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L. Banszerus, K. Hecker, L. Wang, S. Möller, [K. Watanabe](https://orcid.org/0000-0003-3701-8119), [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), G. Burkard, C. Volk, C. Stampfer

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[Phonon-limited valley lifetimes in single-particle bilayer graphene quantum dots](https://mdr.nims.go.jp/datasets/e731027d-0b9c-4e70-9343-24a5c1510244)

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Phonon-limited valley lifetimes in single-particle bilayer graphene quantum dotsPHYSICAL REVIEW B 112, 035409 (2025)Editors’ SuggestionPhonon-limited valley lifetimes in single-particle bilayer graphene quantum dotsL. Banszerus ,1,2,*,† K. Hecker ,1,2,* L. Wang ,3 S. Möller ,1,2 K. Watanabe ,4 T. Taniguchi ,5G. Burkard ,3 C. Volk ,1,2 and C. Stampfer 1,21JARA-FIT and 2nd Institute of Physics, RWTH Aachen University, 52074 Aachen, Germany, EU2Peter Grünberg Institute (PGI-9), Forschungszentrum Jülich, 52425 Jülich, Germany, EU3Department of Physics, University of Konstanz, 78457 Konstanz, Germany, EU4Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan5Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan(Received 1 May 2024; accepted 5 June 2025; published 10 July 2025)The valley degree of freedom in two-dimensional (2D) semiconductors, such as gapped bilayer graphene(BLG) and transition metal dichalcogenides, is a promising carrier of quantum information in the emerging fieldof valleytronics. While valley dynamics have been extensively studied for moderate band gap 2D semiconductorsusing optical spectroscopy techniques, very little is known about valley lifetimes in narrow band gap BLG, whichis difficult to study using optical techniques. Here, we report single-particle valley relaxation times T1 exceedingseveral microseconds in electrostatically defined BLG quantum dots using a pulse-gating technique. The ob-served dependence of T1 on perpendicular magnetic field can be understood qualitatively and quantitatively by amodel in which T1 is limited by electron-phonon coupling. We identify the coupling to acoustic phonons via thebond length change and via the deformation potential as the limiting mechanisms.DOI: 10.1103/dkgn-pfjbI. INTRODUCTIONCharge carriers in two-dimensional (2D) materials witha hexagonal crystal lattice have, in addition to the spin, atunable valley degree of freedom. This renders these materialspromising candidates for valleytronics [1–4], where the valleyrelaxation time T1 is an important figure of merit, allowing usto assess the potential for valley-based information storage. In2D transition metal dichalcogenides with moderate band gaps,the valley degree of freedom is very accessible for opticalmanipulation and readout. This allowed pump-probe spec-troscopy experiments [5,6], in particular time-resolved Kerrrotation experiments [7–11], which revealed relaxation timesranging from nanoseconds to microseconds, depending onthe material and excitation conditions. However, such opticaltechniques are not readily applicable to narrow gap bilayergraphene (BLG), resulting in a lack of knowledge about valleylifetimes in BLG. Recent advances in the confinement of car-riers in BLG using quantum dots (QDs), however, are openingup new avenues for the study of valley lifetimes in BLG.Bernal-stacked BLG comes as a gapless semimetal, inwhich electrons and holes can be described as massiveDirac fermions [12]. However, when an out-of-plane electricdisplacement field D is applied, the inversion symmetry of the*These authors contributed equally to this work.†Present address: Faculty of Physics, University of Vienna, Boltz-manngasse 5, 1090 Vienna, Austria.Published by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.crystal lattice is broken, as the on-site energy of carbon atomsof the top layer becomes different from that of the atoms of thebottom layer [see Fig. 1(a)] [13]. This leads to the opening ofa band gap at the two valleys, K and K ′ [see Fig. 1(b)], whichdepends on the strength of the symmetry breaking potential,i.e., on D [14–16], resulting in a tunable band structure thatallows for gate-defined charge carrier confinement [17–19].The broken inversion symmetry also leads to a finite Berrycurvature � near the K points, where � has opposite signs atthe K and K ′ points and has mirror symmetry for electronsand holes [13,20] [see Fig. 1(c)]. The Berry curvature givesrise to a valley-dependent anomalous velocity term leadingto the valley-Hall effect in bulk BLG [21,22] and to finiteout-of-plane magnetic moments in BLG QDs. These topo-logical orbital magnetic moments, which have opposite signsfor K and K ′, couple to an external out-of-plane magneticfield and are the origin of the valley Zeeman effect in BLGQDs [23,24].To create gate-defined QDs in BLG, the electronic wavefunction needs to be confined by a potential U (r) in realspace [see Fig. 1(a)] and will be distributed near the K andK ′ points in k space. A single-electron or single-hole QD canthen be described by the Hamiltonian HQD = HBLG + HZ +HSO + U (r) [25]. Here, HBLG represents the effective 4 × 4Hamiltonian of bulk BLG near the K and K ′ points basedon the sublattice and layer degrees of freedom and includesthe bulk valley Zeeman effect (see Appendix A), which willbe further modified by the confinement U (r). HZ denotes thespin Zeeman coupling. HSO describes the intrinsic Kane-Melespin-orbit (SO) coupling, which lifts the zero B field degener-acy of the four single-particle states, leading to the formationof two Kramers pairs with opposite spin and valley quantumnumbers (|K↑〉, |K ′↓〉) and (|K ′↑〉, |K↓〉), separated by theSO gap �SO. In BLG devices �SO typically has values in the2469-9950/2025/112(3)/035409(8) 035409-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-1855-1287https://orcid.org/0000-0002-7398-8131https://orcid.org/0000-0002-5037-4217https://orcid.org/0000-0002-6237-6762https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0001-9053-2200https://orcid.org/0000-0002-9527-317Xhttps://orcid.org/0000-0002-4958-7362https://ror.org/04xfq0f34https://ror.org/02nv7yv05https://ror.org/0546hnb39https://ror.org/026v1ze26https://ror.org/026v1ze26https://crossmark.crossref.org/dialog/?doi=10.1103/dkgn-pfjb&domain=pdf&date_stamp=2025-07-10https://doi.org/10.1103/dkgn-pfjbhttps://creativecommons.org/licenses/by/4.0/L. BANSZERUS et al. PHYSICAL REVIEW B 112, 035409 (2025)(b)xUQDr0 0.1 0.2 0.3 0.4 0.500.4-0.4E (meV)B  (T)ΔEV+ΔESΔSOΔEV +ΔSOΔES +ΔSO(a)0.2-0.2DtopbottomEelectronsholesΩelectronsholese s(c) (d)FIG. 1. (a) Lattice structure of BLG highlighting the top andbottom layers and a symmetry breaking displacement field D. Theelectrostatic confinement potential U allows us to form a QD.(b) Band structure of gapped BLG. Close to the K and the K ′valleys a band gap opens. (c) The broken inversion symmetry re-sults in a finite Berry curvature � near the K and K ′ points, whichhas opposite signs for the two valleys and for electrons and holes.(d) Single-particle spectrum of a BLG QD. At B⊥ = 0, �SO leads tothe formation of two Kramers doublets. A finite B⊥ results in a spinand valley Zeeman effect, leading to additional energy splittings of�Es = gsμBB and �Ev = gvμBB. The arrows depict the transitionenergies between the ground state |K ′↑〉 and the three excited states.range of 40−80 µeV [18,20,23,26,27]. In Fig. 1(d) we showthe single-particle spectrum of a BLG hole QD as a functionof the out-of-plane magnetic field B⊥. As B⊥ couples to boththe spin and valley magnetic moments, we observe linearenergy shifts given by E (B⊥) − E (0) = (±gs ± gv)μBB⊥/2[28]. Here, μB is the Bohr magneton, gs ≈ 2 is the spin gfactor, and the valley g factor gv quantifies the strength of theBerry curvature induced valley magnetic moment, which canbe tuned by the confinement potential of the QD in a rangetypically between gv ≈ 10 and 70 [24,29,30]. All this makesthe valley degree of freedom in BLG QDs highly tunable,in stark contrast to its behavior in Si, Ge, and SiGe QDs,enabling significant valley polarization at relatively low B⊥fields, which is evident from the separation between the |K ′〉and excited |K〉 states shown in Fig. 1(d).II. RESULTSThe fabricated device consists of a flake of BLG encap-sulated by two crystals of hexagonal boron nitride (hBN)placed on a graphite flake acting as a back gate. On top ofthe van der Waals heterostructure, split gates (SGs) are usedto gap out the BLG underneath, resulting in a narrow n-typeconductive channel connecting the source (S) and drain (D)leads [see Fig. 2(a)]. To confine single charge carriers, theband edge profile along the channel can be adjusted using twolayers of interdigitated finger gates (FGs) [17,19]. One of theFGs is used as a plunger gate (PG) to tune the QD, locallyovercompensating for the channel potential set by the backgate. The width of the PG measures about 70 nm, and theseparation of the SGs is around 80 nm, setting an upper limitof the QD radius r to around 30–40 nm. The DC potentialapplied to the plunger gate VPG allows us to control the chargecarrier occupation down to the last hole [see Fig. 2(b)]. Tostudy transient transport through the QD, an AC potential,VAC, can be applied to the PG via a bias tee [see Fig. 2(a)].To maximize the transient currents and to study the relaxationdynamics of the QD states, the FGs adjacent to the PG (yellowand blue) are used to reduce the tunnel coupling between theQD and the left and right reservoirs, �L and �R [31–34]. Thedetails of the device fabrication and the experimental setup aregiven in Appendixes B and C.To study the relaxation dynamics of an excited valley state,we first investigate the single-particle spectrum of the QD.For that purpose, we perform excited state transient currentspectroscopy measurements by applying a square pulse witha frequency f (duty cycle 50%) to the PG [see Fig. 2(c)].Figure 2(d) shows the average number of charge carriers tun-neling through the QD per pulse cycle, 〈n〉/pulse = I/( f e),with the elementary charge e, as a function of the pulse ampli-tude VAC and the DC plunger gate voltage �VPG relative to theCoulomb peak position at VAC = 0. At finite VAC, transportvia the ground state (GS) may occur when the GS resideswithin the bias window during either part of the square pulse(τi, τm), resulting in a splitting of the GS Coulomb peak (|K↑〉iand |K↑〉m) [32,35]. Once VAC becomes large enough that anexcited state (ES) enters the bias window, a transient currentvia the ES contributes to the overall current and shows up as aresonance in Fig. 2(d) (see dashed lines). From the positionsof the two prominent ES resonances we can extract their en-ergies. Figure 2(e) depicts the energy difference between theground state (|K ′↑〉) and the first spin ES (yellow data points,|K ′↓〉) and the valley ESs (red data points, |K↓〉 and |K↑〉)determined from measurements in Fig. 2(d) as a function ofB⊥. The energy splitting of the spin ES and the GS increaseslinearly with B⊥ due to the spin Zeeman effect. A fit to thedata yields a spin g factor of gs = 2.0 ± 0.2 and a zero-fieldsplitting of �SO = 75 µeV, in agreement with the slightlyproximity-enhanced Kane-Mele SO coupling [20,23,26]. Dueto the finite peak width, the energy of the two valley ESscannot be determined independently. Thus, the data were fitconsidering the average energy splitting with a slope cor-responding to gv + gs/2. A valley g factor of gv = 30.2 ±0.2 was determined, similar to values reported in earlierworks [23,24].Next, to investigate the relaxation dynamics of theobserved single-particle valley ES, we apply a three-levelpulse scheme to the PG [see Fig. 3(a)] and measure thetunneling current through the QD [33]. During τi, the QDis initialized in the empty state. Subsequently, during τh, theGS |K ′↑〉, the spin ES |K ′↓〉, and the two valley ESs, |K↑〉and |K↓〉, are tuned below the electrochemical potentials ofthe source and drain leads. After the characteristic tunnelingtime, on the order of 1/(�L + �R), either the GS or one ofthe three ESs will be occupied by a single charge carrier.035409-2PHONON-LIMITED VALLEY LIFETIMES IN … PHYSICAL REVIEW B 112, 035409 (2025)(c) Timeτm τi initialize measure01.5ΔE(meV)(e)g  = 2   0.2sg  = 30.2   0.2vVPG-7.4 -7.2 -7.0 -6.8 -6.60.00.8I(nA)0 h1 h2 h3 h4 h(b)0.4 (V)SGSGFGsVAC0.51.00.1500.3ΔSO0VPGVACVmVi-2 0 2ΔVPG (mV)0246m m m m i 0 0.4B  (T)0.2 0.6  n/pulseΓL ΓRΓL ΓR5713V    = 0 VACV    = 0.38 VAC(d)(a)S DFIG. 2. (a) False-color scanning electron microscopy image of the gate structure of the device. The plunger gate (PG, red) is connectedto a bias tee for applying AC and DC signals to the same gate. (b) Current through the device as a function of VPG at a source-drain biasvoltage of VSD = 200 µV. The n-type channel is pinched off close to VPG = −6.8 V. Upon further decreasing VPG, a hole QD is formed, andCoulomb resonances appear when additional holes are added to the QD (see labels). (c) Top: Schematic of the square pulse applied to the PGcharacterized by the voltages Vi and Vm and the times τi and τm. Bottom: Schematic of the QD states relative to the electrochemical potentialsof the leads. (d) Excited state (ES) spectroscopy using transient current measurements. The average number of charge carriers 〈n〉 tunnelingthrough the QD per pulse is plotted as a function of �VPG for different VAC ( f = 5 MHz, B⊥ = 300 mT). Traces are offset for clarity. |K ′↑〉denotes the current via the ground state. Orange and yellow dashed lines highlight transient currents via excited states (|K ′↓〉, |K↑〉, and |K↓〉).(e) Energy �E of the ES relative to the GS as a function of B⊥. Fitting �SO/2 + (gv + gs/2)μBB (orange dashed line) yields gv. The solidlines indicate the energies of the valley ESs deduced from the fits. The inset shows a close-up of the low-B⊥ regime.A charge carrier in an ES has the chance to relax intoan energetically lower-lying state by either spin or valleyrelaxation with a characteristic relaxation time T1. Finally,during τm, we perform a valley-selective readout, measuredby aligning the |K〉 states in the bias window. Only chargecarriers occupying one of the two |K〉 states, which havenot relaxed into a |K ′〉 state, can tunnel out and contributeto the transient current. Figure 3(b) shows the current Ithrough the QD as a function of VPG while applying thepulse sequence depicted in Fig. 3(a). The three peaks labeled|K ′↑〉i, |K ′↑〉h, and |K ′↑〉m correspond to GS transportduring each of the three pulse steps. Furthermore, transientcurrents via the three ESs, |K↑〉m, |K↓〉m, and |K ′↓〉m, canbe observed during τm. The relaxation time T1 of the |K〉states into an energetically lower-lying state can be probed byextracting the maximum value of the combined |K↑〉m and|K↓〉m peaks, which at a fixed value of B⊥ is at a constantenergy difference [see Fig. 1(d)], as a function of the holdingtime τh (for more information on the peak analysis seeAppendix D) [31,33,34]. We convert the current I into thenumber of charge carriers tunneling through the QD per pulsecycle, 〈n〉/pulse = I (τi + τh + τm)/e. The number of chargecarriers 〈n〉|K〉mtunneling via the excited |K〉 states is directlyproportional to the probability of |K〉 remaining occupiedafter τh, P|K〉(τh). The relative occupation probability of |K〉mas a function of τh decays exponentially with the characteristicrelaxation time T1, 〈n〉|K〉m(τh)/〈n〉|K〉m(0) =P|K〉(τh)/P|K〉(0) = e−τh/T1 [31,33].Figure 3(c) depicts P|K〉(τh)/P|K〉(0) as a function of τh forthree different out-of-plane magnetic fields. The datasets showan exponential decay of the occupation probability as a func-tion of τh. An exponential fit (solid line) yields, for example,T1 = 4.0 µs at B⊥ = 0.175 T. T1 decreases with increasingB⊥ and reaches a value of 845 ns at B⊥ = 0.45 T. A singlecharge carrier occupying |K↑〉 or |K↓〉 may relax into a lower-lying state either by pure valley relaxation (|K↑〉 → |K ′↑〉 and|K↓〉 → |K ′↓〉) or by additionally flipping the spin (|K↑〉 →|K ′↓〉 and |K↓〉 → |K ′↑〉). Relaxation processes requiring asingle valley flip are expected to be faster than processes thatrequire both a spin flip and a valley flip. This is supported byrecently reported spin relaxation times between hundreds ofmicroseconds and 50 ms for energy splittings �ES > 200 µeV[33,34]. For the pure spin relaxation between |K ′↓〉 and theGS, no relaxation could be observed over the whole range ofinvestigated τh and B⊥ (the amplitude of |K ′↓〉m is constantas a function of τh; see Appendix D). Hence, we concludethat T1 extracted from Fig. 3(c) must be limited by the valleyrelaxation time.In Fig. 4, we plot the valley relaxation time T1 extractedfrom exponential fits, as exemplarily shown in Fig. 3(c) as afunction of B⊥ and as a function of the energy splitting �Ev.When decreasing B⊥ from 0.7 to about 0.15 T, T1 increasesfrom below 0.5 to about 7 µs, while at even lower B⊥, therelaxation rate decreases again to T1 ∼ 2 µs at 80 mT (graysymbols).035409-3L. BANSZERUS et al. PHYSICAL REVIEW B 112, 035409 (2025)(µs)h τ(c)00.512 50 1 43-6.390-6.394 -6.3866measureholdinitializeT1ΓL ΓRΓL ΓR ΓRTimeτi VACVmViVhτh τm VPG (V)-6.392 -6.388 -6.384i m hI (pA)012m m m (a)7(b)1.5 B = 175 mTB = 250 mTB = 450 mTh P   (τ  ) / P(0)FIG. 3. (a) Top: Schematic of the three-level pulse scheme ap-plied to the PG, which is characterized by the voltages Vi,Vh, andVm and the times τi, τh, and τm. Bottom: Schematic of the QD statesrelative to the electrochemical potentials in the leads (see text fordetails). (b) Current I as a function of VPG while the pulse sequencein (a) is applied (B⊥ = 0.22 T, VSD = 10 µV). The valley T1 time isderived from the amplitude of |K↑〉m. Dashed curves are Lorentzianfits to the peaks. (c) Relative occupation probability of |K↑〉 after theholding pulse P|K〉(τh)/P|K〉(0) as a function of the holding time, τh.The traces are offset for clarity.III. THEORETICAL MODELTo gain a better understanding of the experimental results,we compare them with theory. We model the system usingthe Hamiltonian H = HQD + HEPC + HKK ′ , where HQD de-scribes a single electron or hole in the BLG QD and HEPC =∑λq HλqEPC is the electron-phonon coupling. Furthermore, weallow mixing between the two valleys described by the inter-valley coupling term HKK ′ = �KK ′τx/2, with the Pauli matrixτx acting on the valley degree of freedom. For simplicity,we model electrostatic confinement using a finite circularlysymmetric step potential U (r) with potential depth U0 ≈39.6 meV and r = 25 nm. This yields a valley g factor of gv =30, in good agreement with the experiment [see Fig. 2(e)].We consider transitions between states with equal spin butopposite valley degrees of freedom mediated by coupling101T   (µs)1500ΔE  (µeV)v0.1 10.510002000.20.0525200.50.2deformation potentialbond lengthchangeB (T)cB  100FIG. 4. Valley relaxation time T1 as a function of B⊥ (bottomaxis) and the valley splitting �E = gvμBB⊥ (top axis). Error barsindicate the 1σ confidence interval of an exponential fit to the dataas in Fig. 3(c). The black curve represents a fit assuming T1 islimited by electron-phonon coupling arising from the deformationpotential and from bond length change. The blue (red) curve showsthe contribution of the deformation potential (bond length change)separately.to in-plane acoustic phonons arising either from the defor-mation potential (coupling strength g1) or from bond lengthchange (g2) [36]. Since we operate in the low-energy limit,we consider only acoustic phonons, while out-of-plane acous-tic (ZA) phonons are supposed to be quenched in graphenesupported on a substrate, especially in encapsulated graphene[25,37]. The Hamiltonian describing coupling to phonons inmode λ with wave vector q has the form HλqEPC = cq(g1a1σ0 +g2a′2σx + g2a′′2σy)(eiq·rb†λq − e−iq·rbλq), with σx,y,z being thePauli matrices for the sublattice degree of freedom [38,39]and cq = √q/Aρvλ, with A being the area of the BLG sheet,ρ being the mass density of BLG, and vλ being the soundvelocity; a1,2 are phase factors, and bλq and b†λq are the phononladder operators [25]. Using Fermi’s golden rule, we calculatethe valley relaxation times T1 between the initial and finaleigenstates |i〉 and | f 〉 of the Hamiltonian HQD + HKK ′ withopposite valley quantum numbers and eigenenergies εi andε f ,1T1= 2πA∑λ∫d2q(2π )2| 〈i| HλqEPC | f 〉 |2δ(ε f − εi + vλq).We take into account only the emission of phonons (with en-ergy vλq) because the thermal energy is significantly smallerthan the valley splitting. To quantify the electron-phononcoupling strength, we perform the least squares fit to theexperimental data using g1 and g2 as free fit parameters. Ourmodel is in good qualitative and quantitative agreement withthe data taken above B⊥ = 0.1 T, where increasing B⊥ resultsin decreasing T1, but it cannot explain the decrease in T1observed for B⊥ < 0.1 T, suggesting that other mechanismsdominate in this regime. We speculate that the discrepancybetween our model and the data in the regime B⊥ < 0.1 Tmay be due to a hot spot [40], thermal broadening induced035409-4PHONON-LIMITED VALLEY LIFETIMES IN … PHYSICAL REVIEW B 112, 035409 (2025)charge noise [41,42], or 1/ f charge noise, which is expectedto become dominant in the small energy range, as discussedin more detail in Ref. [25]. Consequently, we have restrictedthe fit to the data for B⊥ � 0.1 T, which yields couplingparameters of g1 = 50 eV and g2 = 5.4 eV. It is noteworthythat both parameters are in good agreement with the literature,which includes values in the range of 20–50 eV for g1 [43–47]and values in the range of 1.5–5 eV for g2 [36,47], wherethe wide range of values is partly due to the dependence ofthe deformation potential on screening and doping [43,44].The black solid line in Fig. 4 corresponds to the contributionsfrom both the deformation potential coupling and bond lengthchange coupling, while the dashed lines show the individualcontributions (see labels). In the calculation, the intervalleycoupling, which is mainly responsible for the absolute val-ues of T1 but does not enter the functional B⊥ dependence,was set to �KK ′ = 50 µeV. At larger magnetic fields, B⊥ �0.5 T, T1 is predominantly limited by electron-phonon cou-pling via the deformation potential, whereas at smaller fieldsit is limited by the coupling due to bond length change. Thistransition occurs due to the crossover between the dipoleand the higher multipole regimes for the bond length changecoupling if the phonon wavelength λ ≈ 2π h̄vλ/(gvμBB⊥)is comparable to the QD radius r, where qr = 2πr/λ ≈1. Hence, the crossover occurs around the critical fieldBc⊥ ≈ h̄vλ/(gvμBr) ≈ 0.3 T. The assumed QD radius of r =25 nm is in agreement with the lithographic device dimensionsas well as with the confinement size, giving rise to a valleyg factor of gv = 30, in excellent agreement with experiment.The gray bar in Fig. 4 depicts the range of Bc⊥ assuming theestimate of r deviates by a factor of 2, highlighting that thetransition region is well within the experimentally investigatedB⊥ range.IV. DISCUSSION AND RESULTSThe long single-particle valley lifetimes in BLG QDsof up to 7 µs make BLG a promising candidate for val-leytronic applications and confirm that the valley degree offreedom is, indeed, interesting for the realization of qubits,where the T1 time sets an upper limit on the coherencetime T ∗2 . This potential is furthermore underlined by a re-cent experiment showing long relaxation times from valleytriplet to valley singlet states [48]. Fits to the experimentaldata confirm that electron-phonon coupling mediated by thebond length change and the deformation potential limit therelaxation time over a wide magnetic field range. As the valleymagnetic moment is typically 1 to 2 orders of magnitudelarger than the magnetic moment associated with the electronspin, we anticipate that gate operation times of a valley qubitare much faster than those of a spin qubit, potentially compen-sating for the shorter relaxation times. The magnitude of thevalley magnetic moment can be adjusted all electrically [30],which could provide a way to realize control over a singlevalley without the need for microwave bursts, micromag-nets, or electron spin resonance strip lines and could enablewell-controlled addressability. Crucial follow-up experimentsinclude the determination of the coherence times (T ∗2 and T2)and insights into their limiting mechanisms such as chargenoise, as well as the understanding of the upper bound of therecently reported long relaxation times of Kramers states atsmall energy splittings [49].ACKNOWLEDGMENTSThe authors thank A. Hosseinkhani for fruitful discussionsand F. Lentz, S. Trellenkamp, M. Otto, and D. Neumaierfor help with sample fabrication. This project has receivedfunding from the European Research Council (ERC) underGrant Agreement No. 820254, the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation) underGermany’s Excellence Strategy - Cluster of ExcellenceMatter and Light for Quantum Computing (ML4Q) EXC2004/1-390534769 through DFG (STA 1146/11-1), and theHelmholtz Nano Facility [50]. K.W. and T.T. acknowledgesupport from the JSPS KAKENHI (Grants No. 21H05233and No. 23H02052), CREST (JPMJCR24A5), JST, and theWorld Premier International Research Center Initiative (WPI),MEXT, Japan. L.W. and G.B. acknowledge support fromDFG Project No. 425217212, SFB 1432.C.S. designed and directed the project. L.B., K.H., andS.M. fabricated the device. L.B., K.H., and C.V. performed themeasurements and analyzed the data. K.W. and T.T. synthe-sized the hBN crystals. L.W. and G.B. performed calculationsof the T1 time. G.B., C.V., and C.S. supervised the project.L.B., K.H., L.W., G.B., C.V., and C.S. wrote the manuscriptwith contributions from all authors.The authors declare no competing interests.DATA AVAILABILITYThe data that support the findings of this article are openlyavailable [51].APPENDIX A: HAMILTONIAN OF BILAYER GRAPHENEThe Hamiltonian HBLG used to describe the band structureof bulk BLG is given byHBLG(k) =⎡⎢⎢⎣� γ0 p γ4 p∗ γ1γ0 p∗ � γ3 p γ4 p∗γ4 p γ3 p∗ −� γ0 pγ1 γ4 p γ0 p∗ −�⎤⎥⎥⎦, (A1)with the displacement field being 2� and the hoppingparameters γ0 = 2.6 eV, γ1 = 0.339 eV, γ3 = 0.28 eV, andγ4 = −0.14 eV. The momentum p(k) = −√3a(τkx − iky −ixB⊥e/2 − τyB⊥e/2)/2, with valley index τ = ±1 and lat-tice constant a = 2.46 Å, includes the valley Zeeman effect[13,52,53].APPENDIX B: DEVICE FABRICATIONThe device is composed of a van der Waals heterostructure,where a BLG flake is encapsulated between two hBN flakesapproximately 25 nm thick and placed on a graphite flakewhich acts as a back gate (BG). Cr/Au split gates on top ofthe heterostructure define an 80 nm wide channel. Across thechannel, two layers of 70 nm wide interdigitated Cr/Au fingergates are fabricated. Two 15 nm thick layers of atomic layerdeposited Al2O3 act as a gate dielectric. For the details of thefabrication process, we refer to Ref. [54].035409-5L. BANSZERUS et al. PHYSICAL REVIEW B 112, 035409 (2025)APPENDIX C: EXPERIMENTAL SETUPAND PARAMETERSIn order to perform RF gate modulation, the sample ismounted on a home-built printed circuit board. All DC linesare low pass filtered (10 nF capacitors to ground). All FGsare connected to onboard bias tees, allowing for AC and DCcontrol on the same gate [see Fig. 2(a)]. All AC lines areequipped with cryogenic attenuators of −26 dB. VAC refersto the AC voltage applied prior to attenuation. The measure-ments are performed in a 3He/4He dilution refrigerator at abase temperature of approximately 20 mK and at an electrontemperature of around 60 mK. The current through the deviceis amplified and converted into a voltage with a home-builtI-V converter at a gain of 108.Throughout the experiment, a constant back gate voltageof VBG = 5.025 V and a split gate voltage of VSG = −5.435 Vare applied to define an n-type channel between the sourceand the drain. The four gates acting as barrier gates to the QD[see Fig. 2(a), yellow and blue color coding] are biased by−6.05 ± 0.1, −4.95, −5.18, and −6.15 ± 0.1 V, respectively.The voltages are adjusted to compensate for the influence ofB⊥ on the tunnel coupling.APPENDIX D: PEAK ANALYSIS AND EXTRACTINGRELAXATION TIMESFigure 5(a) shows an exemplary measurement of the aver-age number of charge carriers 〈n〉 tunneling through the QDper pulse cycle as a function of the duration of the holdingpulse τh and the plunger gate voltage VPG. The colored la-bels mark the probed states. The measurement was recordedat an out-of-plane magnetic field of B⊥ = 0.25 T [the mea-surement corresponds to the results depicted in Fig. 3(c)].Exemplarily, line cuts of the measurement in Fig. 5(a) areshown in Fig. 5(b). The peak heights of |K ′↑〉m and |K ′↓〉mare displayed in Fig. 5(c) as a function of the pulse durationτh. The maximum value of the peak attributed to |K↑〉m and|K↓〉m [red data points in Fig. 5(c)] is extracted in a smallvoltage window at a fixed distance from the position of thepeak |K ′↑〉m [visualized by the red shaded area in Fig. 5(b)].It is important to note that the observed exponential decaydoes not depend on details of the peak analysis. The line cutsin Fig. 5(b) reveal that the finite peak linewidth comparedto the small energy spacing between the |K〉 states makes itdifficult to identify peaks |K↓〉m and |K↑〉m independently.Therefore, we extracted the maximum value of the com-bined peak contributions [red shaded area in Fig. 5(b)]. Tohighlight that the obtained result is robust, we determine thevalues at fixed energy splittings from the ground state peak|K ′↑〉m, �EV + �ES [red arrow in Fig. 5(b)] and �EV + �SO[orange arrow in Fig. 5(b)], which correspond to the ex-pected positions of peaks |K↓〉m and |K↑〉m, respectively. TheFIG. 5. (a) Average number of charge carriers 〈n〉 tunnelingthrough the device per pulse cycle (derived from the current I)as a function of VPG and τh while the pulse sequence depicted inFig. 3(a) is applied. The color-coded arrows mark the positions ofthe peaks |K ′↑〉m (yellow), |K ′↓〉m (blue), and |K↓〉m and |K↑〉m(red). An out-of-plane magnetic field of B⊥ = 0.25 T was applied.(b) Exemplary line traces of the measurement in (a), where 〈n〉/pulseis shown as a function of VPG for different values of τh (black arrowon the right side). The traces are offset by an arbitrary noffset forbetter visibility. (c) Maximum values of the labeled peaks in (a) as afunction of the pulse duration τh. The maximum value of the peakthat we attribute to |K↓〉m and |K↑〉m was extracted in a voltagewindow at a fixed energy difference from the peak |K ′↑〉m [redshaded area in (b)]. 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