Dissipation-enabled hydrodynamic conductivity in a tunable bandgap semiconductor

Electronic transport in the regime where carrier-carrier collisions are the dominant scattering mechanism has taken on new relevance with the advent of ultraclean two-dimensional materials. Here, we present a combined theoretical and experimental study of ambipolar hydrodynamic transport in bilayer graphene demonstrating that the conductivity is given by the sum of two Drude-like terms that describe relative motion between electrons and holes, and the collective motion of the electron-hole plasma. As predicted, the measured conductivity of gapless, charge-neutral bilayer graphene is sample- and temperature-independent over a wide range. Away from neutrality, the electron-hole conductivity collapses to a single curve, and a set of just four fitting parameters provides quantitative agreement between theory and experiment at all densities, temperatures, and gaps measured. This work validates recent theories for dissipation-enabled hydrodynamic conductivity and creates a link between semiconductor physics and the emerging field of viscous electronics.


Fabrication of Devices
Of the five total devices measured in this work, four were made with graphite gates, and one with metallic gates. All showed closely similar behavior. To fabricate dual graphite gate devices, we first assemble a stack with hBN, top graphite, hBN, graphite contacts (optional), BLG, hBN, and bottom graphite in that order. The stack is then etched twice with 40 sccm CHF 3 + 4 sccm O 2 to first shape the top gate and then the channel; Cr/Pd/Au (2 nm/40 nm/50 nm) is evaporated to make contact to the gates and channel. The metallic gate device was made by placing down a hBN, BLG, hBN stack on to a pre-patterned Pd back gate. Cr/Pd/Au (2 nm/20 nm/50 nm) was evaporated as top gate before the stack was then etched to shape the device. Finally, Cr/Pd/Au (2 nm/20 nm/50 nm) was evaporated to make contacts to the channel. We list in Table 1 the device dimensions and figures they correspond to. A typical device with dual graphite gates and graphite contacts is shown in Fig. S1, and the cross section schematic is presented in Fig. S2B. Figure S1: Device schematic. A fabricated device with dual graphite gates and graphite contacts. The measurement scheme is shown in red. Scale bar is 10 µm.  2 Two-fluid model for bilayer graphene in the hydrodynamic regime

XX XY
The transport properties of bilayer graphene in the hydrodynamic can be determined using a two-fluid model (see e.g. Ref. (12,15)). The two-fluid model describes the evolution of the average drift velocity of electrons and holes ⃗ u e/h . It consists of two equations of motion, one for each carrier species: where e > 0 is the magnitude of the electron charge. Here, τ −1 e(h),eh is the average rate of collisions with holes (electrons) per electron (hole), while τ −1 e(h),dis is the average rate of collisions with surrounding impurities and phonons per electron (hole). m * e/h are the effective masses and ⃗ E is an external electric field. We detail the method for calculating the various relaxation times τ e/h in the subsection below. The electron and hole drift velocities ⃗ u e,h are the average velocities of the electrons and holes and given by where f (ϵ) is the standard Fermi distribution function 1/ (exp[(ϵ − µ)/(k B T )] + 1), with ϵ ⃗ k = ℏ 2 k 2 /2m * , and the integrals are over the entire Brillouin zone. The above equations are obtained from the Boltzmann kinetic equations using standard techniques (12). Physically, they describe the electrons (holes) as moving with one effective drift velocity ⃗ u e(h) This description works well in the hydrodynamic regime where the electron-electron collision rate is the largest scattering rate in the system. It is likely to hold even when electron-hole scattering rate is larger than but still comparable to that of electron-electron (36).
Note that since electron-hole collisions preserve total momentum density, m * e n e d⃗ u e /dt + m * h n h d⃗ u h /dt = 0 must be true in the absence of external forces (i.e. τ −1 e/h,dis = 0 and ⃗ E = 0).
Combining equations (S1) with this condition yields this constraint on electron-hole relaxation times: n e m * e τ h,eh = n h m * h τ e,eh . Taking τ e,eh = m * e ne+m * h n h m * h n h τ 0 and τ h,eh = m * e ne+m * h n h m * e ne τ 0 ensures that the constraint is satisfied, where τ −1 0 ≡ τ −1 e,eh + τ −1 h,eh evaluated at charge neutrality n e = n h .
To obtain the conductivity, one may solve Eqs. (S1) for the steady-state ⃗ u e/h , substitute these into the total current density ⃗ j = n e (−e)⃗ u e + n h e⃗ u h , and read off the conductivity σ in ⃗ j = σ ⃗ E.
It is more instructive however to work instead in terms of the center-of-mass (COM) velocity ⃗ u ≡ n e m * e ⃗ u e + n h m * h ⃗ u h n e m * e + n h m * h (S3) and the relative velocity Using these variables, current density becomes ⃗ j = − n e n h n e m * e + n h m * h (m * e + m * h )e⃗ v + (n h − n e )e ⃗ u, in which the first term represents the contribution from the electrons and holes moving in opposite direction due to the opposite forces exerted on them by the electric field and the second represents the contribution from electrons and holes moving in unison in the same direction due to the Coulomb drag "friction" between electrons and holes, the strength of which is quantified by τ −1 0 . Rewriting Eqs. (S1) in terms of ⃗ u and ⃗ v and performing rearrangements to make the time-derivatives of ⃗ u and ⃗ v the subjects, we find . (S10) One may substitute these equations into Eq. (S5) and directly read off the conductivity from To simplify the expression, we make the common assumption of equal electron and where the first and second terms represent contributions from the relative and COM motions respectively of the electron-hole plasma.
Thus far, our calculation has been exact and no approximations have been made. We now consider the limit of strong electron-hole scattering τ 0 /τ e,dis , τ 0 /τ h,dis → 0, and Taylor expand the current density to zeroth order in τ 0 /τ e/h,dis and obtain ⃗ j represents the Coulomb drag conductivity arising from the Coulombic friction between electrons and holes, and (S13) represents the conductivity due to external dissipative forces. Eqs. (S12) and (S13) are used for all the plots calculated in the main text.
Evidently from Eq. (S12), the drag conductivity σ c is determined by electron-hole scattering time τ 0 and the ratio of electron and hole scattering times from external dissipative mechanisms (i.e. if τ e,dis = τ h,dis , the dependence on both τ e,dis and τ h,dis vanishes). A particular exception is at the CNP n e = n h , at which the dissipative times drop out and σ c depends only on τ 0 . We where σ 0 = (e 2 /h) × 8 log(2)/α 0 . Near charge neutrality, the term in large brackets above and this is what we show in the main text for simplicity since it is in good agreement with experimental data (See e.g. Fig. 3A).

Scattering times in gapless bilayer graphene
In this section we discuss the scattering times used in the main text. Electronic scattering times in bilayer graphene have been thoroughly studied for well over a decade and comprehensive reviews may be found in Refs. (35,37). In this section we summarize the relevant aspects of the subject for our experiment.
Electrons in gapless bilayer graphene are well-described by a parabolic band dispersion at energies below 0.4 eV (38), corresponding to density ∼ 10 13 cm −2 and temperature ∼ 4600 K.
The system has been studied experimentally in two configurations-mounted on a hexagonal boron nitride substrate or suspended between supports, the former of which is the focus of this work.
Experimental measurements of charge conductivity in hBN-supported samples have been explained in terms of charged impurity scattering (21) as well as in-plane acoustic phonons (39).
More recently, samples of hBN-supported bilayer graphene were reported (18) to have such high levels of purity as to be in the hydrodynamic regime (2,10,11), in which the scattering rate of electrons with one another exceeds that with impurities and phonons. Here, we give a detailed comparison of the above mentioned scattering rates. We will in-turn discuss (i) Acoustic phonon scattering, (22,32,39) (ii) Optical phonons (22,23) , (iii) Charged-impurity scattering (35), and (iv) Electron-hole scattering (12,14,40) (note that the electron-electron scattering rate is not relevant (12) for current relaxation and is included only for completeness). Our comparison is summarized in Fig. S3, where we find that at low carrier density (n ≲ 3 × 10 11 cm −2 ), for all temperatures considered (50-300K), electron-hole scattering dominates as the largest relevant (i.e. current-relaxing) scattering rate. The forms of the equations used to fit to the scattering rates from experimental data in the main text are consistent with those found in the large body of calculations in the literature on electron scattering rates in bilayer graphene (which we simply summarize and reproduce briefly below).

Acoustic phonon scattering
We calculate the scattering time for electron-acoustic-phonon scattering using the inverse quasiparticle lifetime (see e.g. Ref. (22,41)). The quasiparticle lifetime τ (ϵ k,γ ) of a quasiparticle at energy ϵ k,γ with acoustic phonons is given by where ϵ k,γ = sign(γ) ℏ 2 k 2 2m (S16) with γ = ±1 for conduction and valence band respectively, v s = 13.6km s −1 is the speed of sound for transverse acoustic phonons (42) and the deformation potential matrix element equals (32) |g k+q,k | 2 = (D ac ) 2 ℏq 2ρv s 1 + cos(2θ k,k+q ) 2 (S17)  Figure S3: Comparison of the relevant scattering rates. Here we use n imp = 10 10 cm −2 , D ac = 15eV throughout (see text for details). A sizable hydrodynamic density window in which τ −1 ee dominates occurs at all temperatures above 50K. At low densities, τ −1 eh dominates over a window of densities that grows in size with temperature. bilayer graphene. N q is the Bose-Einstein distribution The scattering time τ (ϵ k,γ ) above is easily computed numerically. To obtain a representative scattering rate for all the electrons relevant to charge transport (i.e. electrons within k B T of the Fermi surface), we perform a thermal average (43) by inserting τ (ϵ k,γ ) (with γ = 1 for where the ±ϵ and e/h correspond to γ = ±1 for electrons and holes respectively, with f e/h = (exp((ϵ ± µ)/k B T ) + 1) −1 .
We plot the inverse of the resulting thermally-averaged lifetime in Fig. S4(a) below as a function of temperature for different densities. Consistent with previous results in the literature (22,39), we find that in the high temperature regime (k B T ≥ µ), the scattering time (i.e. inverse lifetime) is 1/τ (ϵ k,γ ) = α ac k B T /ℏ, with the proportionality constant α ac determined by the value taken for the deformation potential. The estimated range for the deformation potential based on the literature is D ac = 15-30 eV, corresponding to α ac = 0.028-0.11. In Fig. S3 we used D ac = 15 eV (or α ac = 0.03). In the main text, α ac is left as a fitting parameter and we found 0.030 ± 0.008 for electrons and 0.041 ± 0.008 for holes (see Table 2 below) which are both well within the expected range.

Optical phonon scattering
The scattering rate of electrons with bilayer graphene optical phonons is three orders of magnitude lower than with acoustic phonons for all electron energies below 125 meV (23), corresponding to a temperature of 1400 K and a density of 3.5 × 10 12 cm −2 . Therefore intrinsic optical phonon scattering can be ignored for the purposes of this study.
We also consider scattering from polar optical phonons within the boron nitride substrate, using methods reported previously (23,44,45). The scattering rate of an electron of energy ϵ k = ℏ 2 k 2 /2m ⋆ with each of the two polar phonon modes of hBN (denoted ν = 1, 2) is given where Above, A is the area of the bilayer graphene and d = 3.5Å (46) is the van der Waals distance between bilayer graphene and hBN and the substrate phonon energies are given by ℏω to the experiments, the acoustic phonon scattering rate is more than an order of magnitude greater than the hBN substrate phonon scattering rate. Likewise, the resistivity due to acoustic phonon scattering far exceeds greatly exceeds the substrate phonon limited resistivity (Fig.   S4c). The insignificance of both intrinsic and substrate optical phonon scattering is easily understood qualitatively because the optical phonon energies of ∼ 100 meV ( ∼ 1200 K) lead to a small population of these modes within the temperature range considered in this work (23, 44).

Impurity scattering
We calculate the impurity scattering time using the standard expression for inverse quasiparticle lifetime due to charged impurities (24, 35) where n imp is the charged impurity density, θ k,k−q is the angle between k and k − q, and ε refers to the RPA dielectric function defined in Eq. (S26) below. We set n imp = 10 10 cm −2 , γ = 1 above for electrons (−1 is for holes) and perform the thermal average shown in Eq. (S19). We find that the scattering rate is largely independent of carrier density and temperature over the ranges relevant to this experiment (see In fitting the density-and temperature dependent resistivity data, τ imp is kept as a free parameter. We find that the best match to the data is provided by τ imp = 5 ps, matching the value calculated above using the charged impurity density derived from the Hall measurements.  Figure S5: Electron-impurity scattering. Electron-impurity scattering rates as a function of (a) density and (b) temperature. The scattering rates change gradually with respect to both quantities.

Electron-hole scattering
The finite temperature quasiparticle lifetime τ (ϵ k,γ ) of a quasiparticle at energy ϵ k,γ due to scattering with quasiparticles in the opposite band γ ′ is given within the Random Phase Ap-proximation (RPA) by the expression (47) 1 where is the bare Coulomb interaction and κ = 3.5 is the dielectric constant for bilayer graphene mounted on hBN. µ denotes the chemical potential and T denotes temperature. ε(q, ω, µ, T ) is the RPA dielectric function (not to be confused with the energy ϵ k,λ ) given by and is the Lindhard polarizability function and the summation indices run over the carrier species e and h, with components with g = 4 being the degeneracy factor for bilayer graphene. The first subscript of Imχ to −γ in Eq. (S24) for collisions between carriers from different bands. (To get the scattering rate for carriers in the same band, simply set ν to γ instead.) The A γ,γ ′ (k, q, ω) term is given by where and θ k,k−q is the angle between k and k − q. We substitute the above τ (ϵ k,γ ) into the thermal average Eq. (S19) for γ = 1 and display our results in black in Fig. S3. At charge neutrality µ = 0, one can show by substitution of Eq. (S24) for both γ = ±1 into Eq. (S19) that 1/⟨τ ⟩ is linearly proportional to temperature in both cases, leading to 1 in Fig. S2. However, it is clear that device 1 has a slightly larger FWHM, indicating somewhat larger disorder. In addition, sample 1 shows a small extra peak in resistance at small negative density. This may be due to a superlattice moire pattern that modifies the low-energy bandstructure. For these reasons, the low-T data for device 1 may not precisely reflect the intrinsic behavior of bilayer graphene (at higher temperatures these effects are less important).
We also note that the apparent anomalous behavior (i.e. slight rise in conductivity) of device 2 below 100 K is likely an experimental artifact. Its behavior was measured using a fixed grid of gate voltage points, which were not fine enough to precisely capture the exact point of charge neutrality when the resistivity peak becomes extremely narrow at low T. Device 3 was measured by simultaneously sweeping top and bottom gates to vary the net density more finely, and was able to precisely follow the charge neutrality point even at low temperature. Based on the above reasoning, we determine the value of α 0 from the observed conductivity of devices 2 and 3 between 100 K and 300 K. This gives a value of α 0 = 0.225 ± 0.002.
We next measured σ(µ) for sample 3 at fixed temperatures of 50, 100, 175, and 300 K. This data is shown as discrete points in Figure 2B of the main text and Fig. S7. A least squares fit of hydrodynamic conductivity to the experimental conductivity is then performed at each temperature, with the two τ −1 e/h,dis as fit parameters and α 0 fixed at 0.225. The resulting values obtained for τ −1 e/h,dis are shown in the left half of Table 2 below, and the resulting theoretical conductivities using these extracted parameters are displayed in Fig. S7A above. For comparison, we repeat in Fig. S7B the same fitting procedure for τ −1 e/h,dis using a phenomenological Matthiessen's rule where τ eh = τ 0 (n e + n h )/(2n h ), τ he = τ 0 (n e + n h )/(2n e ) and τ 0 = ℏ/(α 0 k B T ). It is clear that Matthiesen's rule is unable to produce good agreement with experiment. Only hydrodynamic conductivity is able to reproduce the experimental data. Note that because the hydrodynamic Eqs. (S12) and (S13) agree with Matthiessen's rule at charge neutrality (up to terms  Figure S7: Additional data for the gapless conductivity. (A) Comparison of theoretical conductivity (solid lines) using hydrodynamic theory with experimental data (symbols). (B) The same using Matthiessen's rule. In both cases, we use the value α 0 = 0.225 obtained from fitting to σ 0 in Fig. 2A of main text, and for τ −1 e/h,dis we use the values obtained by least square fitting to the data at each temperature. The values obtained from fitting using hydrodynamic theory are displayed in Table 2 below.
of O(τ 0 /τ e/h,dis )) where τ −1 eh/he ≫ τ −1 e/h,dis , and also in the opposite high density regime where τ −1 eh/he ≪ τ −1 e/h,dis , it is possible to obtain agreement with experiment at charge neutrality or high density, but not across the the entire range.
In Fig. 3B of the main text, we perform a least squares fit of α (e/h) ac

Ruling out ballistic transport
Given that our samples are ultraclean, it is a valid concern as to whether electrons are simply traveling across the sample ballistically without undergoing any current-relaxing scattering events. Indeed, Nam et al (14) measured a negative bend resistance indicating ballistic transport 0.442 ± 0.002 0.519 ± 0.004 100 0.567 ± 0.002 0.685 ± 0.003 175 0.790 ± 0.002 0.961 ± 0.003 300 1.408 ± 0.002 1.869 ± 0.004 for Fermi energies E F above k B T in their suspended samples that were ∼ 2µm in both width and length. At T = 50 K (the lowest temperature used for analysis of hydrodynamic transport) this corresponds to a density of about 3 × 10 11 cm −2 and at at T = 300 K this corresponds to a density of about 2 × 10 12 cm −2 .
The electron (hole) mean free path l at carrier density |n| can be extracted from the measured conductivity according to σ = 2e 2 /h π|n|l. Since the low-temperature conductivity increases linearly with |n|, the mean free path increases roughly as |n| and is greatest at high carrier density. At 50 K and at density of |n| = 3 × 10 11 cm 2 , the measured conductivity of 150 2e 2 h gives a mean free path of roughly 0.75µm. This is smaller than the smallest dimension of any of the samples studied here (see Table 1 of Section 1 above). Likewise, at 300 K and density of |n| = 1 × 10 12 cm 2 (the highest density studied here), the conductivity is also roughly 150 2e 2 h , corresponding to a mean free path of 0.4µm. We note that at charge neutrality, |n| is not given by the net charge but the total density of electrons and holes due to thermal excitation and static charge disorder. At 50 K this value is roughly |n| = 10 11 cm 2 , leading to a mean free path of 0.2µm. Therefore we can conclude that the samples in this study are diffusive in the regime in which hydrodynamic transport was analyzed.

Determination of the hydrodynamic transport regime
While much experimental effort has been dedicated to the study of the electronic viscosity in the hydrodynamic regime, comparatively little has been dedicated to the influence of hydrodynamics on electrical transport, presumably because carrier-carrier scattering conserves total carrier momentum and thus cannot by itself yield a finite electrical resistance in general (see below for the exception). To put it more clearly, carrier-carrier scattering amounts to an internal force within the gas of charge carriers that cannot prevent its center-of-mass from accelerating to infinite velocity under a non-zero net force caused by an external electric field. The current induced by an external field in the presence of only carrier-carrier scattering is thus expected to be infinite.
An exception to this rule occurs in the case where electrons and holes co-exist in equal number, because the net force exerted on the gas of carriers (i.e. the combined gas of electrons and holes) by the external electric field is zero. In this special case, the internal forces within the combined gas due to electron-hole friction (i.e. electron-hole scattering) are sufficient to prevent the charges from accelerating to infinite speed, thus leading to finite electrical resistance. In detail, upon switching on an external electric field, the individual electron and hole gases initially accelerate in opposite directions due to the field, but since the electron-hole frictional force increases with the difference in their relative velocity, this acceleration continues only up to the point where each gas experiences a frictional force that cancels the external field. The velocity of each gas then remains at this constant value, yielding a finite current and conductivity. The hydrodynamic regime is thus expected to have a direct impact on electrical transport whenever electrons and holes are present in equal density, a situation that is naturally realised in bilayer graphene at the charge neutrality point.
Nam et al. (14) were the first to experimentally realise this situation using ultraclean suspended bilayer graphene. This work found a novel collapse of conductivity as a function of the ratio of Fermi energy to temperature over a range of finite but low charge densities near charge neutrality and at various temperatures, which the authors attributed solely to electron-hole scattering. This conclusion was however problematic because as explained above, electron-hole collisions alone can only yield a well-defined resistance at precise charge neutrality. Subsequent theoretical work (12) showed that the conductivity collapse was in fact due to an interplay of electron-hole scattering and the next fastest scattering process-that of scattering between electrons and in-plane acoustic phonons. Using the model developed in this paper, we illuminate the relationship between electronhole scattering and disorder. As in the main text, we use Eqs. (S12) and (S13) to calculate the reduction in conductivity due to electron-hole scattering ∆σ = σ ac+i − σ total , where σ ac+i is the conductivity incorporating only acoustic phonons and impurities, and σ total is the total conductivity.
Here we have kept all electron-hole and phonon scattering parameters consistent with those used in Fig. 5  The window for electron-hole scattering limited transport is greatly diminished across all temperatures with higher n imp as impurity-limited transport dominates. This result highlights the importance of ultra-clean bilayer graphene for hydrodynamic transport, and the degree to which hexagonal boron nitride encapsulation is needed to provide such a platform. The electron-hole limited regime vanishes for impurity concentrations ≳ 10 11 cm -2 , explaining why this regime was never observed in early samples possessing higher levels of disorder. Figure S2A shows the schematic bandstructure of bilayer graphene, which can be approximated as two hyperbolic bands, with dispersion ϵ ± (k) = ± (ℏ 2 k 2 /(2m * )) 2 + (∆/2) 2 , where ± denote the conduction and valence bands, ℏ is Planck's constant, k the wave vector, and m * the effective mass. Three relevant energy scales are shown: the field-tunable bandgap ∆, and the chemical potential µ and thermal energy k B T, which determine the density of thermally excited electrons and holes (n e and n h ). This dispersion is known to be valid for electrons of energy less than 0.4 eV (38, 51), corresponding to density n = 10 13 cm −2 and temperature 4600 K, making it applicable to our experiment that is restricted to n ≲ 10 12 cm −2 .

Device characterization and experimental control of carrier density and bandgap
The devices were measured by biasing a small current (10 -100 nA) between two outer leads, then measuring the voltages between the center longitudinal and transverse leads V xx and V xy , respectively, as shown in Fig. S1. The currents ∼ 10 nA ensure we are well within the range in which electrons may be considered to be in thermal equilibrium with the lattice even in the presence of strong electron-hole scattering (19). We calculate the resistances by dividing the voltages by the current bias, using the results to derive the conductivity via the tensor relation: Where L and W are device dimensions. The measured charge neutrality resistances of ≈ 10 5 Ω in a current bias scheme show the system to be free from conducting defects such as strained soliton networks (52) . We introduce two parameters V eff and ∆ ext , which tune µ and ∆ with one-to-one correspondence, respectively. We calculate the electrostatic potential V eff at a given constant displacement field D (and therefore band gap ∆) as: Here s = −C TG /C BG is the negative ratio of the top and bottom gate capacitances and can be extracted from the slope of the charge neutrality point (CNP) (where ∆n = 0) when plotting R xx against V TG and V BG . V TG(CNP) is the CNP offset at V BG = 0. We also independently tune ∆ with ∆ ext , which is calculated as: where ∆ ext = 0 at V TG0 , V BG0 , e is the electron charge, ϵ TG(BG) is the top (bottom) boron nitride dielectric constant, t TG(BG) is the top (bottom) boron nitride thickness, and c 0 is the BLG interlayer spacing. The relation between ∆ ext and ∆ is monotonic but not straightforward, and theoretical studies of this relation vary depending on the model and details considered. In the ranges considered in this paper, we can approximate the relation between the two as linear, ∆ ext ≈ 2.6∆ as determined experimentally from Arrhenius fittings, in good agreement with tight-binding models (37). We note that in the calculation of V eff with various ∆ ext the charge neutrality points do not align unless s = −1, and we've renormalized V eff to V eff − V eff(CNP) for ease of comparison. temperatures. This is due to the thermal population of minority carriers, even at considerably large band gaps. The single carrier model is therefore insufficient at higher temperatures, the region of most interest for hydrodynamic transport. Instead, we turn to the ambipolar transport model to extract the Hall density. The conductivity for such a system is given by: where the n e (h) and u e (h) are the electron (hole) density and mobility, respectively, and the mobility can be understood as u = e⟨τ ⟩ m , where ⟨τ ⟩ is the carrier relaxation time. Here we use u instead of the standard µ for mobility as to avoid confusion with the chemical potential. The Hall coefficient for the ambipolar system also becomes: e(n e u e + n h u h ) 2 (S36) By assuming the carrier mobilities to be dominated by electron-hole scattering, we can then calculate the mobilities per: Where u 0 = e⟨τ ⟩ 0 m is the carrier mobility at charge neutrality, ⟨τ ⟩ 0 is calculated below. We can then use the above equations to calculate the electron and hole densities from the measured conductivity and Hall coefficient, provided that ∆n = n e − n h remains consistent with the single carrier Hall density measured at T = 2K. We note that while this holds for the gapless case due to the parabolic approximation, it is an extremely rough approximation at large ∆ ext , due to quantum capacitance. Nonertheless, this allows us to calculate the minority carrier concentration without the assumption of a band structure. The resulting equation is then: from V eff with a hyperbolic band gap. The total capacitance of the system can be written as giving the relation between V eff and µ as C eff is the effective geometric capacitance as a function of V eff , extracted from Hall measurements. C q is the quantum capacitance, calculated from the density of states and derivative of the Fermi-Dirac distribution: The density of states g(E, ∆) is dependent on the bandgap ∆ and given as: Where we approximate the effective mass m * as 0.033m e . Combining Equations S41, S42, and S43, we can then solve numerically for µ given V eff . We plot in Fig. S10 the calculated µ as a function of V eff for ∆ ext = 134 meV, considering temperatures T = 100, 175, 300 K. At high temperatures, µ is approximately linear with V eff due to thermalization of carriers. From µ we can then calculate the carrier densities. Fig. S10 plots the calculated carrier densities as a function of V eff for ∆ ext = 134 meV, temperatures T = 50, 100, 175, 300 K.
Compared with the experimentally determined carrier densities in Fig. S9, the calculated carrier densities are higher at each V eff when compared to the gapless case. We can then calculate the charge density and check the breakdown of the initial assumption of ∆n = n e − n h and its temperature dependence. Fig. S11 shows that taking quantum capacitance into account, the charge density indeed varies with increasing temperature, as expected due to thermal excitation.

Self-consistent determination of gap from displacement field
When a transverse displacement field D is applied across bilayer graphene, a band gap of size ∆ opens up and the resulting dispersion is approximately hyperbolic, given by (37) In this work, we determine the size of the gap for a given displacement field D and density n = n e − n h self-consistently using the method of Ref. (37). This involves evaluating the potential drop across the two graphene layers ∆ ext = ec 0 D, where c 0 = 3.35 Angstrom is the interlayer separation, and determining ∆ numerically in the equation where n ⊥ = 1.1 × 10 13 cm −2 is a characteristic density scale. Solving this equation self-consistently at charge neutrality for ∆ reveals the almost perfectly linear relationship between ∆ and ∆ ext in Fig. S11 below. Experimentally, the relation between ∆ ext and ∆ is extracted via a linear fit, yielding the ∆ ext ≈ 2.6∆ relation used in the main text.

Modification of scattering times in the presence of a bandgap
Within the relaxation time approximation, the non-equilibrium electron distribution g( ⃗ k) in the presence of a driving field ⃗ E is where τ (ϵ k ) is the transport scattering or momentum relaxation time for a charge carrier at energy ϵ k whose form depends on the scattering mechanism, ⃗ v(ϵ k ) the group velocity at ⃗ k and . Following standard steps (53), we integrate over the Brillouin zone to find the total current density ⃗ j and read off the conductivity σ from j = σE to obtain the semiclassical conductivity as before (Eq. S50). We separate the integral into two parts ranging from −∞ to 0 and 0 to ∞ and refer to these as σ h and σ e respectively. The total conductivity σ is then given by the sum σ h +σ e . The momentum relaxation time for each carrier species is obtained by manipulating σ e/h into the standard Drude form (Eq. S51) and reading off τ e/h . Here, with f e/h (ϵ) = 1/ (exp[(ϵ ∓ µ)/k B T ] + 1). Following this procedure, we obtain In this manner, we are able to evaluate collision times τ e/h (∆) numerically at arbitrary gap ∆ given their values at ∆ = 0. The validity of the above procedure is validated by the good agreement with experiment demonstrated in the main text. We remind that for electron-hole scattering, Eq. (S49) may only be applied to the electron-hole scattering time τ 0 at CNP. The electron-hole scattering time away from CNP is then obtained by τ e/h = (n e + n h )/(n h/e ) × τ 0 , where n e/h are given by Eq. (S47).
The calculation for the gapped case ∆ = 51 meV is performed as follows. For electron-impurity and electron-acoustic-phonon scattering, we insert the gapless momentum relaxation time τ as given by the fit parameters in Table 2 into Eq. S49 to find τ e/h (∆). This is the generalisation of Eq. (S19) to the case of finite gap. Physically, this corresponds to assuming that τ (ϵ) for electron-phonon, and electron-impurity are determined primarily by the phonon population and impurity concentration respectively. For electron-hole scattering, we use Eq. (S49) at charge neutrality to obtain τ 0 (∆). From there, we obtain the electron-hole momentum relaxation time at finite densities using the usual τ e/h = (n e + n h )/(n h/e ) × τ 0 , where n e/h are given by Eq. S47. We plot in Fig. S12 the value of ∆σ/σ total against chemical potential µ and temperature T for band gaps ∆ = 0, 51 meV (see below for details of the calculation for ∆) and impurity densities n imp = 10 9 , 10 10 , 10 11 , 10 12 cm -2 . Figure S12: Hydrodynamic window in a semiconductor. Normalized reduction in conductivity due to electron-hole scattering ∆σ/σ total as a function of chemical potential µ and temperature T for ∆ = 0 and 51 meV at various impurity densities n imp . Increasing impurity density shrinks and eventually closes the window for hydrodynamic conductivity.

Evidence of universality
The term "universal" is used in different contexts to mean different things. The purpose of this section is to define how our hydrodynamic transistor is universal. To situate this issue in a broader context, we note that in some communities the term universal is reserved for phenomena that are metrologically precise -for example, the quantum Hall effect is now used for the SI definition of the Ohm (54). The driving force for this switch was after the observation of the quantum Hall effect in monolayer graphene, after which it was understood that the phenomena remained universal regardless of whether the material had linear or quadratic bands. However, in the mesoscopic community, phenomena is considered universal when it becomes independent of impurity concentration. Since the concentration of defects varies from sample-to-sample, in this context, universal implies that the phenomena would not exhibit sample-to-sample fluctuations. For example, "universal conductance fluctuations" (UCF) (17) are considered universal because the variance of the conductance is independent of impurity concentration (provided the temperature is sufficiently low that phase-coherence length is larger than the sample-size). In practice, the magnitude of UCF does depend on factors like the degree of spin-orbit coupling and the applied magnetic field, and since many experiments are done at temperatures where the phase-coherence length is smaller than the sample size, the observed UCF depends on material parameters like the effective mass and impurity concentrations through the phase-coherence length (55). Another example is the minimum conductivity of graphene. The first experimental transport measurements showed remarkable insensitivity to disorder (56), and for this reason many at the time believed that a universal mechanism was responsible. However, we showed that this apparent universality arose from a delicate cancellation between carriers induced by impurities and the scattering of these carriers off these impurities (48). We predicted that the mechanism was not universal, and that there would be a logarithmic increase of the minimum conductivity with decreasing disorder, and effect largely confirmed experimentally (49).
It is in this context that the hydrodynamic conductivity should be understood. As the impurity concentration is further reduced such that electron-hole scattering becomes the dominant scattering mechanism, there emerges two unconventional contributions to the hydrodynamic conductivity. The first contribution depends on extrinsic electron scattering mechanisms such as charged impurities or acoustic phonons. However, it is not the usual scattering of electrons (or holes) off impurities, but the collective scattering of the electron-hole plasma. Far from neutrality it reduces to the usual Drude diffusive transport. This dissipative contribution, while unusual, is not universal. It dominates away from charge neutrality and depends sensitively on both the choice of platform (bilayer graphene in our case) as well as impurity concentration. For the second contribution, however, all the extrinsic factors, such as impurity concentration and electron-phonon coupling constant drop out, and we also demonstrate that the σ = (e 2 /h)(8 log 2/α 0 ) at charge neutrality. As in the UCF example, that independence of the phenomena on impurity concentration (or sample-to-sample variation), is an example of a universal phenomena. However, the universality in our work is even stronger since in addition to no sample-to-sample variation, material parameters like effective mass also drop out.
In Figure S13, we show the inverse lifetime by temperature, i.e. ℏ/(τ k B T ) from G 0 W-RPA calculation for gapless parabolic bands as a function of m ⋆ e 4 /(ℏ 2 κ 2 k B T ). In the limit of sufficiently strong Coulomb interaction and large effective mass, i.e m ⋆ e 4 /(ℏ 2 κ 2 k B * T ) ≫ 1, ℏ/⟨τ ⟩ = 0.35k B T . This is a stronger example of universal in which the phenomena is independent of material parameters, and it is common in the literature to call such phenomena universal (e.g. Planckian resistivity (57)) once it becomes independent of effective mass.

Theoretical evidence of universality: generalization to all strongly interacting hydrodynamic materials with a hyperbolic dispersion
For elastic processes such as electron-impurity and electron-acoustic phonon scattering, the collision operator in the Boltzmann equation may be manipulated to a "relaxation time form" i.e.
−(f (ϵ)−f 0 (ϵ))/τ (ϵ), where f (ϵ) and f 0 (ϵ) are the non-equilibrium and equilibrium distribution functions respectively and τ (ϵ) is the transport scattering time at energy ϵ. Then, one can use standard methods (35) to obtain the conductivity in the form of which can then be expressed in the Drude form giving the energy-averaged transport scattering time as shown in Eq. (S19). However, for inelastic processes such as electron-hole scattering, in general, this procedure is not possible. In the absence of a fully convergent numerical solution of the quantum Boltzmann equation, a common approach is to expand the non-equilbrium distribution function using a finite number of modes. This was done for bilayer graphene by Ref. (11) using the Thomas-Fermi approximation and in Ref. (12) using the leading order temperature expansion for the polarizability.
These two contemporaneous studies found α 0 = 0.15 and 0.29, respectively. In our approach, we use another common method which is to approximate the energy-averaged electron-hole transport scattering time using the energy-averaged electron-hole quasiparticle lifetime which can then be obtained using the full Random Phase Approximation (RPA). This is equivalent to assuming a relaxation time of the form −(f (ϵ) − f 0 (ϵ))/τ (ϵ) for the electron-hole collision operator in the Boltzmann equation, and substituting the electron-hole quasiparticle lifetime in place of τ (ϵ) as an effective transport scattering time. In Fig. S13 we show the inverse quasiparticle lifetime calculated using the finite-temperature RPA polarizability. As expected, for sufficiently large Coulomb interactions m(e 2 /κ) 2 /T ≫ 1, our numerics show that it quickly saturates to a constant universal value 0.356, and that bilayer graphene (black dot) is already approaching this limit. All of these different (and contemporaneous) estimates for α 0 agree to within a factor of ∼ 2. An intermediate approach is to map the electron-hole collision operator onto a "relaxation time form" using a generalization of the Bhatnagar-Gross-Krook formalism, where the quasiparticle lifetime is always smaller than the transport scattering time and found (numerically) that they differ by at most a factor of ∼ 3. Given this history, we find it reasonable to use the energy-averaged electron-hole quasiparticle lifetime (see Sec. 3.4) within a full RPA approximation to estimate the energy-averaged transport scattering time.
We note that at charge neutrality and zero gap, there exist only two energy scales in the system -temperature k B T and the Coulomb energy m * e 4 /ℏ 2 κ 2 . Therefore, 1/⟨τ 0 ⟩ must be a func-  Figure S13: Universal limit for the electron-hole lifetime. The inverse energy-averaged electron-hole quasiparticle lifetime for gapless parabolic bands ϵ ± = ±ℏ 2 k 2 /2m * tends to a universal form 1/⟨τ 0 ⟩ = 0.35k B T /ℏ in the limit of strong Coulomb interaction and large effective mass m * e 4 /(ℏ 2 κ 2 ) ≫ k B T . The black circle denotes the bilayer graphene with κ = 3.5, T = 200 K and m * = 0.033m e . tion of only these two energy scales. (Here, ⟨τ 0 ⟩ refers to Eq. (S24) evaluated at charge neutrality and energy-averaged using Eq. (S19). Within the RPA, we find that ℏ/(k B T ⟨τ 0 ⟩) is a function only of the ratio of these two energy scales m * e 4 /(ℏ 2 κ 2 k B T ), and becomes universal when the ratio is large. Using the same argument, we believe that the energy-averaged transport scattering time also shows the same universal behavior as the quasiparticle lifetime. Moreover, we can make the same argument using dimensional analysis. Since the charge-neutral conductivity σ 0 is temperature-independent, it is therefore an experimental fact that the energy-averaged transport scattering time τ 0 goes as 1/T since density increases linearly in T . The only func- is a dimensionless function that can depend on x only through x 0 . In other words, F cannot depend on system-specific parameters such as m * and κ. This convinces us that our experimentally observed value of α 0 = 0.225 ± 0.002 is universal.
Having established that the quasiparticle lifetime is a good estimate of the transport scat-tering time, we substitute τ (ϵ k,γ ) in Eq. (S24) into Eq. (S19) at charge neutrality and zero gap and evaluate the resulting expression over a range of temperatures to read off the prefactor α 0,qp in τ −1 0,qp = α 0,qp k B T /ℏ. This is to be compared against the actual α 0 in the energy-averaged transport scattering time τ 0 , for which τ −1 0 = α 0 k B T /ℏ. We find that α 0,qp = 0.32 for bilayer graphene at 200 K. We note that the α 0,qp obtained from quasiparticle lifetime is 1.5 times larger than that obtained from our experiment (α 0 = 0.225), as expected since transport scattering time is always larger than lifetime as mentioned above. In the limit of strong Coulomb interaction m * e 4 /(κ 2 ℏ 2 ) ≫ k B T and large effective mass m ⋆ , the value of τ −1 0,qp is independent of materialspecific parameters m * and κ, making the value of α 0,qp = 0.356 in τ −1 0,qp universally applicable in all strongly interacting materials with the same dispersion as bilayer graphene. Consistent with our expectations, the value for α 0 resulting from transport scattering time is slightly smaller than that from the quasiparticle lifetime as explained earlier. We therefore expect that the universal conductivity demonstrated in this work will be reproducible in all strongly interacting ambipolar hydrodynamic materials with the same dispersion.

Analytic calculation of α 0
Substituting τ (ϵ) in Eq. (S24) into the energy average formula in Eq. (S19), we numerically evaluate the average scattering rate ⟨τ ⟩ 0 at charge neutrality and find that it is given by 0.356k B T /ℏ as shown in Fig. S13. We can make some approximations to obtain analytical results that shed light on how this limit emerges. In particular, Eq. (S24) is the sum of a quasi-electron term, and a quasi-hole term that is given by the above equation with all the Fermi distribution functions replaced by one minus themselves (i.e. the probability of an electron being present is replaced by the probability of a hole being present). The quantity |W λ,λ " | 2 is the screened Coulomb interaction, W λ,λ " = where we have definedk ′ 2 = ℏ 2 k ′2 /2m * k B T ,q 2 = ℏ 2 q 2 /2m * k B T . Working out the integrals yields 1/τ (eh) qe (0) = (1/8)k B T /ℏ. It can be verified that the quasi-hole term contribution is equal to the quasi-electron term, as it must by electron-hole symmetry. Adding together the quasielectron and quasi-hole terms in Eq. (S24), we obtain the result that 1/τ (eh) (0) = (1/4)k B T /ℏ. This analytical result of dissipation at 1/4 is very close to our full numerical solution of 0.356, which is reasonable because the energy average in Eq. (S19) only involves energies close to zero.
6.2 Experimental evidence of universality: Conductivity collapse curve at µ = 0 Here we derive the collapse curve in Fig. 4(C) of the main text. Upon substituting the appropriate expressions for density and electron-hole momentum relaxation time τ e/h (∆) into Eq. (1) of the main text, the electron-hole limited conductivity at charge neutrality σ eh (∆) normalized by its gapless value σ eh (∆ = 0) collapses as a function of the ratio k B T /∆ according to the fit parameter-free function  Figure S14: Validation of Equation S54. Comparison of the collapse curves of conductivity at charge neutrality against k B T /∆ ext shown in Eqs. S53 (numerical) and S54 (analytic). The two curves are in excellent agreement.
While the integral in the final term has no exact analytical solution, it is trivial to work out numerically and may be excellently approximated using a function of the form F (x) = Ax 2 exp(−Bx), where x ≡ ∆/k B T and A, B are numerical fit parameters. We find that setting A = 1/8 and B = 5/8 reproduces the exact expression to an extent almost indistinguishable to the eye.
Making use of this approximation for the final term in Eq. (S53), we obtain σ eh (∆) σ eh (∆ = 0) = 1 + 1 log(2) log cosh ∆ We note that a similar collapse occurs for the commonly encountered case of gapped parabolic bands ϵ ± (k) = ± (∆/2 + ℏ 2 k 2 /(2m * )). In this case, following the same steps outlined above, the same collapse obtains but with the last term in square brackets removed. That is,  Figure S15: Additional data for the gapped conductivity. (A) Renormalized charge neutral conductivity plotted against k B T /∆ ext . The data is observed to collapse in agreement with ambipolar hydrodynamic conductivity in the strong electron-hole scattering regime, and in disagreement with the Arrhenius fit. (B-D) Comparison of theory (solid lines) in the strong electron-hole scattering limit against experimental data (circles) at four different temperatures and three different band gaps. The same single set of fit parameters is used throughout.

Further comparisons of theory and experiment at different gaps
We present in Fig. S15 the collapse of the charge neutral conductivity as a function of k B T /∆ ext up to room temperature without normalization. Here we have α 0 = 0.225 as obtained by fitting to σ 0 in the main text. Note that all the other scattering parameters α e/h ac , and τ i play no role at charge neutrality regardless of the gap. As seen in the figure, theory collapses in excellent agreement with experiment. We note that this collapse deviates from Arrhenius behavior, as shown from fitting the experimental results to the Arrhenius equation σ = A exp(−∆/2k B T ), where A is a numerical fit parameter and ∆ = ∆ ext /2.6. As stated in the text, we compare theory against experiment at different gaps in Fig. S15 below, using the same single set of four fit parameters featured in Table 2 in Eqs. (S12) and (S13). At all gaps considered, experiment agrees with theory using only the fit parameters extracted at zero gap.