Heterointerface effects on the charging energy of shallow D- ground state in silicon: the role of dielectric mismatch
M.J. Calderon, J. Verduijn, G.P. Lansbergen, G.C. Tettamanzi, S. Rogge, Belita Koiller
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Heterointerface effects on the charging energy of shallow D − ground state in silicon:the role of dielectric mismatch M.J. Calder´on, J. Verduijn, G.P. Lansbergen, G.C. Tettamanzi, S. Rogge, and Belita Koiller Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049 Madrid, Spain Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands Instituto de F´ısica, Universidade Federal do Rio de Janeiro,Caixa Postal 68528, 21941-972 Rio de Janeiro, Brazil (Dated: May 7, 2018)Donor states in Si nanodevices can be strongly modified by nearby insulating barriers and metallicgates. We report here experimental results indicating a strong reduction in the charging energy ofisolated As dopants in Si FinFETs, relative to the bulk value. By studying the problem of twoelectrons bound to a shallow donor within the effective mass approach, we find that the measuredsmall charging energy may be due to a combined effect of the insulator screening and the proximityof metallic gates.
PACS numbers: 03.67.Lx, 85.30.-z, 73.20.Hb, 85.35.Gv, 71.55.Cn
I. INTRODUCTION
For over a decade dopants in Si have constitutedthe key elements in proposals for the implementationof a solid state quantum computer.
Spin or chargequbits operate through controlled manipulation (by ap-plied electric and magnetic fields) of the donor electronbound states. A shallow donor, as P or As in Si, can bindone electron in the neutral state, denoted by D , or twoelectrons in the negatively charged state, denoted by D − .Proposed one and two-qubit gates involve manipulatingindividual electrons or electron pairs bound to donorsor drawn away towards the interface of Si with a barriermaterial. In general, neutral and ionized donor statesplay a role in different stages of the prescribed sequenceof operations.In the proposed quantum computing schemes, donorsare located very close to interfaces with insulators, sepa-rating the Si layer from the control metallic gates. Thisproximity is required in order to perform the manipula-tion via electric fields of the donor spin and charge states.The presence of boundaries close to donors modifies thebinding potential experienced by the electrons in a semi-conductor. This is a well-known effect in Si MOSFETs, where the binding energy of electrons is reduced with re-spect to the bulk value for distances between the donorand the interface smaller than the typical Bohr radius ofthe bound electron wave-function. On the other hand,in free-standing Si nanowires with diameters below 10nm, the binding energy of donor electrons significantlyincreases leading to a strongly reduced doping effi-ciency in the nanowires. The continuous size reduction of transistors alongyears, with current characteristic channel lengths of tensof nanometers, implies that the disorder in the distribu-tion of dopants can now determine the performance, inparticular the transport properties of the devices.
Inspecific geometries, like the nonplanar field effect transis-tors denoted by FinFETs, isolated donors can be iden- tified and its charge states (neutral D , and negativelycharged D − ) studied by transport spectroscopy.The existence of D − donor states in semiconductors,analogous to the hydrogen negative ion H − , was sug-gested in the fifties and is now well established exper-imentally. Negatively charged donors in bulk Si werefirst detected by photoconductivity measurements. Thebinding energies of D − donors, defined as the energy re-quired to remove one electron from the ion ( D − → D +free-electron) E D − B = E D − E D − , are found experimen-tally to be small ( E D − B ∼ . ∼ . E D B (45 meV for P and 54 meV for As).For zero applied magnetic fields, no excited bound statesof D − in bulk semiconductors or superlattices arefound, similar to H − which has only one bound state inthree-dimensions as shown in Refs. 20,21.A relevant characteristic of negatively charged donorsis their charging energy, U = E D − − E D , which givesthe energy required to add a second electron to a neutraldonor. This extra energy is due to the Coulomb repulsionbetween the two bound electrons, and does not contributein one electron systems, as D . The measured values inbulk Si are U bulk , expAs = 52 meV for As and U bulk , expP = 43meV for P.From the stability diagrams obtained from transportspectroscopy measurements we observe that the chargingenergy of As dopants in nanoscale Si devices (FinFETs) isstrongly reduced compared to the well known bulk value.By using a variational approach within the single-valleyeffective mass approximation, we find that this decreaseof the charging energy may be attributed to modifica-tions on the bare insulator screening by the presence of anearby metallic layer. For the same reason, we also findtheoretically that it may be possible to have a D − boundexcited state.This paper is organized as follows. In Sec. II, we in-troduce the formalism for a donor in the bulk in analogywith the hydrogen atom problem. In Sec. III, we studythe problem of a donor close to an interface within a flatband condition. We show experimental results for thecharging energy and compare them with our theoreticalestimations. We also calculate the binding energy of a D − triplet first excited state. In Sec. IV we present dis-cussions including: (i) assessment of the limitations inour theoretical approach, (ii) considerations about themodifications of the screening in nanoscale devices, (iii)the implications of our results in quantum device appli-cations, and, finally, we also summarize our main conclu-sions. II. DONORS IN BULK SILICON
A simple estimate for the binding energies of both D and D − in bulk Si can be obtained using the analogybetween the hydrogen atom H and shallow donor statesin semiconductors. The Hamiltonian for one electron inthe field of a nucleus with charge + e and infinite massis, in effective units of length a B = ¯ h /m e e and energy Ry = m e e / h , h ( r ) = T ( r ) − r , (1)with T ( r ) = −∇ . The ground state is φ ( r , a ) = 1 √ πa e − r /a (2)with Bohr radius a = 1 a B and energy E H = − Ry .This corresponds to one electron in the 1 s orbital.For negatively charged hydrogen (H − ) the two elec-trons Hamiltonian is H Bulk = h ( r ) + h ( r ) + 2 r , (3)where the last term gives the electron-electron interac-tion ( r = | ~r − ~r | ). As an approximation to the groundstate, we use a relatively simple variational two particleswave-function for the spatial part, a symmetrized combi-nation of 1 s atomic orbitals as given in Eq. 2, since thespin part is a singlet, | s, s, s i = [ φ ( r , a ) φ ( r , b ) + φ ( r , b ) φ ( r , a )] . (4)The resulting energy is E H − = − . Ry with a =0 . a B and b = 3 . a B (binding energy E H − B =0 . Ry ). Here we may interpret a as the radius ofthe inner orbital and b of the outer orbital. This approx-imation for the wave-function correctly gives a boundstate for H − but it underestimates the binding energywith respect to the value E H − B = 0 . Assuming an isotropic single-valley conduction bandin bulk Si the calculation of the D and D − energies E D = − Ry ∗ a = 1 a ∗ E D − = − . Ry ∗ a = 0 . a ∗ ; b = 3 . a ∗ E B = E D - E D − = 0 . Ry ∗ U = E D − -2E D = 0 . Ry ∗ TABLE I: Bulk values of energies and orbital radii for theground state of neutral and negatively charged donors withinour approximation (see text for discussion). Effective unitsfor Si are a ∗ = 2 .
14 nm and Ry ∗ = 31 . r − − z=−d z=0 r+ Barrier Si z FIG. 1: (Color online) Schematic representation of a nega-tively charged donor in Si (solid circles) located a distance d from an interface. The open circles in the barrier (left)represent the image charges. The sign and magnitude ofthese charges depend on the relation between the dielectricconstants of Si and the barrier given by Q = ( ǫ barrier − ǫ Si ) / ( ǫ barrier + ǫ Si ). For the electrons, Q <
Q >
0, see Eq. 6). reduces to the case of H just described. Within this ap-proximation, an estimation for E D − B can then be obtainedby considering an effective rydberg Ry ∗ = m ∗ e / ǫ ¯ h with an isotropic effective mass (we use ǫ Si = 11 . m ∗ = 0 . m e so that the ground state energyfor a neutral donor is the same as given by an anisotropicwave-function in bulk: within a single valley approxima-tion E D B = − Ry ∗ = − . a = 1 a ∗ with a ∗ = ¯ h ǫ Si /m ∗ e = 2 .
14 nm. Inthis approximation, E D − B = 0 .
84 meV. In the same way,an estimation for the charging energy can be made fordonors in Si: U = 0 . Ry ∗ = 30 .
35 meV. Even though the trial wave function in Eq. (4) underes-timates the binding energy, we adopt it here for simplic-ity, in particular to allow performing in a reasonably sim-ple way the calculations for a negatively charged donorclose to an interface reported below. In the same way, wedo not introduce the multivalley structure of the conduc-tion band of Si. The approximations proposed here leadto qualitative estimates and establish general trends forthe effects of an interface on a donor energy spectrum.The limitations and consequences of our approach arediscussed in Sec. IV. d (a*) -1-0.9-0.8-0.7-0.6 E D ( R y* ) Q=-0.5Q=0Q=0.5
FIG. 2: (Color online) Energy of the neutral donor versusits distance d from an interface for different values of Q =( ǫ barrier − ǫ Si ) / ( ǫ barrier + ǫ Si ). III. DONORS CLOSE TO AN INTERFACEA. D and D − ground states We consider now a donor (at z = 0) close to an inter-face (at z = − d ) (see Fig. 1). Assuming that the interfaceproduces an infinite barrier potential, we adopt varia-tional wave-functions with the same form as in Eqs. (2)and (4) multiplied by linear factors ( z i + d ) ( i = 1 , D − , we brieflypresent results for the neutral donor D which are in-volved in defining donor binding and charging energies.For this case, the Hamiltonian is H ( r ) = h ( r ) + h images ( r ) (5)with h ( r ) as in Eq. 1 and h images ( r ) = − Q z + d ) + 2 Q p x + y + ( z + 2 d ) , (6)where Q = ( ǫ barrier − ǫ Si ) / ( ǫ barrier + ǫ Si ). ǫ barrier is thedielectric constant of the barrier material. The first termin h images is the interaction of the electron with its ownimage, and the second is the interaction of the electronwith the donor’s image. If the barrier is a thick insulator,for example SiO with dielectric constant ǫ SiO = 3 . Q < Q = − . ),which prevents charge leakage, plus metallic electrodeswhich control transport and charge in the semiconductor.This composite heterostructure may effectively behave asa barrier with an effective dielectric constant larger thanSi, since ǫ metal → ∞ , leading to an effective Q > Q , the net image potentialswill be repulsive or attractive, which may strongly affectthe binding energies of donors at a short distance d fromthe interface.Using a trial wave-function φ D ∝ e − r/a ( z + d ), mostof the integrals involved in the variational calculation of E D can be performed analytically. E D is shown inFig. 2 for different values of Q and compare very well withthe energy calculated by MacMillen and Landman for Q = − d . For Q < d ∼ a ∗ . This minimum arisesbecause the donor image attractive potential enhancesthe binding energy but, as d gets smaller, the fact thatthe electron’s wave-function is constrained to z > − d dominates, leading to a strong decrease in the bindingenergy. Q = 0 corresponds to ignoring the images. Q =1 would correspond to having a metal at the interfacewith an infinitesimal insulating barrier at the interface toprevent leackage of the wave-function into the metal. We show results for Q = 0 . E = − Ry ∗ is reached at long distances for all values of Q .Adding a second electron to a donor requires the inclu-sion of the electron-electron interaction terms. The neg-ative donor Hamiltonian parameters are schematicallypresented in Fig. 1, and the total two electrons Hamilto-nian is H = H ( r ) + H ( r ) + 2 r − Q p ( x − x ) + ( y − y ) + (2 d + z + z ) , (7)where H ( r i ) includes the one-particle images (Eq. 5) andthe last term is the interaction between each electron andthe other electron’s image.In Figs. 3 and 4, we plot E D − and the binding en-ergy E D − B = E D − E D − assuming a trial wave-function ∝ [ φ ( r , a ) φ ( r , b ) + φ ( r , b ) φ ( r , a )] ( z + d )( z + d ) withvariational parameters a and b , for Q = − . Q = 0 . a ∼ a ∗ while b , the radius of the outer orbital, depends verystrongly on Q and d and is shown in Figs. 3(c) and 4(c).We have done calculations for several values of Q , rang-ing from Q = +1 to Q = −
1. The general trends andqualitative behavior of the calculated quantities versusdistance d are the same for all Q > Q ≤ Q ≤ Q = − . D − is not bound for small d (for d < a ∗ in the case of Q = − . d ’s, thebinding energy is slightly enhanced from the bulk value.The radius of the outer orbital b is very close to the bulkvalue for d ≥ a ∗ . For Q > Q = 0 . D − is bound at all distances d , though thebinding energy is smaller than in bulk. The radius of theouter orbital b is very large and increases linearly with -1.05-1-0.95-0.9 e n e r gy ( R y* ) D D - (1s,1s,s)D - (1s,2s,t) E B ( R y* ) d(a*) b ( a * ) E(D )E(D - (1s,1s,s))E B (D - (1s,1s,s)E B (D - (1s,2s,t)=0 FIG. 3: (Color online) Results for Q = − .
5. (a) Energy fora neutral donor D , and for the ground D − | s, s, s i and firstexcited D − | s, s, t i negatively charged donor. (b) Bindingenergies of the D − states. (c) Value of the variational param-eter b for the D − ground state. For d < D − | s, s, s i is notstable and the energy is minimized with b → ∞ . Bulk valuesare represented by short line segments on the right. d up to d crossover ∼ . a ∗ [see Fig. 4 (c)]. For larger d , b is suddenly reduced to its bulk value. This abruptbehavior of the b that minimizes the energy is due to twolocal minima in the energy versus b : for d < d crossover the absolute minimum corresponds to a very large (butfinite) orbital radius b while for d > d crossover the abso-lute minimum crosses over to the other local minimum,at b ∼ b bulk . As d increases from the smallest values and b increases up to the discontinuous drop, a “kink” in theD − binding energy is obtained at the crossover point [seeFig. 4 (b)], changing its behavior from a decreasing toan increasing dependence on d towards the bulk value as d → ∞ . B. Charging energy: experimental results
The charging energy of shallow dopants can be ob-tained by using the combined results of photoconductiv-ity experiments to determine the D − binding energy and direct optical spectroscopy to determine the bindingof the D state. It was shown recently that the chargingenergy in nanostructures can be obtained directly fromcharge transport spectroscopy at low temperature. Sin-gle dopants can be accessed electronically at low temper-ature in deterministically doped silicon/silicon-dioxideheterostructures and in small silicon nanowire field ef-fect transistors (FinFETs), where the dopants are posi-tioned randomly in the channel. Here we will fo-cus in particular on data obtained using the latter struc- -1-0.95-0.9-0.85-0.8-0.75-0.7 e n e r gy ( R y* ) D D - (1s,1s,s)D - (1s,2s,t) E B ( R y* ) d(a*) b ( a * ) E(D )E(D - (1s,1s,s))E B (D - (1s,1s,s))E B (D - (1s,2s,t))=0 FIG. 4: (Color online) Same as Fig. 3 for Q = 0 . tures. FinFET devices in which single dopant transport havebeen observed typically consist of crystalline silicon wirechannels with large patterned contacts fabricated onsilicon-on-insulator. Details of the fabrication can befound in Ref. 15. In this kind of samples, few dopantsmay diffuse from the source/drain contacts into the chan-nel during the fabrication modifing the device character-istics both at room and low temperatures. Insome cases, subthreshold transport is dominated by asingle dopant. Low temperature transport spectroscopy relies on thepresence of efficient Coulomb blockade with approxi-mately zero current in the blocked region. This requiresthe thermal energy of the electrons, k B T , to be muchsmaller than U , a requirement that is typically satisfiedfor shallow dopants in silicon at liquid helium temper-ature and below, i.e. ≤ . K . At these temperaturesthe current is blocked in a diamond-shaped region in astability diagram, a color-scale plot of the current – ordifferential conductance d I/ d V b – as a function of thesource/drain, V b , and gate voltage, V g .In Fig. 5, the stability diagram of a FinFET with onlyone As dopant in the conduction channel is shown. Atsmall bias voltage ( eV b ≪ k B T ), increasing the voltageon the gate effectively lowers the potential of the donorsuch that the different donor charge states can becomedegenerate with respect to the chemical potentials in thesource and drain contacts and current can flow. The dif-ference in gate voltage between the D + / D and D / D − degeneracy points (related to the charging energy) de-pends, usually in good approximation, linearly on thegate voltage times a constant capacitive coupling to thedonor. Generally a more accurate and direct way todetermine the charging energy is to determine the biasvoltage at which the Coulomb blockade for a given charge
Differential Conductance [μS]Vg [mV] V b [ m V ]
220 240 260 280 300 320 340 360 380 400 420−50−40−30−20−1001020304050 25 20 15 10 5 0
T = 0.3 K SD A G V b V g U = 36 meV
FIG. 5: (Color online) Differential conductance stability di-agram showing the transport characteristics of a single Asdonor in a FinFET device. The differential conductance isobtained by a numerical differentiation of the current with re-spect to V b at a temperature of 0 . D charge state in this case) is lifted for all V g . The transition point is indicated by the horizontal arrow,leading to a charging energy U = 36 meV, as given by thevertical double-arrow. The inset shows the electrical circuitused for the measurement. state is lifted for all gate voltages, indicated by the hor-izontal arrow in Fig. 5. This method is especially usefulwhen there is efficient Coulomb blockade. For the partic-ular sample shown in Fig. 5, U = 36 meV. This is similarto other reported values in the literature rangingfrom ∼
26 to ∼
36 meV. There is therefore a strongreduction in the charging energy compared to the bulkvalue U bulk = 52 meV. The ratio between the observedand the bulk value is ∼ . − . d for Q = − .
5, 0, and 0 . U of the order of theone observed occurs at d ∼ a ∗ for 0 . < Q < Q = 0 . U is consistent with apredominant influence of the metallic gates material inthe D − energetics. On the other hand, for Q ≤ U is slightly enhanced as d decreases and, for the smallestvalues of d considered, the outer orbital is not bound. Atvery short distances d , the difference in behavior betweenthe insulating barrier ( Q <
0) and the barrier with moremetallic character (
Q >
0) is in the interaction betweeneach electron and the other electron’s image, which is re-pulsive in the former case and attractive in the latter.Although this interaction is small, it is critical to lead toa bound D − for Q > D − for Q < d . d(a*) U ( R y* ) Q=-0.5Q=0Q=0.5
FIG. 6: (Color online) Charging energy U of the D − groundstate for three different values of Q . For Q ≤
0, the chargingenergy is nearly constant with d . For these cases, the nega-tively charged donor is not bound for small d . For Q > d . The latter is consistentwith the experimental observation. C. D − first excited state. It is well established that in 3 dimensions (with nomagnetic field applied) there is only one bound stateof D − . Motivated by the significant changes in theground state energy produced by nearby interfaces, weexplore the possibility of having a bound excited state ina double-charged single donor. Like helium, we expectthe D − first excited state to consist of promoting one 1 s electron to the 2 s orbital. The spin triplet | s, s, t i state(which is orthogonal to the singlet ground state) has alower energy than | s, s, s i . As a trial wave-functionfor | s, s, t i we use the antisymmetrized product of thetwo orbitals 1 s and 2 s and multiply by ( z + d )( z + d )to fulfill the boundary condition, namely,Ψ s, s,t = N h e − r a e − r b (cid:16) r b − (cid:17) − e − r a e − r b (cid:16) r b − (cid:17)i × ( z + d )( z + d ) (8)with a and b variational parameters and N a normaliza-tion factor. Note that, for a particular value of b , theouter electron in a 2 s orbital would have a larger effec-tive orbital radius than in a 1 s orbital due to the differentform of the radial part.For Q <
0, the outer orbital is not bound and theenergy reduces to that of D (see Fig. 3). Surprisingly,for Q > | s, s, t i state is bound and, as d increases,tends very slowly to the D energy as shown in Fig. 4.Moreover, its binding energy is roughly the same as theground state | s, s, s i for d ≤ a ∗ , another unexpectedresult. The existence of a bound D − triplet state opensthe possibility of performing coherent rotations involvingthis state and the nearby singlet ground state. IV. DISCUSSIONS AND CONCLUSIONS
Our model for D − centers involves a number of sim-plifications: (i) the mass anisotropy is not included; (ii)the multivalley structure of the conduction band of Si isnot considered; (iii) correlation terms in the trial wave-function are neglected. These assumptions aim to de-crease the number of variational parameters while allow-ing many of the integrals to be solved analytically.Qualitatively, regarding assumption (i), it has beenshown that the mass anisotropy inclusion gives an in-crease of the binding energy for both D and D − (seeRef. 29); regarding (ii), inclusion of the multivalley struc-ture of the conduction together with the anisotropy of themass would lead to an enhancement of the binding energyof D − due to the possibility of having intervalley config-urations in which the electrons occupy valleys in ’per-pendicular’ orientations, (with perpendicularly oblatedwave-functions), thus leading to a strong reduction ofthe electron-electron repulsive interaction. Regard-ing point (iii), more general trial wave-functions for D − have been proposed in the literature. For example, theone suggested by Chandrasekar models correlation effectsby multiplying Eq. 4 by a factor, (1 + Cr ), where C is an additional variational parameter. In the bulk, theeffect of this correlation factor is to increase the bind-ing energy of D − from 0 . Ry ∗ if C = 0 (our case) to0 . Ry ∗ . We conclude that all three simplificationsassumed in our model lead to an underestimation of thebinding energy of D − , thus, the values reported here areto be taken as lower bounds for it.As compared to experiments, an important differencewith respect to the theory is that we are assuming a flat-band condition while the actual devices have a built-inelectric field due to band-bending at the interface be-tween the gate oxide and the p-doped channel. Ifan electric field were included, the electron would feel astronger binding potential (which results from the addi-tion of the donor potential and the triangular potentialwell formed at the interface) leading to an enhancementof the binding energy of D and D − (with an expectedstrong decrease of the electron-electron interaction in thiscase for configurations with one electron bound to thedonor at z = 0 and the other pulled to the interface at z = − d ).The presented results are dominated by the presenceof a barrier, which constrains the electron to the z > − d region, and the modification of the screening due to thecharge induced at the interface, a consequence of the di-electric mismatch between Si and the barrier material.This is included by means of image charges. Effects ofquantum confinement and dielectric confinement arenot considered here: we believe these are not relevantin the FinFETs under study. Although the conductionchannel is very narrow (4 nm ) the full cross section ofthe Si wire is various tens of nm and quantum and di-electric confinement is expected to be effective for typicaldevice sizes under 10 nm. Both quantum and dielectric confinement lead to an enhancement of the binding en-ergy with respect to the bulk, which is the opposite towhat we obtain for small d .Neutral double donors in Si, such as Te or Se, have beenproposed for spin readout via spin-to-charge conversion and for spin coherence time measurements. The nega-tive donor D − also constitutes a two-electron system,shallower than Te and Se. In this context, investigationof the properties of D − shallow donors in Si affectingquantum operations as, for example, their adequacy forimplementing spin measurement via spin-to-charge con-version mechanism, deserve special attention. Ourtheoretical study indicates that, very near an interface(for d < a ∗ ), the stability of D − against dissociationrequires architectures that yield effective dielectric mis-match Q >
0, a requirement for any device involvingoperations or gates based on D − bound states.In conclusion, we have presented a comprehensivestudy of the effects of interface dielectric mismatch inthe charging energy of nearby negatively charged donorsin Si. In our study, the theoretical treatment is based ona single-valley effective mass formalism, while transportspectroscopy experiments were carried out in FinFET de-vices. The experiments reveal a strong reduction on thecharging energy of isolated As dopants in FinFETs ascompared to the bulk values. Calculations present, be-sides the charging energy, the binding energy of donor inthree different charge states as a function of the distancebetween the donor and an interface with a barrier. Theboundary problem is solved by including the charge im-ages whose signs depend on the difference between thedielectric constant of Si and that of the barrier material[the dielectric mismatch, quantified by the parameter Q defined below Eq. (6)].Typically, thin insulating layers separate the Si chan-nel, where the dopants are located, from metallic gatesneeded to control the electric fields applied to the de-vice. This heterostructured barrier leads to an effectivescreening with predominance of the metallic components,if compared to a purely SiO thick layer, for which Q < Q = 0 . ǫ barrier = 3 ǫ Si ), we obtain a reduction ofthe charging energy U relative to U bulk at small d , consis-tent with the experimental observation. We did not at-tempt quantitative agreement between presented valueshere, but merely to reproduce the right trends and clar-ify the underling physics. It is clear from our results thatmore elaborate theoretical work on interface effects indonors, beyond the simplifying assumptions here, shouldtake into account the effective screening parameter as acombined effect of the nearby barrier material and theadjacent metallic electrodes. From our calculations andexperimental results, we conclude that the presence ofmetallic gates tend to increase ǫ effectivebarrier above ǫ Si , leadingto Q >
Acknowledgments
M.J.C. acknowledges support from Ram´on y CajalProgram and FIS2009-08744 (MICINN, Spain). B.K. ac-knowledges support from the Brazilian entities CNPq, In-stituto Nacional de Ciencia e Tecnologia em Informa¸c˜aoQuantica - MCT, and FAPERJ. J.V., G.P.L, G.C.T and S.R. acknowledge the financial support from the EC FP7FET-proactive NanoICT projects MOLOC (215750) andAFSiD (214989) and the Dutch Fundamenteel Onderzoekder Materie FOM. We thank N. Collaert and S. Biese-mans at IMEC, Leuven for the fabrication of the dopantdevice. B. E. Kane, Nature , 133 (1998). R. Vrijen, E. Yablonovitch, K. Wang, H.-W. Jiang, A. Ba-landin, V. Roychowdhury, T. Mor, and D. DiVincenzo,Phys. Rev. A , 012306 (2000). A. J. Skinner, M. E. Davenport, and B. E. Kane, Phys.Rev. Lett. , 087901 (2003). S. D. Barrett and G. J. Milburn, Phys. Rev. B , 155307(2003). L. C. L. Hollenberg, A. S. Dzurak, C. Wellard, A. R. Hamil-ton, D. J. Reilly, G. J. Milburn, and R. G. Clark, Phys.Rev. B , 113301 (2004). M. J. Calder´on, B. Koiller, X. Hu, and S. Das Sarma, Phys.Rev. Lett. , 096802 (2006). D. MacMillen and U. Landman, Phys. Rev. B , 4524(1984). M. J. Calder´on, B. Koiller, and S. Das Sarma, Phys. Rev.B , 125311 (2007). C. Delerue and M. Lannoo,
Nanostructures. Theory andmodelling (Springer-Verlag (Berlin), 2004). M. Diarra, Y.-M. Niquet, C. Delerue, and G. Allan, Phys.Rev. B , 045301 (2007). M. T. Bj¨ork, H. Schmid, J. Knoch, H. Riel, and W. Riess,Nature Nanotechnology , 103 (2009). P. M. Voyles, D. A. Muller, J. L. Grazui, P. H. Citrin, andH.-J. L. Grossmann, Nature , 827 (2002). T. Shinada, S. Okamoto, T. Kobayashi, and I. Ohdomari,Nature (London) , 1128 (2005). M. Pierre, R. Wacquez, X. Jehl, M. Sanquer, M. Vinet,and O. Cueto, Nature Nanotechnology , 133 (2009). H. Sellier, G. P. Lansbergen, J. Caro, S. Rogge, N. Collaert,I. Ferain, M. Jurczak, and S. Biesemans, Phys. Rev. Lett. , 206805 (2006). M. A. Lampert, Phys. Rev. Lett. , 450 (1958). M. Taniguchi and S. Narita, Solid State Communications , 131 (1976). D. M. Larsen and S. Y. McCann, Phys. Rev. B , 3485(1992). J. M. Shi, F. M. Peeters, and J. T. Devreese, Phys. Rev.B , 7714 (1995). C. L. Pekeris, Phys. Rev. , 1470 (1962). R. Hill, Phys. Rev. Lett. , 643 (1977). H. Bethe and S. Salpeter,
Quantum mechanics of one andtwo electron atoms (Dover publications, NY, 2008). A. Ramdas and S. Rodriguez, Reports on Progress inPhysics , 1297 (1981). A. F. Slachmuylders, B. Partoens, F. M. Peeters, andW. Magnus, Appl. Phys. Lett. , 083104 (2008). P. Dean, J. Haynes, and W. Flood, Physical Review ,711 (1967). A. Morello, J. J. Pla, F. A. Zwanenburg, K. W.Chan, H. H. M. Mottonen, C. D. Nugroho, C. Yang,J. A. van Donkelaar, A. Alves, D. N. Jamieson, et al.,arXiv:1003.2673 (2010). G. P. Lansbergen, R. Rahman, C. J. Wellard, I. Woo,J. Caro, N. Collaert, S. Biesemans, G. Klimeck, L. C. L.Hollenberg, and S. Rogge, Nature Physics , 656 (2008). B. Bransden and C. Joachain,
Physics of atoms andmolecules (Prentice Hall, 2003). J.-i. Inoue, J. Nakamura, and A. Natori, Phys. Rev. B ,125213 (2008). D. M. Larsen, Phys. Rev. B , 5521 (1981). S. Chandrasekhar, Rev. Mod. Phys. , 301 (1944). R. Rahman, G. P. Lansbergen, S. H. Park, J. Verduijn,G. Klimeck, S. Rogge, and L. C. L. Hollenberg, Phys. Rev.B , 165314 (2009). H. Sellier, G. P. Lansbergen, J. Caro, S. Rogge, N. Collaert,I. Ferain, M. Jurczak, and S. Biesemans, Appl. Phys. Lett. , 073502 (2007). B. E. Kane, N. S. McAlpine, A. S. Dzurak, R. G. Clark,G. J. Milburn, H. B. Sun, and H. Wiseman, Phys. Rev. B , 2961 (2000). M. Calder´on, B. Koiller, and S. Das Sarma, Phys. Rev. B , 161304(R) (2007). F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Han-son, L. H. W. van Beveren, I. T. Vink, H. P. Tranitz,W. Wegscheider, L. P. Kouwenhoven, and L. M. K. Van-dersypen, Science309