Charge transport in ferroelectric (FE) gated graphene far from the Dirac point (DP) was studied in the temperature range 300 K < T < 350 K. A non-monotonic/monotonic/non-monotonic behavior in the conductivity [σ(T)] was observed as one moved away from the DP. As the gate polarization increased, additional impurity charges were compensated, which reduced charge scattering. The uncompensated charges doped graphene and σ(T) switched to a monotonic increase with increasing T. However, far from the DP, the polarization reached saturation, which resulted in still lower impurity charge scattering. The carrier concentration increased, and a non-monotonic response in σ(T) reappeared, which was attributed to phonon scattering. A theoretical model is presented that combined impurity charge and phonon scattering conduction mechanisms. The top gate polarizable FE provided a novel approach to investigate charge transport in graphene via controlled compensation of impurity charges, and in the process revealed non-monotonic behavior in σ(T) not previously seen in SiO2 back gated graphene devices.

Graphene is a two-dimensional thin film made of covalently bonded C atoms arranged in a hexagonal lattice.1 Pure graphene has a zero bandgap and can be n- or p-type doped by an external electric field.2–4 Charged impurities unintentionally adsorbed on the graphene surface during synthesis and post processing typically lead to p-type doping. Adsorbed impurity charges also influence charge transport, affecting the most important transport mechanisms that include activation across potential fluctuations created by charge impurities and scattering of charge carriers by these impurities and phonons. The impurity charge concentration may be temperature dependent, thus bringing changes into the temperature dependence of the graphene conductivity (σ). Temperature dependent conductivity [σ(T)] studies of graphene deposited on Si/SiO2 substrates with back gating show that the conductivity could switch from monotonous to non-monotonous behavior depending on the back gate voltage.5–10 The significance of having a monotonous to non-monotonous behavior in σ(T) shows that the charge transport mechanism has changed, and this can provide insight regarding charge dynamics in graphene.

The graphene investigated in this work was grown by chemical vapor deposition and transferred on to a p-doped Si/SiO2 substrate. It was then electrically wired in a field effect transistor (FET) configuration where the gate material was a top coated ferroelectric (FE) polymer thin film. Figures 1(a) and 1(b) show a schematic of the device configuration and a top view of the actual device itself. By controlling the gate voltage (VG), the gate polymer polarization increases or decreases. This results in a surface bound charge of one sign in the polymer at the polymer/graphene interface. Thus, impurity charges of the opposite sign at the polymer/graphene interface could be compensated or uncompensated by the polarizable FE polymer, leading to p- or n-type doping.11 The impurity charge concentration (ni) and hence the induced carrier charge concentration (n) depend on VG and temperature (T). Our device is therefore different from Si/SiO2 back gated devices where the linear dielectric does not permit a bound charge at the SiO2/graphene interface. The graphene used in this work had a high ni, so the activation transport mechanism was negligible and diffusive transport predominated. In a previous study, we investigated charge transport in the same graphene at and near the Dirac point (DP) in the temperature range 300 K < T < 350 K and found that impurity charge scattering dominated charge transport.12 Our results, however, showed that the temperature dependence of the conductivity [σ(T)] exhibited a non-monotonic behavior at the DP, i.e., the conductivity increased with increasing temperature, reaching a peak at 326 K above which it began to decrease. Such behavior has not been observed in SiO2 back gated graphene. A temperature dependent impurity charge separation at the graphene/SiO2 interface and subsequent scattering with charge carriers was proposed to explain this effect.12 

FIG. 1.

(a) 3-D drawing of the device and external connections. S = source, D = drain, and G = gate terminals. (b) Microscope image of the graphene used in the device. The dotted line indicates the location of the graphene flake. (b) is reproduced with permission from Figueroa et al., AIP Adv. 11, 085015 (2021). Copyright 2021 Author(s), licensed under a Creative Commons Attribution 4.0 License.

FIG. 1.

(a) 3-D drawing of the device and external connections. S = source, D = drain, and G = gate terminals. (b) Microscope image of the graphene used in the device. The dotted line indicates the location of the graphene flake. (b) is reproduced with permission from Figueroa et al., AIP Adv. 11, 085015 (2021). Copyright 2021 Author(s), licensed under a Creative Commons Attribution 4.0 License.

Close modal

We now expand our study to include charge transport far from the DP. As we shift away from the DP, the conductivity increases monotonically with increasing temperature. This behavior is attributed to an increase in gate polarization, which, in turn, increases graphene doping and hence n. Impurity charge scattering is still the dominant transport mechanism. Far from the DP, where the gate polarization reaches saturation, σ(T) once again shows a slow decrease as T increases. A non-monotonic/monotonic/non-monotonic dependence of σ(T) as we shift away from the DP has not been reported before. As the polarization nears saturation ni decreases, and fewer impurity charges result in diminishing contribution from scattering on these charges into the diffusive transport of charge carriers. It is shown below that this effect becomes more pronounced when the temperature rises. At the same time, the scattering on phonons does not depend on VG and it moderately increases following the increase in the temperature. Thus, the relative contribution of the impurity charge scattering to phonon scattering is lowered as temperature increases, although the impurity charge scattering still remains the dominant transport mechanism. Therefore, at high temperatures and far from the DP, the phonon scattering becomes more important, and its contribution manifests itself in a decrease in σ(T) at higher temperatures as observed in the present experiment. The purpose of this study combined with our earlier work12 is to show that gating graphene with a ferroelectric polymer leads to impurity charge compensation. This, in turn, affects the scattering by carriers not only with the uncompensated impurity charges adsorbed on both graphene interfaces, i.e., graphene/polymer and graphene/SiO2, but also with phonons. The net result is a change in charge transport mechanisms and a re-entrant non-monotonic-monotonic-nonmonotonic behavior in σ(T) not previously reported in graphene.

Graphene was grown via chemical vapor deposition (CVD) and transferred onto the substrate as mentioned earlier. Adsorption of unintended impurity charges during the growth and handling process leads to doping that was typically p-type, since the DP in all our devices (with SiO2 back gating) lays in the first quadrant of the σ − VG curves. The room temperature ferroelectric co-polymer, poly(vinylidene fluoride/trifluoro ethylene) (PVDF/TrFE:75/25) was used as the gate insulator. This polymer was spun-cast from a 9 wt. % solution in 1-methyl-2-pyrrolidinone. After drying, a Ag gate electrode was thermally evaporated through a shadow mask to cover the graphene channel between the source and drain electrodes of the device. Keithley instruments were used for current–voltage data acquisition of the FEFET in vacuum. The drain-source voltage (VD) was fixed at 100 mV, while VG was scanned as follows: −40 V → +40 V → −40 V at a fixed scan rate of 100 mV/s for each set point temperature. VG was fixed at −40 V for 5 min before a fresh scan was initiated. The conductivity was calculated from σ = LW−1R−1, where R = resistance, L = length, and W = width of the graphene channel. See Ref. 12 for further details on the device fabrication and electrical characterization.

Figure 2(a) shows σ vs VG at 326 K. The arrows indicate the direction of the gate scan and show a counterclockwise σVG response in the first quadrant and a clockwise response in the second quadrant. Such hysteresis behavior is consistent with the ferroelectric polarization of the gate polymer, where the current minima correspond to the DP in graphene, while the maximum current remains high upon gate voltage reversal due to polarization retention in the gate polymer.11,13–18 Dipole switching at the coercive voltage between two polarized states (↓↑) results in two DP’s. At these points, the majority of charge carriers switch from holes to electrons and vice versa in crossing each DP. Apart from the doping due to gate polarization (P), the electric field (E) produced by VG can also dope graphene.13 The slope of the dotted straight line fits to the data shown in Fig. 2(a) is used to calculate the charge (hole) mobility (μ), while their intersection gives the gate voltage at the DP (VGD).12Figure 2(b) shows the σVG curves during the forward sweep at all recorded temperatures from 300 to 350 K in 2 K steps. Figure S1 in the supplementary material shows the σVG curves for the forward and reverse VG sweep at all temperatures. Figure 2(b) shows that the conductivity at the DP increases with increasing temperature, reaching a maximum value at 326 K, above which it decreases (pink symbols) due to a temperature dependent impurity charge separation at the graphene/SiO2 interface as discussed in Ref. 12. In this paper, we analyze the temperature dependent charge transport extracted from Fig. 2(b) as one shifts away from the DP.

FIG. 2.

(a) Conductivity (σ) vs VG at 326 K for the graphene FE-FET under forward and reverse gate voltage sweep. VD = 0.1 V. The arrows indicate the direction of the sweep starting at −40 V. (b) σ vs VG for the forward gate sweep only (−40 V → +40 V) at all measured temperatures: 300 K (red) to 350 K (green) in steps of 2 K. The blue curve corresponds to T = 326 K. The pink curve represents the conductivity values at the DP corresponding to each temperature and shows a non-monotonic behavior. (b) has been reproduced with permission from Figueroa et al., AIP Adv. 11, 085015 (2021). Copyright 2021 Author(s), licensed under a Creative Commons Attribution 4.0 License.

FIG. 2.

(a) Conductivity (σ) vs VG at 326 K for the graphene FE-FET under forward and reverse gate voltage sweep. VD = 0.1 V. The arrows indicate the direction of the sweep starting at −40 V. (b) σ vs VG for the forward gate sweep only (−40 V → +40 V) at all measured temperatures: 300 K (red) to 350 K (green) in steps of 2 K. The blue curve corresponds to T = 326 K. The pink curve represents the conductivity values at the DP corresponding to each temperature and shows a non-monotonic behavior. (b) has been reproduced with permission from Figueroa et al., AIP Adv. 11, 085015 (2021). Copyright 2021 Author(s), licensed under a Creative Commons Attribution 4.0 License.

Close modal

Figure 3 shows the σ(T) plots for different values of ΔV (=VGVGD), where the plot for ΔV = 0 corresponds to the DP and is the same as the pink curve seen in Fig. 2(b). From Fig. 3, we see that σ(T) is clearly non-monotonic at and near the DP and that the non-monotonicity disappears as one shifts away from the DP. At large ΔV values (>−30 V), there appears once again a weak non-monotonic change in σ(T). The data and fitting lines for ΔV = 0, −1, and −2 V are reproduced with permission from Figueroa et al., AIP Adv. 11, 085015 (2021). Copyright 2021 Author(s), licensed under a Creative Commons Attribution 4.0 License. In this work, we theoretically analyze the data for ΔV = −6 V up to −40 V.

FIG. 3.

Temperature dependence of the conductivity for different values of ΔV=VGVGD shown to the right of each curve. These data points were extracted from Fig. 2(b) for each ΔV. The conductivity measured for ΔV = 0 corresponds to the conductivity at the DP. The data and fitting lines for ΔV = 0, −1, and −2 V are reproduced with permission from Figueroa et al., AIP Adv. 11, 085015 (2021). Copyright 2021 Author(s), licensed under a Creative Commons Attribution 4.0 License. The solid pink symbol curve is the same as that seen in Fig. 2(b).

FIG. 3.

Temperature dependence of the conductivity for different values of ΔV=VGVGD shown to the right of each curve. These data points were extracted from Fig. 2(b) for each ΔV. The conductivity measured for ΔV = 0 corresponds to the conductivity at the DP. The data and fitting lines for ΔV = 0, −1, and −2 V are reproduced with permission from Figueroa et al., AIP Adv. 11, 085015 (2021). Copyright 2021 Author(s), licensed under a Creative Commons Attribution 4.0 License. The solid pink symbol curve is the same as that seen in Fig. 2(b).

Close modal

In this section, we elucidate the origin of nontrivial temperature dependencies of densities of both impurities and charge carriers, which occur due to the presence of the ferroelectric gate polymer and affect temperature dependencies of σ(T). Figure 4(a) shows the schematic model of the impurity charge dynamics at the graphene/polymer (ni1) and graphene/SiO2 (ni2) interfaces for some fixed gate voltage (−18 V). The black arrows represent the polarization growth in the PVDF-TrFE film as T increases. The basic figure shows graphene lying over SiO2, which is covered with a thin film of PVDF-TrFE. The three panels in the schematic Fig. 4(a) show how as the temperature rises, the polarization in PVDF-TrFE increases [inset in Fig. 4(b)]. This leads to increased impurity charge compensation of one sign at the graphene/polymer interface and the impurity charge of the other sign dopes graphene increasing the carrier density n1. At the same time, as the temperature increases, the impurity charges at the graphene/SiO2 interface get closer to graphene, which also increases the carrier density n2. The carrier concentration in graphene due to doping by ni2 increases initially as T increases. However, at high T, there is increased compensation between the carriers and the impurity charges at the graphene/SiO2 interface due to their proximity to each other. This reduces n2, resulting in a Gaussian shape for n2(T), as shown in Fig. 4(b). Since the impurity charge concentration (ni2) is not affected by VG, its contribution toward the carrier concentration in graphene (n2) is unaffected and retains the Gaussian shape for all values of ΔV. Thus, we see in Fig. 4(b) that the carrier concentration (n) in graphene reveals a non-monotonic decrease with increasing temperature, which, in turn, is reflected in the measured σ(T) plots of Fig. 3. The inset in Fig. 4(b) shows how the saturation polarization of the polymer increases with increasing temperature.12 

FIG. 4.

(a) Schematic model of impurity charges at the graphene/polymer (ni1) and graphene/SiO2 (ni2) interfaces for arbitrary fixed ΔV = −18 V. Red (Blue) dots represent positive (negative) impurity charges, and P is the polymer polarization. P increases with increasing temperature and so does the bound surface density. The dotted oval shows charge compensation of the impurity charge by the bound surface charge in the polymer. (b) Carrier concentration (n) vs temperature for ΔV = −6 V (red) and −40 V (blue). The dashed lines represent carrier concentration n1 due to doping by ni1, and the Gaussian curve (black) represents carrier concentration n2 due to doping by ni2, which is independent of ΔV. Inset: PVDF-TrFE saturated polarization as a function of temperature measured in a PVDF-TrFE capacitor.12 

FIG. 4.

(a) Schematic model of impurity charges at the graphene/polymer (ni1) and graphene/SiO2 (ni2) interfaces for arbitrary fixed ΔV = −18 V. Red (Blue) dots represent positive (negative) impurity charges, and P is the polymer polarization. P increases with increasing temperature and so does the bound surface density. The dotted oval shows charge compensation of the impurity charge by the bound surface charge in the polymer. (b) Carrier concentration (n) vs temperature for ΔV = −6 V (red) and −40 V (blue). The dashed lines represent carrier concentration n1 due to doping by ni1, and the Gaussian curve (black) represents carrier concentration n2 due to doping by ni2, which is independent of ΔV. Inset: PVDF-TrFE saturated polarization as a function of temperature measured in a PVDF-TrFE capacitor.12 

Close modal

Some charge transport mechanisms in graphene include activation across potential fluctuations created by charged impurities (which are responsible for inhomogeneous electron–hole puddle formation) and diffusive transport controlled by scattering of charge carriers on impurity charges and phonons.5,19–23 Activation plays an important role at low impurity concentration where potential fluctuations are higher and, hence, lead to higher activation energy. When the activation energy is greater than the thermal energy, then this mechanism must be considered, for example, in the experiment carried out in Ref. 5.

In our current experiments, the energy of carriers’ activation across potential fluctuations near DP is a few meV and is much smaller than the thermal energy in the range 300 K < T < 350 K.12 This is typical for “dirty” graphene samples, i.e., large ni (>1012 cm−2), where the effect of potential fluctuations is rather small and negligible. For this reason, we focus on the diffusive transport only, which is controlled by impurity charge scattering and phonon scattering. The purpose of this analysis is to separate out the contributions from impurity and phonon scattering. By this, we show that scattering on acoustic phonons could rather noticeably contribute to charge transport in our sample at higher temperatures and at ΔV values far from the DP.

The diffusive conductivity in graphene is given by5,20–22

σDT=e22dEDEvF2τE,TfEE.
(1)

Here, DE=2Eπ(vF)2 is the density of states of graphene, vF is the graphene Fermi velocity, f(E) is the Fermi distribution function, and τ is the scattering time that takes into consideration the relevant mechanisms of transport,

1τ=1τi+1τaph+1τoph.
(2)

Here, characteristic times τi, τaph, and τoph correspond to scattering on charged impurities and on acoustic and optical phonons, respectively.

To find the best approximations for these times, we estimate EF for the graphene sample. As known, EF = ℏkFvF and the Fermi wave vector is related to the charge carrier’s density n:kF2=nπ. The Fermi velocity was vF = 0.8 × 106 m/s. Now, we can derive the expression for the Fermi temperature as TF=EFkB:TFK1500n×1012, and the charge density is expressed in 1 cm2.22 The charge carrier density (n) in our graphene film depends on the temperature dependent ni and VG. At each value of VG, we can roughly estimate n from the ratio of measured conductivity (σ) and mobility (μ). For the sample used in the experiment, we obtain EF ≈ 300–450 meV, which corresponds to TF ≈ 4000–6000 K.12 Therefore, we see that T ≪ TF remains valid over considered temperatures used in this work, which leads to the following approximations:22,23

1τi=ninEF(2rs)2I0:1τaph=π4nΛ2ρmvph2TTF,
(3)

where ρm, vph, and Λ are the graphene mass density, sound velocity in graphene, and the deformation potential, respectively.

Using the expression for the characteristic time τoph given in earlier works,3,24 we may get an approximation as follows:

1τoph4κI11ε+11ε0+1lΩlexpΩlkBT.
(4)

Here, the summation is carried over all optical modes. Integrals I0 and I1 have the following form:

I0=01dxx21x2(x+2rs)2,I1=01dxx1x2(x+2rs),
(5)

where rs is the Wigner–Seitz radius, which, in the considered case, takes on values 0.8–0.9. At rs = 0.88, I0 = 0.0377 and I1 = 0.1433.

To estimate 1τoph given by Eq. (4), we assume that for SiO2, ϵ = 2.4 and ϵ0 = 3.9,8 and the frequencies of the optical modes in graphene are ℏΩ1 = 63 meV and ℏΩ2 = 116 meV.10 Then, we get 1τoph0.05ps1 at room temperature kBT = 26 meV. In the considered case of a sample with high impurity density ni, the charge carrier’s density n may be expected to take on values of the order or greater than of 1012 cm−2.9 Assuming that n ∼ 1012 cm−2 and the deformation potential Λ ∼ 10 eV, we may estimate: 1τaph2ps1; therefore, the scattering of charge carriers on acoustic phonons predominates. In further calculations, we omit contribution from the scattering on the optical phonons from the total scattering time. Then, the expression for the conductivity accepts the form

σσ0=σiσ011+ασiσ0TTo+Δσσ0.
(6)

Here, T0 = 1500 K, σ0=e2h, and the term σi=Aσ0nni21 represents the diffusive conductivity in the case when the effect of scattering on the phonons may be neglected.22 Solving Eq. (6) for σi, we get

σiσ0=Anni=σDσ011ασDσ0TTo,
(7)

where σD = σ − ∆σ, and Δσ=1.16e2h represents the intrinsic conductivity in graphene.12 σ represents the measured conductivity, as shown in Fig. 3. The dimensionless coefficient α equals

α=π8T0Λ2nρmvph2EFTF.
(8)

Figures 5(a) and 5(b) show the temperature dependence of the total conductivity σ and σι, respectively, for different ΔV values. Again, σi is the diffusive conductivity as it should be in the absence of phonons when the scattering of the charge carriers in graphene solely occurs on charged impurities. Note that this contribution to the conductivity could be directly extracted from the experimental data for the conductivity assuming that ∆σ is known.12 The coefficient α was estimated by using the graphene mass density ρm = 7.6 × 10−8 g/cm2, the sound velocity vph = 2 × 106 cm/s, and Λ ≈ 35 eV and calculated to be 0.027. Note that the ratio nTFEF does not depend on n and T. Equations for the fitting curves presented in Figs. 5 and 6 were obtained by using the least mean squares method and are given in the supplementary material.

FIG. 5.

(a) Measured conductivity-σ and (b) conductivity due to scattering by charge impurities-σi as functions of temperature for several values of ΔV shown to the right of each plot. The experimental data in Fig. 5(b) were obtained from Fig. 5(a) using σ = σD + ∆σ, α = 0.027, and T0 = 1500 K, and Eq. (7) where Δσ=1.16e2h. Fitting curves for Fig. 5(b) were obtained using Eqs. (1)–(3) in the supplementary material.

FIG. 5.

(a) Measured conductivity-σ and (b) conductivity due to scattering by charge impurities-σi as functions of temperature for several values of ΔV shown to the right of each plot. The experimental data in Fig. 5(b) were obtained from Fig. 5(a) using σ = σD + ∆σ, α = 0.027, and T0 = 1500 K, and Eq. (7) where Δσ=1.16e2h. Fitting curves for Fig. 5(b) were obtained using Eqs. (1)–(3) in the supplementary material.

Close modal
FIG. 6.

Ratios (a) n/ni and (b) ρi/ρaph as functions of temperature plotted for several values of ΔV. (c) Explicit plots of the temperature dependencies of the resistivity (for different ΔV) due to impurity charge scattering (colored dotted line) and acoustic phonon scattering (aph) (black dotted line); the solid line fits to the data, which is the sum of the two scattering mechanisms (i.e., impurity and phonon). The open circles represent the experimental data. The solid lines fit to the data in (a) were obtained by using Eq. (7) and the fitting parameter A. The solid lines for (b) and (c) were obtained by using Eqs. (7) and (9).

FIG. 6.

Ratios (a) n/ni and (b) ρi/ρaph as functions of temperature plotted for several values of ΔV. (c) Explicit plots of the temperature dependencies of the resistivity (for different ΔV) due to impurity charge scattering (colored dotted line) and acoustic phonon scattering (aph) (black dotted line); the solid line fits to the data, which is the sum of the two scattering mechanisms (i.e., impurity and phonon). The open circles represent the experimental data. The solid lines fit to the data in (a) were obtained by using Eq. (7) and the fitting parameter A. The solid lines for (b) and (c) were obtained by using Eqs. (7) and (9).

Close modal

To better separate out the contribution from the phonon scattering, we introduce a “resistivity” ρD=σD1, which equals the sum of terms ρi=σi1 and ρaph=σaph1. These terms are contributions from the scattering on the charged impurities and acoustic phonons. Note that the phonon contribution does not depend on VG. We have

ρD=ρi+ρaph=1Anin1σ0+αTT01σ0.
(9)

The ratio ρiρaph strongly depends on T, especially for lower values of ΔV (∼−6 V), as shown in Fig. 6(b). We observe that this ratio takes on smaller values at higher ΔV (>−30 V). This is because as we move away from the DP, impurity charge scattering decreases due to increased gate polarization, which compensates impurity charges of one sign (i.e., positively charged impurities) and leads to a decrease in the charged impurities density (ni) and to the corresponding lengthening of the scattering time (τi). Graphene is then doped by the uncompensated charges (i.e., n increases) and partly by the electric field produced by VG. The result is an increase in σ, as shown in Fig. 2(b), as we move away from the DP. In addition, at each ΔV, ρiρaph falls as temperature rises, and the fall is significantly more pronounced at lower ΔV. As shown in Fig. 6(b), ρi is significantly greater than ρaph (i.e., ΔV = −6 V), which agrees with the theory of electron transport in graphene within the high carrier density transport regime.20–22 However, the effect of scattering on acoustic phonons to the diffusive conductivity does not appear to be negligible, especially at high ΔV, and it becomes more significant when the temperature increases. At T ≈ 350 K and ΔV = −40 V, ρaph reaches about 10% of ρi. Therefore, one may conjecture that a slight decrease in the conductivity accompanying the temperature rise as T approaches 350 K occurring at high ΔV originates from the scattering of charge carriers on acoustic phonons. Figure 6(c) shows the explicit temperature dependencies of the resistivities due to impurity and phonon scatterings as a function of temperature and how the sum of the two compares to the experimentally observed total resistivity.

Note that in the present analysis, we did not use independently obtained experimental data for n and/or ni. The relevant data for the quantity σiσ0=Anni, which is proportional to the ratio nni, were extracted solely from the experimental data concerning the conductivity σ by means of Eq. (7). This differs from the commonly used methods, which require separate determining of n and ni. As we explore charge transport away from the DP, the charge carriers’ density may be expressed in terms of the graphene conductivity σ and mobility μ,

nT=σ(T)eμ(T).
(10)

At the same time, we have a relationship19,20

ni(T)5×1015Vs1μ(T).
(11)

Using these equations, we can express the ratio nni in terms of the conductivity σ, which was experimentally explored, and obtain the data shown in Fig. 6(a). Then, we apply Eqs. (1)–(3) given in the supplementary material and attain a reasonably good fit when the constant A is close to 20, which is the theoretically predicted value for graphene on a SiO2 substrate.19,20 This is illustrated in Fig. 6(a), and Table I gives the values of A for each fit. We think it likely that the proposed method of determining nni maybe also applied near the DP provided that the considered sample is sufficiently “dirty,” and diffusive transport predominates.

TABLE I.

Tabulated values of A for each ΔV. The parameter A was obtained using the fitting curves for σi.

ΔV (V)A
−6 19.18 
−10 19.79 
−18 20.33 
−30 20.66 
−40 20.79 
ΔV (V)A
−6 19.18 
−10 19.79 
−18 20.33 
−30 20.66 
−40 20.79 

We analyzed charge transport in graphene in the temperature range 300 K < T < 350 K far from the DP where the effect of electron–hole “puddling” was negligible, and diffusive transport mechanisms predominated. At and near the DP (ΔV ∼ 0 V), σ(T) was non-monotonic. We focused our analysis of the temperature dependent conductivity at moderate and strong gating. It was observed that at moderate gate voltages (ΔV ∼ −10 V), the conductivity increased monotonically as temperature increased. However, at strong gating (ΔVg ∼ −40 V), it showed a non-monotonic temperature dependency. Initial growth of σ with increasing temperature was replaced by a slow decrease as T further increased and approached 350 K. Such non-monotonic/monotonic/non-monotonic behavior of σ within a narrow temperature range has not been reported before.

In our previous work,12 we reported a non-monotonic σ(T) at and near the DP. In that case, σ was determined by charge carrier scattering off charged impurities, and the non-monotonicity was explained as the effect of the FE polymer trapping the impurity charges at the polymer/graphene interface and compensating them via polarization. This effect along with the impurity charges separation at the graphene/SiO2 interface resulted in the specific temperature dependencies of nni ratio responsible for the appearance of the non-monotonicity in the σ vs T curves.

To explain the non-monotonicity in σ(T) far from the DP observed in this work, we have estimated and compared contributions to the charge transport from different diffusive transport mechanisms. We showed that although scattering on impurities remained the predominating mechanism, the corresponding scattering time τi strongly depended on VG. The VG induced polarization reduced ni, thus increasing τi and diminishing the part taken by scattering on the charged impurities in the diffusive transport. This made the part taken by scattering on phonons, which did not depend on ΔV, more significant. At ΔV ∼ −40 V, the contribution to the graphene resistivity from scattering on acoustic phonons was about 10% of the contribution from scattering on charged impurities, and its influence may not be neglected. As known, scattering on phonons becomes more intensive as the temperature increases, leading to the fall of conductivity. We believe that non-monotonic temperature dependencies of σ observed far away from the DP are manifestations of the scattering on the acoustic phonons whose part in relation to the scattering on charged impurities becomes enhanced at strong gating. Finally, our device structure with a top gate polarizable FE provided a novel approach to investigate charge transport in graphene via controlled compensation of impurity charges, and in the process revealed non-monotonic behavior in σ(T) not previously seen in SiO2 back gated devices.

See the supplementary material for a complete plot of the device curves at all temperatures and for additional equations as mentioned in the text.

This work was supported by the National Science Foundation under Grant Nos. DMR-PREM-2122102, DMR-RUI-1800262, and DMR-MRSEC-1720530.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Supplementary Material