Spatially inhomogeneous electron state deep in the extreme quantum limit of strontium titanate

When an electronic system is subjected to a sufficiently strong magnetic field that the cyclotron energy is much larger than the Fermi energy, the system enters the extreme quantum limit (EQL) and becomes susceptible to a number of instabilities. Bringing a three-dimensional electronic system deeply into the EQL can be difficult however, since it requires a small Fermi energy, large magnetic field, and low disorder. Here we present an experimental study of the EQL in lightly-doped single crystals of strontium titanate. Our experiments probe deeply into the regime where theory has long predicted an interaction-driven charge density wave or Wigner crystal state. A number of interesting features arise in the transport in this regime, including a striking re-entrant nonlinearity in the current–voltage characteristics. We discuss these features in the context of possible correlated electron states, and present an alternative picture based on magnetic-field induced puddling of electrons.


SUPPLEMENTARY FIGURES
3.57 x 10 17 cm - 3 2.57 x 10 17 cm -3 6.1 x 10 16 cm -3 3.44 x 10 16 cm -3 7.67 x 10 15 cm -3 ρ xx (Ω -cm) Supplementary Figure 3. Zero-field resistivity for our lowest-density sample as a function of temperature. The inset shows a detailed view of the data around T = 105 K. There is no sign of any anomaly in the transport at this temperature, which corresponds to a structural transition in the STO lattice.  Figure 5. Independence of the nonlinearity on the field direction. The differential resistance, dV /dI, is plotted as a function of the bias current I and normalized to the value of dV /dI at I = 0. The green curve corresponds to the case where the applied magnetic field is perpendicular to the bias current, while the orange curve corresponds to the case where the magnetic field and current are parallel. Measurements in this plot correspond to T = 25 mK and B = 35.1 T.

SUPPLEMENTARY NOTE 1: ZERO-FIELD MOBILITY AND ESTIMATE OF IMPURITY CONCENTRATION
As mentioned in the main text, as-grown STO crystals generally have a relatively large number of impurities that act as deep acceptors. [1,2] We therefore expect that the concentration of impurities N i significantly exceeds the concentration n of free electrons. Here we show that this expectation is consistent with measurements of the zero-field, low-temperature mobility.
In particular, at low enough temperatures that the mobility saturates at a constant value, one can expect that the electron mobility µ e is limited primarily by scattering from ionized impurities (Rayleigh scattering). For such scattering processes, screening of the impurity potential by conduction electrons is essentially irrelevant in our samples. This can be seen by examining the Thomas-Fermi screening radius where ν is the electron density of states. In our samples r s ∝ a B /n 1/3 ∼ 60 nm, which is much longer than the distance between electrons or between charged impurities. (Here, a B denotes the effective Bohr radius at zero magnetic field.) Consequently, the screening of impurities can be ignored for calculating the scattering cross section. In this case, [3] .
Solving Supplementary Eq. (2) for N i gives an estimate of the impurity concentration for a given mobility µ e and carrier density n. For the samples presented here, this estimate yields values of N i of a few times 10 18 cm −3 , which is consistent with our assumptions in the main text and with previous studies.[1, 2,4] Of course, this value can be considered an upper-bound estimate for N i , since the presence of other, short-ranged scatterers will also decrease the mobility. The Coulomb potential created by impurities is also not perfectly described by a constant dielectric function ε, since the dielectric function is dispersive and thus the dielectric response is not fully developed at short distances from the impurity. Hence, scattering by impurities may be somewhat stronger than implied by Supplementary Eq. (2), which would further reduce the estimate for N i . Theoretical estimates for the dispersive nature of the dielectric function, however, suggest that this effect is not too large for isolated monovalent charges. [5]

SUPPLEMENTARY NOTE 2: SHUBNIKOV-DE HAAS ANALYSIS
In the main text we present the results of a Shubnikov-de Haas (SdH) analysis of our doped STO samples. In particular, in this analysis we identify oscillations of the longitudinal resistivity ρ xx that are periodic in 1/B. As mentioned in the main text, this is done by subtracting a smooth, fourth-order polynomial from the curve ρ xx (B) and identifying the field values B N corresponding to the maxima of the background-subtracted curve ∆ρ xx (B).
In Fig. 1 we show full details of this data analysis procedure for five of our studied samples, each of which is driven into the extreme quantum limit at large field. In particular, for each sample we show the raw curve ρ xx (B), the background-subtracted curve ∆ρ xx (B), and the same data ∆ρ xx plotted against the inverse field 1/B.

A. Resistivity Scaling
At high magnetic fields, the resistivity ρ exhibits a significant nonlinearity, such that ρ is a function of both the bias voltage V and the magnetic field B. We observe that the resistivity can in fact be scaled in such a way that different curves for the differential resistance, dV /dI, as a function of V collapse on top of each other at small V . This is shown in Fig. 4(b). The collapse of the curves suggests that one can write dV /dI = f (V )h(B), where f and h are scaling functions.
From the data in Supplementary Fig. 4 one can extract a characteristic electric field scale F 0 for the nonlinearity in multiple different ways. For example, one can define F 0 as the value of the electric field above which the scaling shown in Supplementary Fig. 4(b) is lost. This definition gives F 0 ≈ 10 mV cm −1 . Alternatively, one could define F 0 as the value for which the differential resistance drops to half its V = 0 value. Such a definition gives F 0 on the order of ≈ 100 mV cm −1 , depending on the value of B.

B. Nonlinearity in the perpendicular versus parallel field directions
Our analysis of the nonlinearity has largely focused on the case where the magnetic field direction is perpendicular to the current direction. Here we briefly show results for the case where the magnetic field and current directions are co-linear. As one can see in Supplementary Fig. 5, the two cases give essentially identical results for the nonlinear differential resistance.

C. Tetragonal domain walls
At temperatures below T = 105 K, STO is known to undergo a transition from cubic to tetragonal crystal symmetry. [6] Consequently, at T < 105 K the sample contains domain walls between differently-oriented tetragonal domains, and these can potentially influence the electron transport. (For example, such influence has recently been studied at the STO-LaAlO 3 interface. [7,8]) These domains are typically tens of microns in size, and the domain walls have a width of about 2 nm and are associated with an electronic energy scale of about 3.2 meV. [9] In our experiments, however, we find it unlikely that these domain walls are related to the observed nonlinearity in the electron transport. Most tellingly, the features we observe in the electron transport are strongly magnetic field-dependent, while the domain structure at fixed low temperature is completely insensitive to magnetic field. In addition, we see no sign of any anomalies in the resistivity at T = 105 K, at which the structural transition occurs. This is shown in Supplementary Fig. 3.

D. Nonlinearity in the Electron Puddle Picture
In the picture of "electron puddles" presented in the main text, there is a natural electric field scale for the nonlinearity of the resistivity, which can be derived as follows.
For an almost-completely-compensated semiconductor in the EQL, electrons become localized in wells of the disorder potential with typical radius [10] and typical concentration n p given by Eq. (4) of the main text. One can estimate the typical distance R between puddles by noting that the total number of electrons within a puddle is equal to Q p (4π/3)n p r 3 p , and therefore the volume-averaged concentration of electrons is n Q p /R 3 . Rearranging this expression for R gives a typical distance between puddles R (4πn p /3n) 1/3 r p .
In the absence of a bias electric field, the typical activation energy between neighboring puddles is E a = cγ, where γ is the typical amplitude of the disorder potential [see Eq. (2) of the main text] and c is a numerical factor that is typically ≈ 0.15. [11] One can define the typical field scale F 0 as the value of the electric field for which the difference in electric potential between puddles due to the applied field becomes equal to the activation energy E a . In other words, eF 0 R E a . Rearranging this equation for F 0 and substituting R (4πn p /3n) 1/3 r p gives where the second equality is reached by substituting Eq.