in this direction as well. Further, it has been proposed to use the valley degree of freedom in silicon double quantum dots to encode the states of a qubit that can be fully electrically controlled.39 The gating demonstrated here is very similar in that it provides control over the valley degree of freedom. Even though there is to date no equivalent proposal for a donor-interface system such as the one studied here, it may be possible to use a subset of the donor-interface states to encode and manipulate quantum information in a useful way. The excited states with the rich valley structure of this system can readily be tuned in and out of resonance with each other using an electric field.12 Depending on the valley composition, the magnitude and direction of the field states can hybridize or do not interact at all.40 Some explorations of these effects and the associated physics are presented by Debernardi and Fanciulli41 and Baena et al.13 For the feasibility of this idea, the robustness of the dopant-interface (excited) state spectrum to disorder needs further investigation.42,43 The experiments presented here demonstrate electrical control over single donor wave functions. This result offers
function, the delocalization of the wave function can be observed in the magnitude of the tunnel current. We develop a model to quantify this effect and to perform a more detailed analysis of the data. We solve a single valley effective mass Schrödinger equation for a donor located in the center of the channel on a grid. The channel is modeled as a (40 × 40 × 20) nm3 box (x, y, and z dimensions) subject to a linear electric field in the z-direction. For the moment the influence of the Bloch wave functions is ignored, and we obtain an envelope wave function of the ground state.28 We compute the tunnel coupling to one of the contacts by integrating the envelope wave function over a plane perpendicular to the transport direction (in the x-direction), similar to Bardeen’s approximation; see the Supporting Information for more details. Figure 3a shows the result of the calculation. The computed curve is matched in two different regimes: the low field regime and the high field regime, labeled i and ii, respectively. Around the flatband voltage, curve i matches the data very well. This is a strong indication that the donor is indeed approximately located in the center of the channel, i.e., at 10 nm (∼5 Bohr radii) from the interface. Furthermore, the tunneling amplitude is in the low field region (curve i) as well as in the high field region (curve ii) regime symmetric around the flatband voltage (dashed vertical line), at a backgate voltage of (2.50 ± 0.02) V. We find that the symmetry of the tunnel coupling versus electric field is very sensitive to the location of the donor in the channel. Already when using a donor depth of 8 nm instead of 10 nm, we find that the asymmetry would have a significant effect on the tunnel current; see also the Supporting Information. When the tunnel current has a minimum, exactly at the flatband point, the donor wave function is closest to bulk-like. Here, the electron is tightly bound to the donor nucleus since the electric field is approximately zero and therefore the tunnel coupling small. Note that these results are consistent with the electrostatic analysis of the device as presented in the Supporting Information, which independently predicts the same trend for the electric field. At larger backgate voltage, however, the experimental data show a shoulder at positive as well as negative backgate voltage. Here the data does not match the model in the low-field nor in the high-field limit. These shoulders turn out to be caused by the nontrivial way in which the valley population shifts from the x-valleys and y-valleys to the z-valleys as an electric field is applied. To allow further quantitative analysis of the data, we assume a phenomenological description of the valley population as a function of backgate voltage (electric field). In this way we introduce some of the valley physics in the model that we have ignored in our analysis so far. To quantify the valley population versus backgate voltage we assume a smooth phenomenological function which qualitatively reproduces the expected behavior;13 see Supporting Information for the chosen functional form. This function is then used in a fit to “weight” the model curves i and ii to the data (Figure 3a). The fit translates the shoulders at Vbg ∼ −2.4 V and Vbg ∼ 7.3 V in Figure 3a into a transition of the valley population. In this we have assumed that only the x-valleys contribute significantly to the current. The result of this fit, which fixes all valley occupations at the same time, is shown in Figure 3b. A positive (or negative) applied field in the z-direction moves the valley population for the xvalleys and y-valleys into the z-valleys. The reason is that, due to the anisotropic effective mass, the electric field has a smaller 1479
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(15) Caldeŕon, M.; Koiller, B.; Das Sarma, S. Phys. Rev. B 2006, 74, 081302. (16) Lansbergen, G. P.; Tettamanzi, G. C.; Verduijn, J.; Collaert, N.; Biesemans, S.; Blaauboer, M.; Rogge, S. Nano Lett. 2010, 10, 455. (17) Hoehne, F.; Lu, J.; Stegner, A.; Stutzmann, M.; Brandt, M.; Rohrm̈ uller, M.; Schmidt, W.; Gerstmann, U. Phys. Rev. Lett. 2011, 106, 196101. (18) Park, S.; Rahman, R.; Klimeck, G.; Hollenberg, L. Phys. Rev. Lett. 2009, 103, 106802. (19) Bradbury, F.; Tyryshkin, A.; Sabouret, G.; Bokor, J.; Schenkel, T.; Lyon, S. Phys. Rev. Lett. 2006, 97, 176404. (20) Khalafalla, M. A. H.; Ono, Y.; Nishiguchi, K.; Fujiwara, A. Appl. Phys. Lett. 2007, 91, 263513. (21) Prati, E.; Belli, M.; Cocco, S.; Petretto, G.; Fanciulli, M. Appl. Phys. Lett. 2011, 98, 053109. (22) Roche, B.; Voisin, B.; Jehl, X.; Wacquez, R.; Sanquer, M.; Vinet, M.; Deshpande, V.; Previtali, B. Appl. Phys. Lett. 2012, 100, 032107. (23) Lansbergen, G. P.; Rahman, R.; Wellard, C. J.; Woo, I.; Caro, J.; Collaert, N.; Biesemans, S.; Klimeck, G.; Hollenberg, L. C. L.; Rogge, S. Nat. Phys. 2008, 4, 656. (24) Wacquez, R.; Vinet, M.; Pierre, M.; Roche, B.; Jehl, X.; Cueto, O.; Verduijn, J.; Tettamanzi, G.; Rogge, S.; Deshpande, V.; Previtali, B.; Vizioz, C.; S. Pauliac-Vaujour, Comboroure, C.; Bove, N.; Faynot, O.; Sanquer, M. 2010 Symposium on VLSI Technology (VLSIT), Honolulu, Hawaii, June 15−18, 2010; Vol. 193. (25) Beenakker, C. Phys. Rev. B 1991, 44, 1646. (26) Caldeŕon, M.; Koiller, B.; Hu, X.; Das Sarma, S. Phys. Rev. Lett. 2006, 96, 096802. (27) Madelung, O. Semiconductors. Data Handbook; Springer: Verlag, 2004. (28) Kohn, W.; Luttinger, J. Phys. Rev. 1955, 98, 915. (29) Hüttel, A.; Ludwig, S.; Lorenz, H.; Eberl, K.; Kotthaus, J. Phys. Rev. B 2005, 72, 081310. (30) Pla, J. J.; Tan, K. Y.; Dehollain, J. P.; Lim, W. H.; Morton, J. J. L.; Jamieson, D. N.; Dzurak, A. S.; Morello, A. Nature 2012, 489, 541. (31) Wilson, D.; Feher, G. Phys. Rev. 1961, 124, 1068. (32) Pereira, R.; Stegner, A.; Andlauer, T.; Klein, K.; Wiggers, H.; Brandt, M.; Stutzmann, M. Phys. Rev. B 2009, 79, 161304. (33) Debernardi, A.; Baldereschi, A.; Fanciulli, M. Phys. Rev. B 2006, 74, 035202. (34) Rahman, R.; Wellard, C.; Bradbury, F.; Prada, M.; Cole, J.; Klimeck, G.; Hollenberg, L. Phys. Rev. Lett. 2007, 99, 036403. (35) Dreher, L.; Hilker, T.; Brandlmaier, A.; Goennenwein, S.; Huebl, H.; Stutzmann, M.; Brandt, M. Phys. Rev. Lett. 2011, 106, 037601. (36) Koiller, B.; Hu, X.; Das Sarma, S. Phys. Rev. Lett. 2001, 88, 027903. (37) Fang, A.; Chang, Y.; Tucker, J. Phys. Rev. B 2002, 66, 155331. (38) Wellard, C.; Hollenberg, L. Phys. Rev. B 2005, 72, 085202. (39) Culcer, D.; Saraiva, A.; Koiller, B.; Hu, X.; Das Sarma, S. Phys. Rev. Lett. 2012, 108, 126804. (40) Smit, G. D. J.; Rogge, S.; Caro, J.; Klapwijk, T. M. Phys. Rev. B 2004, 70, 035206. (41) Debernardi, A.; Fanciulli, M. Phys. Rev. B 2010, 81, 195302. (42) Srinivasan, S.; Klimeck, G.; Rokhinson, L. P. Appl. Phys. Lett. 2008, 93, 112102. (43) Rahman, R.; Verduijn, J.; Kharche, N.; Lansbergen, G.; Klimeck, G.; Hollenberg, L.; Rogge, S. Phys. Rev. B 2011, 83, 195323.
encouraging prospects for the control of the coupling between donors and control over the strength of the hyperfine coupling of the electron−nuclear spin system on single donors.
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ASSOCIATED CONTENT
S Supporting Information *
Experimental methods, comparison of doped and undoped devices, capacitive coupling, modeling of the tunnel coupling, and fit to obtain the valley composition. 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 The devices have been designed and fabricated by the AFSiD Project partners, see http://www.afsid.eu. J.V. acknowledges Jan Mol for discussions and help with the finite difference calculations. All authors acknowledge the valuable comments of Lloyd Hollenberg on the manuscript. This work was supported by the EC FP7 FET-proactive NanoICT project AFSiD (214989) and by the Australian Research Council (ARC) Centre of Excellence for Quantum Computation and Communication Technology (CE110001027). S.R. acknowledges an ARC Future Fellowship (FT100100589) and G.C.T. an ARC DECRA Fellowship (DE120100702) and the UNSW under the GOLDSTAR Award 2013 scheme.
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REFERENCES
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