Time, momentum, and energy resolved pump-probe tunneling spectroscopy of two-dimensional electron systems

Real-time probing of electrons can uncover intricate relaxation mechanisms and many-body interactions in strongly correlated materials. Here, we introduce time, momentum, and energy resolved pump-probe tunneling spectroscopy (Tr-MERTS). The method allows the injection of electrons at a particular energy and observation of their subsequent decay in energy-momentum space. Using Tr-MERTS, we visualize electronic decay processes, with lifetimes from tens of nanoseconds to tens of microseconds, in Landau levels formed in a GaAs quantum well. Although most observed features agree with simple energy-relaxation, we discovered a splitting in the nonequilibrium energy spectrum in the vicinity of a ferromagnetic state. An exact diagonalization study suggests that the splitting arises from a maximally spin-polarized state with higher energy than a conventional equilibrium skyrmion. Furthermore, we observe time-dependent relaxation of the splitting, which we attribute to single-flipped spins forming skyrmions. These results establish Tr-MERTS as a powerful tool for studying the properties of a 2DES beyond equilibrium.

MERTS as a powerful tool for studying the dynamics and properties of a two-dimensional electronic system beyond equilibrium.
Nonequilibrium response in a material driven by laser and electrical pulses can reveal many-body effects involving strong interactions between electrons and their surrounding medium.In superconductors, the relaxation dynamics of excited quasiparticles reflects the strength of electron interactions with phonons or other collective excitations, such as spin fluctuations, that may give rise to unconventional superconductivity [1][2][3][4] .The measurement of relaxation times in other correlated states also permits investigation of their electronic origins [5][6][7] .Furthermore, pumping a system can tune electron-lattice interactions 8,9 and magnetic couplings 10,11 , providing a new pathway for exploring many-body correlations that are undetectable in an equilibrium state.However, despite this promise, there are severe limitations in applying standard optical pump-probe spectroscopy 3,12 to low-dimensional correlated materials.First, a high-intensity laser easily heats up the sample and prevents the study of temporal dynamics at millikelvin temperatures, where a variety of delicate correlated phases emerge.In addition, the laser pulse excites carriers into any available states that have transition energy equal to the photon energy, making it difficult to pump carriers only into a specific state.An alternative method, pump-probe scanning tunneling microscopy 13 , can be utilized for investigating the nonequilibrium properties in two-dimensional (2D) materials.However, this method is sensitive to the surface of a sample and is not easily applicable to highquality 2D materials that are often encapsulated with insulating dielectrics.Moreover, for 2D conductors, local pumping from a scanning tip would lead to rapid lateral charge spreading, making it difficult to study dynamics that do not involve transport.
Here, we present a new method, "time, momentum, and energy resolved pump-probe tunneling spectroscopy (Tr-MERTS)" that uses planar tunneling and allows visualization of nonequilibrium states in a two-dimensional electronic system (2DES).Tr-MERTS employs short duty cycle pulses and functions in the millikelvin temperature range.In addition, fine control of electrical pulses utilized in Tr-MERTS permits high-temporal and high-energy resolution 14 and precision tunability of pumping electron densities.
Finally, since the pumping energy can be tuned by the height of an applied pulse, electrons can be pumped into a specific energy state even for a system with equidistant energy levels (Fig. 1c).In this study, we demonstrate the capability of Tr-MERTS via its application to interacting electrons in Landau levels (LLs) and explore the spin-dependent temporal dynamics.Surprisingly, we uncover a transient level splitting that arises from the formation of a locally excited electronic droplet triggered by the addition of a minorityspin into a ferromagnetic ground-state.
To perform Tr-MERTS measurements, we apply a sequence of three pulses to a bilayer tunnel device (Fig. 1a and b).The first "pump" pulse ejects electrons from one of the two layers (source) and pumps these electrons into the other layer (target).We set the energy of the electrons injected into the target during pumping by varying the height of a pump pulse.Immediately after the pump pulse, we apply a second "delay" pulse to induce an out-of-resonance condition, in which the pumped states in the target and the available states in the source are misaligned in energy.In such a condition, the pumped electrons cannot flow back to the source and remain in the target (see Extended data Fig. 1 for details).After the delay pulse, we apply a short "probe" pulse and measure the tunneling current (I) by detecting an initial rise in the image charge of tunneled electrons [14][15][16] .We use precisely defined pulses with 1%-99% rise time of ~4ns, using specialized home-built electronics that eliminates all ringing.Unlike optical pump-probe experiments, the applied pulses in Tr-MERTS do not overlap in time, and we thus measure current solely driven by the probe pulse (see Extended data Fig. 1).Finally, by controlling the time delay (t) between the pump and probe pulses, we monitor time-dependent changes in tunneling spectra.We simultaneously acquire momentum space information by applying an in-plane magnetic field B|| that shifts the momentum Δk = eB||d/ħ of the tunneling electrons (Fig. 1c), where d is the physical separation between the target and the source layers 16 .Fig. 1d shows a sequence of Tr-MERTS spectra measured before (t < 0) and after (t ≥ 0) pumping the target in an applied perpendicular magnetic field B⊥ = 5.1 T and at a temperature T = 50 mK.The Landau level filling factors of the source and target are  =0.35 and  =1 respectively, and the electrons injected from the source are nearly fully spin-polarized 17,18 .At t < 0, the spectra show quantized LLs in energy and momentum space 16 .When we drive the target out of equilibrium by pumping electrons into the N = 1 LL, we observe an upward energy shift ΔE in the spectrum.This ΔE arises from a charging energy, proportional to the density of the pumped electrons n pump (see Methods).Furthermore, the spectrum shows a transient negative current at energy below the N = 1 LL (see yellow features in Fig. 1d).
This transient current arises from the pumped electrons flowing back to the source under a specific condition: When the pumped energy level in the target matches available energy states in the source, the pumped electrons tunnel back to the source (see Extended data Fig. 2 for a detailed explanation).The Δk distribution of the transient current reveals the information about pumped energy states (i.e. the N = 1 LL), and its intensity is proportional to the number of the excited electrons.At large t, the transient current disappears as the pumped electrons decay to the N = 0 LL in the target.
To study the relaxation dynamics of electrons between the two lowest (N = 0 and 1) LLs, we measured the target filling factor (ν) dependence of Tr-MERTS spectra.Fig. 2a shows TR-MERTS spectrum measured as functions of E, Δk, and ν immediately after pumping the N = 1 LL.Each surface in the figure displays a constant E, Δk, or ν cut, and the surface color represents the intensity map of the tunneling current.In a constant momentum cut taken at Δk = 0.014 Å -1 , we observe a transient negative current at energy slightly below the N = 1 LL at ν = 1 and ν ≥ 1.5.At other filling factors in the target, these transient features are absent because electrons can relax between the LLs at a rate faster than the tunneling rate.For more quantitative analysis, we measured the t dependence of the spectrum for different values of ν (see Figs 2b to d).At ν = 1 and 5/3, the transient current persists up to t ≈ 1 μs and 10 μs, respectively.On the other hand, at ν = 3/2, the transient current disappears at t ≈ 0.1 μs.This behavior suggests that the relaxation time is substantially slowed down at integer and fractional fillings, where the ground-state properties change significantly owing to strong electronic interactions 17,18 .
Given that the spectral weight of the transient current is proportional to the density of excited electrons, we deduce the decay constant () by fitting an exponential function e -t/  to the time dependence of the energy integrated transient current (see Extended data Fig. 3).Fig. 2e shows  for 1 ≤  < 2. A simple model based on Fermi's golden rule for a two-level system predicts monotonically increasing  because the number of available states (ρf) in the lower LL linearly decreases with increasing ν.However, we observe more than an order of magnitude reduction in  at ν ≈ 1.2 and 1.5.Such behavior suggests that a spin-flip process is responsible for the slow relaxation time observed around ν = 1 and ν ≥ 5/3.In these ranges of ν, electrons in the occupied states form nearly fully spin-polarized states 17,18 , with unoccupied states in the N = 0 LL spin-polarized in the opposite direction.In such a case, the spin of pumped electrons is antiparallel to that of unoccupied states in the N = 0 LL in the target because the electrons are injected from a nearly spin-polarized source 17 .As a result, relaxation requires a slow scattering process involving the emission of phonons that can mediate the spin-flips via the spin-orbit interaction 19,20 .Between ν = 1.2 and 1.5, the unoccupied states in the N = 0 LL are unpolarized.Therefore,  decreases rapidly in these trivial (unpolarized) states because the relaxation process does not require spin-flips.
To corroborate this assertion, we measured the T dependence of the relaxation at ν = 1 because the magnetization of a 2DES is sensitive to temperature changes 21 .Tr-MERTS spectra in Extended data Fig. 4a show a faster decay of transient current (see yellow features) with increasing T.This trend is consistent with the spin-dependent relaxation process described above.As T increases, thermally excited magnons depolarize the 2DES at ν = 1 22 .Likewise, the unoccupied states (holes) in the N = 0 LL become spindepolarized.In this case, the decay no longer requires a spin-flip process and becomes faster.The T dependence at ν = 5/3 also displays a similar behavior (see Extended data Fig. 4b for details).
To investigate the nonequilibrium dynamics in more detail, we reduce the width of the pump pulse and fine-tune n pump injected into the target.Fig. 3b shows a constant wavevector cut of the spectrum measured as a function of n pump immediately after pumping the N = 1 LL in the target at ν = 1.At small n pump , we observe a single tunneling peak in the N = 1 LL, with an unexpected higher energy peak emerging at higher pumping density.A crossover between the two peaks occurs at intermediate n pump ~ 4.6×10 9 cm -2 .Furthermore, we have detected these two split peaks in various B ⊥ , but they are absent away from ν=1 (see Extended data Fig. 5).Lastly, the B ⊥ energy scaling of the splitting between the two peaks (see Fig. 4c) suggests that it originates from electronic interactions in the ν = 1 state.
The formation of other equilibrium states such as skyrmions 17,21,23 cannot be responsible for the splitting because the tunneling spectrum measured without pumping shows no signature of the splitting at static electron density n static tuned away from ν = 1 (see Fig. 3d and Extended data Fig. 6).In addition, after a sufficiently long-time delay t = 300 ~ 500 ns, the split peaks disappear and transform back to a single energy peak (see Fig. 3e).This temporal behavior suggests that the splitting is characteristic to the system driven out of equilibrium.
To understand the transient splitting, we perform exact diagonalization calculations 24,25 for a 9electron system at ν ~ 1 (see Supplementary Section 2 for details).In the equilibrium state, electronic interactions at ν = 1 create an exchange splitting Eex 14 and produce a fully spin-polarized system.The top trace in Fig. 4b shows single electron injection spectrum into the N = 1 LL for this condition.When many electrons are pumped into the N = 1 LL, a fraction of them relax, due to the phonon-induced spin-flip processes [26][27][28] , to the N = 0 LL prior to the application of the probe pulse and create spin-down minorities in the N = 0 LL.The energy-angular momentum diagram in Fig. 4a displays the possible spin configurations that arise when a spin minority electron is introduced into a spin-polarized system containing 8 electrons.First, a high energy state (blue) corresponds to a single minority spin immersed in a fully spin-polarized state, denoted as ν=1+.Second, the global ground-state corresponds to a spindepolarized skyrmion state (black).However, as the transition between ν=1+ and skyrmions requires energy and spin dissipation, the skyrmion state is unlikely to form immediately after the addition of extra electrons.Instead, the system starts with a nonequilibrium population of several of the excited states, including a substantial contribution from the ν=1+ state with single flipped spins.The subsequent spin-up electrons injected into the N = 1 LL probe this new nonequilibrium state, which is manifested by the appearance of the additional peak at energy higher than that of the first injected electrons (bottom trace in Fig. 4b).The double-peak structure observed in Fig. 4b qualitatively agrees with this expectation; the lowenergy peak at 0.6Eex corresponds to the ferromagnetic ν = 1 state (red trace) while the higher-energy peak at 0.8Eex arises from the transient ν = 1+ state (blue trace).In this model, the gradual transition between the ν = 1 and ν = 1+ states in Fig. 3b suggests the coexistence of localized ν = 1 and ν = 1+ domains.
Furthermore, supporting this model, Fig. 4c shows that the measured splitting between the two tunneling peaks is comparable to the calculated splitting between the ν = 1 and ν = 1+ states.
Finally, the transient double-peak injection spectrum emerges at relatively small n pump and reverts to the single-peak form as the system re-equilibrates (see Fig. 3e).Within our model discussed above, the relaxation process would involve smooth rearrangements of the local spins in the ν = 1+ droplet and a transition into the skyrmion state 17,21,23 .However, the critical n pump ≈ 5×10 9 cm -2 , corresponding to an average spacing between injected electrons of ≈ 140 nm, at which the ν = 1+ state grows to dominate the spectrum and the relaxation mechanisms remain to be understood.
In conclusion, Tr-MERTS provides unprecedented access to the nonequilibrium phenomena in a 2DES.The spectra that we measured demonstrate the temporal dynamics of a 2DES in a wide range of tunable parameters such as ν, T, and B⊥.Furthermore, the n pump dependence of Tr-MERTS spectra reveals an unexpected nonequilibrium spin state occurring at ultralow temperatures.These results demonstrate the potential broad applicability of Tr-MERTS for exploring nonequilibrium physics in LLs and other correlated electronic systems realized in 2D materials.tunnel barrier (see Methods).We apply a sequence of electrical pulses across the device and detect the image charge of the tunneled electrons [14][15][16] .b, Schematic representation of the pulse sequence.The first "pump" pulse ejects electrons from the source and pumps these electrons into the target.The second "delay" pulse turns off tunneling between the source and target during the time delay (t).The last "probe" pulse measures the current (I) flowing into the target over a short period of time 14 .c, A cartoon depicting a pumping process.During the pump pulse, electrons in the N = 0 LL in the source are pumped into a higher LL in the target.d, A sequence of Tr-MERTS spectra measured as a function of t before and after pumping the N = 1 LL at B⊥ = 5.1 T and T = 50 mK.Arrows indicate transient tunneling current.ΔE arises from a charging effect that is proportional to the density of pumped electrons (see Methods).Yellow features below ΔE are not transient and result from tunneling between the N = 0 LLs in the source and target (See Extended data Fig. 2 for a detailed explanation).   favor the ferromagnetic state at ν = 1.Shortly after pumping, the formation of ν = 1+ droplets with single-flipped spins gives rise to an additional tunneling peak at higher energy.As electrons in the system rethermalize and form an equilibrium skyrmion phase 17,21 , the double-peak structure evolves into the single-peak form, consistent with the equilibrium tunneling spectrum 14,17 measured by static density tuning.

Floating-gate tunnel device
We employed a floating-gate GaAs bilayer device 17 comprising of high-mobility 180 Å and 280 Å wide GaAs quantum wells grown epaxially.The quantum wells are separated by a 130 Å wide Al0.25Ga0.75Astunnel barrier.The floating-gate method allow us to tune the electron density in each quantum well without contacts to the quantum wells.

Pump-probe pulsed tunneling measurement setup
In Tr-MERTS measurements, we apply multiple voltage pulses to the tunnel device (see Extended data Fig. 1) as described in the main text.After applying a sequence of three pulses, we wait ~1 ms for the pumped electrons to fully tunnel back to the source before applying the next sequence of pulses.To measure the tunneling current 14,16 , we record the image charge of tunneled electrons as a function of time using high electron mobility transistors.We determine the tunneling current from the initial slope of a RC charging curve in the recorded time trace, where R and C are the tunneling resistance and the capacitance of a tunnel junction.

Calibration of pumping electron density
When electrons are pumped into the target, a charging effect gives rise to an energy shift ΔE of the Fermi level of the target relative to the Fermi level of the source.Since the charging effect is proportional to the pumped electron density n pump , we determine n pump using a following algebraic expression: where  and  are the capacitance and area of a tunnel junction, respectively.The physical distance d between the source and target is approximately equal to 36 nm.From the above equation, we calibrate n pump , which ranges between 0 and 1×10 10 cm -2 .

Fig. 1 :
Fig. 1: Tr-MERTS setup and spectra.a, The tunnel device consists of two 2DESs separated by a thin