A valleytronic diamond transistor: electrostatic control of valley-currents and charge state manipulation of NV centers
Nattakarn Suntornwipat, Saman Majdi, Markus Gabrysch, Kiran Kumar Kovi, Viktor Djurberg, Ian Friel, Daniel J. Twitchen, Jan Isberg
11 A valleytronic diamond transistor:electrostatic control of valley-currents and charge state manipulation ofNV centers
N. Suntornwipat , S. Majdi , M. Gabrysch , K. K. Kovi , V. Djurberg , I. Friel , D. J. Twitchen and J.Isberg Division for Electricity, Department of Electrical Engineering, Uppsala University, Box 65, 751 03, Uppsala, Sweden. Element Six, Global Innovation Centre, Fermi Ave, Harwell Oxford, Oxfordshire OX11 0QR, United Kingdom. Center for Nanoscale Materials, Argonne National Laboratory, Argonne, IL-60439, United States. (Dated: 17nd Sep 2020)
The valley degree of freedom in many-valley semiconductors provides a new paradigm for storing andprocessing information in valleytronic and quantum-computing applications. Achieving practical devicesrequire all-electric control of long-lived valley-polarized states, without the use of strong external magneticfields. Attributable to the extreme strength of the carbon-carbon bond, diamond possesses exceptionallystable valley states which provides a useful platform for valleytronic devices. Using ultra-pure single-crystalline diamond, we here demonstrate electrostatic control of valley-currents in a dual gate field-effecttransistor, where the electrons are generated with a short UV pulse. The charge- and the valley- currentmeasured at receiving electrodes are controlled separately by varying the gate voltages. A proposed modelbased on drift-diffusion equations coupled through rate terms, with the rates computed by microscopicMonte Carlo simulations, is used to interpret experimental data. As an application, valley-current charge-state modulation of nitrogen-vacancy (NV) centers is demonstrated.Charge and spin are both well-defined quantum numbersin solids. The same is true for the valley pseudospin, alsoknown as valley polarization, in various multi-valleysemiconductor materials. The valley pseudospin degreeof freedom presents an attractive resource forinformation processing, a subject that has been termed valleytronics in analogy with spintronics for spin-basedtechnology. The topic of valleytronics has attractedconsiderable attention lately because of its fundamentalinterest to the physics community. Due to the largeseparation in momentum space between valleys incertain materials, the valley pseudospin can be robustagainst lattice deformation and low-energy phononscattering. Creation and detection of valley-polarizedelectrons have been achieved in several materials, suchas in AlAs where valley polarization was induced by asymmetry-breaking strain, in MoS by means ofcircularly polarized light and in bulk bismuth by usinga rotating magnetic field to modulate the differentcontributions of different valleys. Previously, it hasbeen demonstrated that valley-polarized states can be created in diamond. This occurs by the hot electronrepopulation effect in a high electric field or byseparating an initially unpolarized population intodifferently polarized electron states by means of crossedelectric and magnetic fields, i.e., by the Hall effect.However, to realize practical valleytronicapplications, establishing a fast, scalable and directelectrical control of valley states is crucial. So far,electrical control of valley transport has not been studiedexperimentally outside the field of topological transportin low-dimensional materials. In this article, wedemonstrate such electrical control of valley transport indouble-gated diamond field-effect transistors (FET),where a short UV pulse is used to generate the valley-polarized electrons. We find that it is possible to controlthe charge- and valley- current separately at receivingelectrodes, thereby enabling rapid and scalable valley-current control. We also demonstrate charge-statemodulation of nitrogen-vacancy (NV) centers indiamond through valley-currents. NV centers in diamondhave important applications in e.g. single-spinmagnetometry and single photon sources Ourtransistors show that it is possible to manipulate thecharge states of NV centers locally in devices solely byapplication of a gate bias voltage. *Corresponding author:
Prof. Jan Isberg
Email: [email protected]
Diamond is unique among solids as the extremerigidity of the lattice leads to suppressed intervalleyscattering and highly stable valley states. Diamond alsoexhibits many other valuable properties for solid-stateQIP, e.g. a high electron mobility and the existence ofthe NV center with ultra-long spin coherence time. Theconduction band structure of diamond is similar to thatof silicon, with six equivalent conduction band valleysoriented along the {100} axes. The minima aresymmetrically situated at 76% of the distance from the (cid:0) point to the X point as indicated in the energy diagram inFigure 1(a). At temperatures below ~ 80 K, theintervalley scattering rates become so low (< 10 s − ) thatelectrons are effectively confined to a given valley. Therefore, electrons exist in six different valleypseudospin states. Valley electrons have a large ratiobetween longitudinal effective mass ( (cid:1865) || ) andtransversal effective mass ( (cid:1865) (cid:2884) ), i.e. (cid:1865) (cid:2884) ≈ 5.5 (cid:1865) || . This, in conjunction with the freeze-out of electrons intodifferent valleys at low temperatures, leads to a break ofthe cubic symmetry of the crystal and to anisotropiccharge transport properties that depend on the specificvalue of the pseudospin. This dependence makes itpossible to separate electrons with different pseudospinsby their drift in a suitably arranged electric field, asillustrated in Figure 1(a). The anisotropic energydispersion results in a conduction anisotropy, differentfor electrons in valleys on different axes. In an appliedfield ( (cid:1831)(cid:3364) ), electrons in separate valleys acquire differentdrift velocities ( (cid:1874)̅ ). If intervalley scattering rates aresmall, this results in a separation of the electronpopulations. Diamond FET devices on selected single-crystallinediamond plates were made to demonstrate electrostaticcontrol of valley pseudospin states. These plates weresynthesized by Element Six Ltd. with ultra-low nitrogenimpurity concentration, below 0.05 ppb. FET:s indiamond have been profusely studied in recent years but these have however not been previously employed tostudy valley polarized currents. The designed transistoris depicted in Figure 1(b) and comprises a sourceelectrode, two gate electrodes isolated from the diamondby a 30 nm Al O dielectric layer and two drainelectrodes at which induced currents can be monitored.The Al O layer serves both as an insulating gate oxideand as a surface passivation layer reducing the surfacescattering rate and the interface trap density. The reasonfor the multiple drain electrodes is that we wish to showthat valley-polarized currents can be directed to differentlocations in the crystal. The double-gate configurationallows for more control of the valley-currents than a b) Figure 1. (a) The conduction band structure ofdiamond. The central image of the left figure depicts thefirst Brillouin zone, together with iso-energy surfacessurrounding the six conduction band minima (“valleys”).The blue dot in the band structure plot (upper left)indicates the position of one conduction band minimum.The remaining images illustrate the energy dispersion indifferent valleys. As different valleys have differentanisotropic effective mass tensors, electrons in differentvalleys drift in different directions when an electric fieldis applied. (b) A schematic of a dual-gate two-draintransistor together with a simulation showing the electrondensity in different valleys a certain time afterillumination. G1 and G2 are two independentlycontrollable gates and D1 and D2 are the two drains. Athin ion-implanted layer with NV centers is incorporated.Electroluminescence from the NV centers can beobserved through a semitransparent back contact. single-gate configuration does. Holding one gate at a constant voltage while varying the bias on the second gate offer improved ability to separate the charge clouds with precision. With the chosen geometry, drain currents are only induced when charges are drifting in close proximity to the contacts. We emphasize, in contrast to (Ref. 15-18), that the electrostatic potential at the gate acts directly through coulomb interaction and is not used to modulate the energy gap or affect the Berry phase. Details of the device processing are described in the supporting information.The charge transport was modelled using drift-diffusion and Poisson equations with several electronconcentrations, one for each valley in the Brillouin zone(BZ). As valleys on the same axis in the BZ have thesame effective mass tensor (to quadratic order) thecontribution from these valley-pairs can be added forsimplicity, resulting in three different electronconcentrations (cid:1866) (cid:3039) ( (cid:1864) = 1,2,3). For low electric fields, theaverage carrier drift velocity for electrons in the (cid:1864) :thvalley, ̅(cid:1874) (cid:3039) , is proportional to the electric field (cid:1831)(cid:3364) , i.e. ̅(cid:1874) (cid:3039) = (cid:3039) for the (cid:1864) :th valley-pair (cid:2246) (cid:3364) . Here, the mobility tensor (cid:1831) (cid:3039) (cid:2246) is related to the effective mass tensor (cid:2169) (cid:3039) by (cid:2246) (cid:3039) ≡ (cid:1869) < (cid:2028) > (cid:3435)(cid:2169) (cid:3039) (cid:3439) (cid:2879)(cid:2869) , where (cid:1869) is the elementary charge and < (cid:2028) > is the average intravalley relaxation time. Since (cid:2246) (cid:3039) isa tensor, the velocity ̅(cid:1874) (cid:3039) and the electric field (cid:1831)(cid:3364) are notparallel in general. As ̅(cid:1874) (cid:3039) depends on the valley index,electrons with different valley pseudospin tend to drift indifferent directions. The effective mass tensor (cid:2169) (cid:3039) isgiven by the curvature of the conduction band ℰ(cid:3435)(cid:1863)(cid:3364)(cid:3439) at therespective band minima, ( (cid:2879) (cid:2169) (cid:2869) ) (cid:3080)(cid:3081) = (cid:2869)ħ (cid:3118) (cid:3105) (cid:3118) ℰ ( (cid:3038)(cid:3364) ) (cid:3105)(cid:3038) (cid:3328) (cid:3105)(cid:3038) (cid:3329) . For acubic semiconductor with the energy minima on theprincipal axes in k-space, such as diamond, this gives: (cid:2246) (cid:2869) = (cid:4684)(cid:2020) || (cid:2020) (cid:2884)
00 0 (cid:2020) (cid:2884) (cid:4685) , (cid:2246) (cid:2870) = (cid:4684)(cid:2020) (cid:2884) (cid:2020) ||
00 0 (cid:2020) (cid:2884) (cid:4685) , (cid:2246) (cid:2871) = (cid:4684)(cid:2020) (cid:2884) (cid:2020) (cid:2884)
00 0 (cid:2020) || (cid:4685) Where the ratio between transversal mobility (cid:2020) (cid:2884) andlongitudinal mobility (cid:2020) || equals the inverse ratio of thecorresponding effective masses, i.e., (cid:2020) (cid:2884) ≈ (cid:2020) || , resulting in a strong charge transport anisotropy.Intervalley f -scattering is incorporated in the model byincluding rate terms in the drift-diffusion equations. Incontrast, g -scattering is inconsequential as it occursbetween valleys on the same axis. Under theseassumptions and with the convention that summation isimplied for repeated Greek indices, the drift-diffusionand Poisson equations read: (cid:2034)(cid:1866) (cid:3039) (cid:2034)(cid:1872) = −(cid:2246) (cid:3080)(cid:3081)(cid:3039) ∇ (cid:3080) (cid:3435)(cid:1866) (cid:3039) ∇ (cid:3081) (cid:2038)(cid:3439) + (cid:1863) (cid:2886) (cid:1846) (cid:3030) ( ∇(cid:2038) ) (cid:2246) (cid:3080)(cid:3081)(cid:3039) (cid:1869) ∇ (cid:3080) ∇ (cid:3081) (cid:1866) (cid:3039) + Γ ( ∇(cid:2038) )2 (cid:3437) (cid:3533) (cid:1866) (cid:3040) − (cid:1866) (cid:3039)(cid:2871)(cid:3040)(cid:2880)(cid:2869) (cid:3441) , (cid:1864) = 1,2,3 ∇ (cid:2870) (cid:2038) = (cid:1869)(cid:2013) (cid:3533) (cid:1866) (cid:3039)(cid:2871)(cid:3039)(cid:2880)(cid:2869) Here, (cid:2038) is the electrostatic potential, Γ is the E-fielddependent intervalley relaxation time (Ref. 19), (cid:1863) (cid:3003) is Boltzmann’s constant and (cid:1846) (cid:3030) is the carrier tempera-ture which is also E-field dependent. The dependence of Γ and (cid:1846) (cid:3030) on the electric field was in the finite element method (FEM) simulation treated by creating a look-up table obtained from Monte Carlo (MC) simu-lations in conjunction with an interpolation scheme (the MC model is described in the supporting informa-tion). (1) (2) Figure 2.
Simulation of valley pseudospin transport ina double-gated transistor. The simulation (a-f) shows theelectron density in different valleys ((001) valleys in redcolor and (010) and (100) valleys in cyan) and at differenttimes (2.5 ns, 8 ns, 13.5 ns, 19 ns, 41.5 ns and 52 nsrespectively) after current injection. The parameters of theapplied bias are: (cid:1848) (cid:2929)(cid:2925)(cid:2931)(cid:2928)(cid:2913)(cid:2915) = − V, (cid:1848) (cid:2891)(cid:2869) = − V, (cid:1848) (cid:2891)(cid:2870) = − V, (cid:1848) (cid:2912)(cid:2911)(cid:2913)(cid:2921)(cid:2926)(cid:2922)(cid:2911)(cid:2924)(cid:2915) = − V, with the drains grounded.The inhomogeneous electric field distribution is in (a)indicated by purple equipotential lines and yellow fieldlines.
An example of a drift-diffusion simulation of thetransistor depicted in Figure 1(b) is shown in Figure 2.Charge is injected at the edge of the source contacts, andstates of different valley pseudospin can be seen to driftto different drain contacts. The simulation show for aspecific set of bias voltages the entire process fromgeneration of the charge, drift and finally detection of theelectrons. For this choice of bias voltages, electrons in(001) valleys predominantly drift to D1 while electronsin other valleys are collected at D2. To further investigatethis, we recorded the simulated current at drains D1 andD2 for a situation where the voltage at G1 was varied andall other bias voltages were held fixed. It can be notedthat the current reaches two separate peaks at differenttimes and at different gate voltages. The contribution tothe current from different valley pseudospins is indicatedin Figure 3(a,b) by red and blue contours. It is clear thatthe two peaks originate from electrons in differentpseudospin states. In this case, the electrons in the (001)valleys travel faster and arrive at the drain contacts earlierthan the electrons in the (010) and (100) valleys.With a suitable choice of gate voltage (cid:1848) (cid:2891)(cid:2869) it ispossible to direct electrons in the (010) and (100) valleysto one contact and (001) electrons to another. Figure 3(c)shows a simulation of how the degree of valleypseudospin arriving at the second drain (D2) can bemodulated by varying the gate voltage (cid:1848) (cid:2891)(cid:2869) . Thisdemonstrates theoretically that in our devices the (001)valleys’ contribution to the total charge can be modulatedwith a very high fidelity, from 10% to 90%. In our first experiment, the edge of the transistorsource contact was illuminated using a passivelyQ-switched 213 nm wavelength laser with a repetitionrate of 300 Hz. This was done to measure the charge drifttime and to compare it with the simulation. To reduceelectron-electron scattering to negligible levels, the pulseenergy was limited to < 1 nJ/pulse by an attenuatorresulting in a peak carrier concentration < 10 cm − .Below this concentration the electric field is influencedby less than 5% by the space charge in the electronclouds. A low power used, < 0.3 µW, ensures negligiblesample heating. Electron-hole pairs were created near theedge of the source electrode by short pulses (800 ps) ofphotons with above-bandgap energy ( ℎ(cid:1874) = 5.82 eV > (cid:1831) (cid:3034)(cid:3028)(cid:3043) = 5.47 eV). The optical excitation creates electron-hole pairs, with the electrons equally populating the sixvalleys. By applying different negative bias voltages atthe source, the gate and the back contacts, the electronsdrift towards the drain contacts with different velocitiesand directions depending on their valley polarization.The holes are rapidly extracted at the source electrodeand their contribution to the drain current is negligible.The induced currents were measured at the draincontacts, which were held at (virtual) ground potential.Further experimental details are given in the supportinginformation.Figure 4 shows measured time-resolved currents fromthe transistor where we have applied the same set of biasvoltages as in the simulation described earlier. Thecurrents originating from longitudinal and transversal Figure 3.
Color plot of the simulated time-resolved total induced currents in (a) D1 and (b) D2. The voltage at G1 isvaried between and − V in steps of
V, and the other bias voltages are fixed: (cid:1848) (cid:2929)(cid:2925)(cid:2931)(cid:2928)(cid:2913)(cid:2915) = − V, (cid:1848) (cid:2891)(cid:2870) = − V, (cid:1848) (cid:2912)(cid:2911)(cid:2913)(cid:2921)(cid:2926)(cid:2922)(cid:2911)(cid:2924)(cid:2915) = − V and drains grounded. The superimposed red and blue contours specify the contribution to the totalcurrent from different valleys. (c) The simulated (001) valley contribution to the total charge collected at drain D2, for thesame bias voltages as for (a) and (b), at 78 K. valleys are observed as two distinguished peaks and arehighlighted by the dashed lines. These two current peakscan be observed for different gate voltages and also atdifferent times. The behavior of the measured currents iswell reproduced in the carrier transport simulations, ascan be noted by comparing with Figure 3. The differencein carrier arrival times shows that these peaks can beattributed to electrons in different states of valleypseudospin arriving at the drain contacts.Experimentally observed arrival times are within 8%of the values obtained from the simulation. The absolutevalues of the gate voltages at peak current are slightlyunderestimated in the simulation (5-20%) which can beexplained by the presence of interface states that are notaccounted for in the simulation. The main differencebetween the experimental and the simulated data is thatthe peaks are broader experimentally. This is presumablydue to small inhomogeneities in the electric fieldoriginating from extended defects such as dislocationbundles. In the second experiment we have used ourtransistors to achieve charge-state manipulation of NVcenters in diamond. This was done by adding a thin (120 nm) layer with NV centers below the top electrodesas in Figure 1(b). This layer was added to demonstratethe charge-state manipulation and to make it possible tooptically monitor current densities byelectroluminescence (EL). The EL was observedthrough the sample and the semitransparent back contactusing a home-built microscope equipped with a cooledsCMOS camera. The rear surface was covered with a thingold layer providing an optically semitransparent backcontact, which provides a well-defined referencepotential, see more details in supporting information.In this experiment electron-hole pairs were generatednear the source electrode by the 213 nm wavelength laserat a repetition rate of 660 Hz and a pulse energy of10 nJ/pulse. The sample was from below imaged with aCCD camera through the semitransparent contact.Exposure times of several seconds ensured that enoughlight was collected to yield clear images. Figure 5(b,c)shows how the observed electroluminescence shiftsbetween different locations in the device solely byvarying the gate voltages. These images are compositeswith two different exposure times, 60 ms inside thedashed yellow circle and 8 s outside. The line inside thecircle is strong luminescence from the region where
Figure 4.
Experimentally observed time-resolved currents in a double-gated transistor. Time-resolved draincurrents, (a) at D1 and (b) at D2, measured in a double-gated transistor at 78 K as a function of applied gate voltage, (cid:1848) (cid:2891)(cid:2869) , and time after carrier injection. The same set of bias voltages used in the simulations has been applied in theexperiment: (cid:1848) (cid:2929)(cid:2925)(cid:2931)(cid:2928)(cid:2913)(cid:2915) = − V, (cid:1848) (cid:2891)(cid:2870) = − V, (cid:1848) (cid:2912)(cid:2911)(cid:2913)(cid:2921)(cid:2926)(cid:2922)(cid:2911)(cid:2924)(cid:2915) = − V and drains grounded. electron-hole pairs were generated. The EL wasobserved at D1 for V (cid:2891)(cid:2869) = − V and V (cid:2891)(cid:2870) = − V(Figure 5b), while it clearly shifted to D2 by changing thegate bias to V (cid:2891)(cid:2869) = − V and V (cid:2891)(cid:2870) = − V(Figure 5c). In Figure 5(b,c), the EL spectrum comesfrom the rightmost part of the device (to the right of G2).The illuminated area was masked off so that light fromthis region did not enter the spectrometer and themeasurements were integrated over time. The conductionband electrons that are generated by illumination(213 nm laser pulses) are excited across the indirect bandgap with a possible (many orders of magnitude smaller)contribution from photoionization of NV centers. Thepresence of a dominant 575 nm peak in the luminescence spectra together with several phonon replicas (Figure 5d)show that the luminescence indeed originates fromneutral charge (NV ) centers. The specific opticalsignature of the different charge states of NV centers hasbeen discussed and identified in (Ref. 9,21-23). The EL can be understood by a three stage process: (i) The NV center traps a conduction band electron and is con-verted to the negative charge state (NV − ). (ii) The NV − center reverts to the neutral state by de-trapping the electron, either by direct tunneling to the contact or by hopping conduction via other defects, leaving the NV center in an exited state. (iii) The NV center re-verts to the ground state and emits a photon.In summary, we have demonstrated electrostaticcontrol of valley-currents in double-gated diamond field-effect transistors as well as charge-state manipulation ofNV centers. This was done by detecting different electrondrift times in time-resolved current measurements andalso by direct observation of electroluminescence fromNV centers near the receiving contacts. Transportsimulations of valley-polarized electrons show goodagreement with the observations. These valleytronicdevices enable electrostatic manipulation of valley-currents and can be used to deliver electrons with a highdegree of valley polarization for, e.g., electrical pumpingof color centers in diamond for single photon sources.They o ff er a possible solid state platform for valleytronic-based information processing and for furtherinvestigations into the physics of spin-valley states. Weanticipate that such devices will play a significant role inquantum information processing and future quantumcomputing. ACKNOWLEDGMENT
This study is supported by the Swedish ResearchCouncil (research grant 2018-04154), the ÅForskFoundation (Grant No. 15-288 and 19-427), the OlleEngkvists Foundation (198-0384) and the STandUP forEnergy strategic research framework. The Monte Carlosimulations were performed on resources provided by theSwedish National Infrastructure for Computing (SNIC)through the Uppsala Multidisciplinary Center forAdvanced Computational Science.
Figure 5.
Electroluminescence of the valley pseudospincurrent. (a) Transmitted light optical micrograph of thetransistor. (b,c) Luminescence images at 78 K for differentgate voltages, with the transistor imaged from above. Thetransistor is near the contact surface equipped with a thinlayer of NV centers that luminesce with a typical NV spectral signature, with a strong zero phonon line (ZPL) at575 nm at 78 K, together with associated phonon replicas,as shown in (d). For comparison, the room-temperaturespectrum is also presented in (d). The spectrum is madewith the leftmost part (left of G2) masked off. The positionof the contacts are indicated in (b,c) by dashed white lines.Note the shift in electroluminescence from D1 to D2 as thegate bias is varied. REFERENCES (1) Gunawan, O.; Shkolnikov, Y. P.; Vakili, K.; Gokmen, T.;De Poortere, E. P.; Shayegan, M. Valley Susceptibility ofan Interacting Two-Dimensional Electron System.
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Monte Carlo simulations:
To compute the electric field dependence of the intervalley relaxation time and the carrier temperature Monte Carlosimulations were performed. Because of the low concentration of impurities in our samples impurity scattering wasneglected and only electron-phonon interactions that are allowed by symmetry selection rules were included in thesimulations. Likewise, carrier-carrier scattering was neglected due to the low carrier concentrations considered. We usea simple conduction band structure, consisting of six parabolic but anisotropic (ellipsoidal) valleys. This simplificationis adequate at the moderate electric fields considered in this work and it reduces the computational effort considerably,compared to full-band simulations. Scattering by acoustic phonons is treated through inelastic deformation-potentialinteraction. In a crystal with the valleys centered on the {100} axes, the deformation potential tensor has two independentcomponents for interactions with transversal and longitudinal phonon modes. However, the effect of this anisotropy issmall so as a further simplification it is possible to average over the phonon wave vector angle, which leaves only oneindependent component of the deformation potential tensor. With these simplifications the scattering rate P ac per unitvolume can be written as:
12 1 12 2 ( , ) exp( ) 1 ( ( ) ( ) )
Aac c cB vq DP k k E k E k vv k T kpk d k kr - ì üæ öï ï± = - + × ± -í ýç ÷è øï ïî þ h m m h (S1)where k is the phonon wave vector and k and k k± are the initial and final state electron wave vectors, respectively.The upper sign is used for phonon absorption and the lower sign for phonon emission. v is an averaged sound velocity lt vvv += and r is the material density (3.515 g/cm ). The acoustic deformation potential D A is assumed to be12.0 eV . Intervalley scattering is included in the model through f - and g -scattering deformation potential interactions, (cid:1830) (cid:3033) = (cid:1830) (cid:3034) = eV/cm and we assume a constant transition phonon energy ħ w f = 110 meV, ħ w g = 165 meV. The f- and g- scattering rates per unit volume are given by:
12, , 1 1, ,2 2, ( , ) exp( ) 1 ( ( ) ( ) ) f g f gf g c c f gf g B D P k k E k E k k T p w k d k wrw - ì üæ ö ï ï± = - + ± × ± - ±í ýç ÷è øï ïî þ h h (S2)For the final state selection in the Monte Carlo simulation we keep to the treatment described in (Ref. 1). The simulateddependence of the f -scattering rate and the electron temperature on the electric field is plotted in Figure S1. These areused together with an interpolation scheme in the drift-diffusion charge transport simulations. Figure S1.
Electron temperature and f -scattering rate vs. E-field from Monte Carlo simulations. (a) Data shown forfour different lattice temperatures and for the E-field in parallel with longitudinal (yellow circles) and transversal(red circles) to the major axis of the valley energy ellipsoid. As the E-field increases, the lattice temperatureincreases more rapidly for the electrons on the transversal axis. (b) Illustrates the scattering rate of the electrons indifferent valleys parallel or orthogonal to the E-field. The simulation results reveal a higher scattering rate ratiobetween the transversal and longitudinal valleys at lower temperatures. Sample processing:
The transistors were made using freestanding single-crystalline CVD (001) plates synthesized under conditions of highpurity by Element Six Ltd. The samples are 4.5 × 4.5 mm, with thicknesses ranging from 390 to 510 µm. In ourexperiments, the influence of ionized impurity scattering is minimized by selecting the purest CVD diamond samplesavailable. The concentration of the dominant impurity, nitrogen, is below 10 cm − in all samples, as determined byelectron paramagnetic resonance (EPR). From previous studies it is known that the concentration of ionized impuritiesis less than 10 cm − . In one sample the top (001) surface was ion-implanted at room-temperature with N using aDanfysik 350 kV high current implanter. Three different energy levels and doses, 30 keV (3×10 cm − ), 60 keV(4×10 cm − ) and 90 keV (3×10 cm − ) were used for the implantation. This was followed by anneal at 800 °C for 2 hto create the desired NV centers. On all samples a 30 nm thick Al O oxide layer was deposited on the top (001) oxygenterminated surface of the diamond using a Picosun R200 system for atomic layer deposition (ALD) fromtrimethylaluminum and ozone as precursors. The deposition temperature was 300 °C. The ALD process is initiated withozone present as a last cleaning step and also to ensure high oxygen termination coverage. Openings in the oxide for thesource and drain contacts were made by lithographic patterning and hydrofluoric (HF) etching. The samples weremetallized by Ti/Al (20 nm/300 nm) evaporation and source, gate and drain contacts were formed by standardlithographic techniques and wet etching. The back (001) surface was metallized by evaporating 10 nm Au covering theentire surface providing an optically semitransparent back contact. Experimental details:
The samples are mounted in a customized Janis ST-300MS vacuum cryostat with UV optical access. The sampletemperature was monitored using a LakeShore 331 temperature controller with a calibrated TG-120-CU-HT-1.4H GaAlAs diode sensor in good thermal contact with the sample to keep the temperature constant within 0.1 K. The currentis measured using broadband low-noise amplifiers with R = 50 Ω input impedance, together with a digital samplingoscilloscope (3 GHz, 10 GS/s). The trigger jitter is < 100 ps, which enables averaging over many (typically 100) pulsesto improve the signal-to-noise ratio. Samples were illuminated with 800 ps (FWHM) 213 nm pulses from a CryLaSFQSS213-Q4-STA passively Q-switched DPSS laser. A 213 nm interference filter ensures that no other wavelengths aretransmitted. Reflective UV optics are used for focusing, in conjunction with an EHD UK1158 UV-enhanced CCDcamera for imaging and positioning. The luminescence was monitored with a Dhyana 400D Peltier-cooled sCMOScamera in combination with standard microscope optics and a longpass filter with 550 nm cut-off. The luminescencespectra were obtained using an Ocean Optics S2000 fibre optic spectrometer. REFERENCES (1) Jacoboni, C.; Reggiani, L. The Monte Carlo Method for the Solution of Charge Transport in Semiconductors withApplications to Covalent Materials.
Rev. Mod. Phys. , (3), 645.(2) Hammersberg, J.; Majdi, S.; Kovi, K. K.; Suntornwipat, N.; Gabrysch, M.; Isberg, J. Stability of Polarized Statesfor Diamond Valleytronics. Appl. Phys. Lett. , , 232105.(3) Isberg, J.; Gabrysch, M.; Tajani, A.; Twitchen, D. J. Transient Current Electric Field Profiling of Single CrystalCVD Diamond. Semicond. Sci. Technol. ,21