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 SiO_{2} back gated graphene devices.

## INTRODUCTION

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/SiO_{2} 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/SiO_{2} 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 (V_{G}), 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 (*n*_{i}) and hence the induced carrier charge concentration (*n*) depend on V_{G} and temperature (T). Our device is therefore different from Si/SiO_{2} back gated devices where the linear dielectric does not permit a bound charge at the SiO_{2}/graphene interface. The graphene used in this work had a high *n*_{i}, 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 SiO_{2} back gated graphene. A temperature dependent impurity charge separation at the graphene/SiO_{2} interface and subsequent scattering with charge carriers was proposed to explain this effect.^{12}

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 *n*_{i} 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 V_{G} 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 work^{12} 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/SiO_{2}, 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.

## EXPERIMENTAL DETAILS

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 SiO_{2} back gating) lays in the first quadrant of the σ − V_{G} 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 (V_{D}) was fixed at 100 mV, while V_{G} was scanned as follows: −40 V → +40 V → −40 V at a fixed scan rate of 100 mV/s for each set point temperature. V_{G} was fixed at −40 V for 5 min before a fresh scan was initiated. The conductivity was calculated from *σ* = *LW*^{−1}*R*^{−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.

## EXPERIMENTAL RESULTS

Figure 2(a) shows σ vs V_{G} at 326 K. The arrows indicate the direction of the gate scan and show a counterclockwise *σ* − *V*_{G} 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 V_{G} 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 ($VGD$).^{12} Figure 2(b) shows the *σ* − *V*_{G} 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 *σ* − *V*_{G} curves for the forward and reverse V_{G} 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/SiO_{2} 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.

Figure 3 shows the *σ*(*T*) plots for different values of ΔV (=V_{G} − $VGD$), 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.

## SPECIFICS OF CHARGE DYNAMICS

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 (*n*_{i1}) and graphene/SiO_{2} (*n*_{i2}) 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 SiO_{2}, 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 *n*_{1}. At the same time, as the temperature increases, the impurity charges at the graphene/SiO_{2} interface get closer to graphene, which also increases the carrier density *n*_{2}. The carrier concentration in graphene due to doping by *n*_{i2} increases initially as T increases. However, at high T, there is increased compensation between the carriers and the impurity charges at the graphene/SiO_{2} interface due to their proximity to each other. This reduces *n*_{2}, resulting in a Gaussian shape for *n*_{2}(*T*), as shown in Fig. 4(b). Since the impurity charge concentration (*n*_{i2}) is not affected by V_{G}, its contribution toward the carrier concentration in graphene (*n*_{2}) 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}

## THEORY AND DISCUSSION

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 *n*_{i} (>10^{12} 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 by^{5,20–22}

Here, $DE=2E\pi (\u210fvF)2$ is the density of states of graphene, *v*_{F} 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,

Here, characteristic times *τ*_{i}, *τ*_{aph}, and *τ*_{oph} correspond to scattering on charged impurities and on acoustic and optical phonons, respectively.

To find the best approximations for these times, we estimate *E*_{F} for the graphene sample. As known, *E*_{F} = *ℏk*_{F}*v*_{F} and the Fermi wave vector is related to the charge carrier’s density $n:kF2=n\pi $. The Fermi velocity was *v*_{F} = 0.8 × 10^{6} m/s. Now, we can derive the expression for the Fermi temperature as $TF=EFkB:TFK\u22481500n\xd710\u221212$, and the charge density is expressed in 1 cm^{2}.^{22} The charge carrier density (*n*) in our graphene film depends on the temperature dependent *n*_{i} and V_{G}. At each value of V_{G}, we can roughly estimate *n* from the ratio of measured conductivity (σ) and mobility (μ). For the sample used in the experiment, we obtain E_{F} ≈ 300–450 meV, which corresponds to T_{F} ≈ 4000–6000 K.^{12} Therefore, we see that T ≪ T_{F} remains valid over considered temperatures used in this work, which leads to the following approximations:^{22,23}

where *ρ*_{m}, *v*_{ph}, and Λ are the graphene mass density, sound velocity in graphene, and the deformation potential, respectively.

Using the expression for the characteristic time *τ*_{oph} given in earlier works,^{3,24} we may get an approximation as follows:

Here, the summation is carried over all optical modes. Integrals I_{0} and I_{1} have the following form:

where *r*_{s} is the Wigner–Seitz radius, which, in the considered case, takes on values 0.8–0.9. At *r*_{s} = 0.88, I_{0} = 0.0377 and I_{1} = 0.1433.

To estimate $1\tau oph$ given by Eq. (4), we assume that for SiO_{2}, ϵ_{∞} = 2.4 and ϵ_{0} = 3.9,^{8} and the frequencies of the optical modes in graphene are ℏΩ_{1} = 63 meV and ℏΩ_{2} = 116 meV.^{10} Then, we get $1\tau oph\u223c0.05ps\u22121$ at room temperature *k*_{B}*T* = 26 meV. In the considered case of a sample with high impurity density *n*_{i}, the charge carrier’s density *n* may be expected to take on values of the order or greater than of 10^{12} cm^{−2}.^{9} Assuming that *n* ∼ 10^{12} cm^{−2} and the deformation potential Λ ∼ 10 eV, we may estimate: $1\tau aph\u223c2ps\u22121$; 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

Here, T_{0} = 1500 K, $\sigma 0=e2h$, and the term $\sigma i=A\sigma 0nni$^{21} represents the diffusive conductivity in the case when the effect of scattering on the phonons may be neglected.^{22} Solving Eq. (6) for *σ*_{i}, we get

where *σ*_{D} = *σ* − ∆*σ*, and $\Delta \sigma =1.16e2h$ represents the intrinsic conductivity in graphene.^{12} *σ* represents the measured conductivity, as shown in Fig. 3. The dimensionless coefficient α equals

Figures 5(a) and 5(b) show the temperature dependence of the total conductivity *σ* and *σ*_{ι}, respectively, for different ΔV values. Again, *σ*_{i} 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 *ρ*_{m} = 7.6 × 10^{−8} g/cm^{2}, the sound velocity *v*_{ph} = 2 × 10^{6} 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.

To better separate out the contribution from the phonon scattering, we introduce a “resistivity” $\rho D=\sigma D\u22121$, which equals the sum of terms $\rho i=\sigma i\u22121$ and $\rho aph=\sigma aph\u22121$. These terms are contributions from the scattering on the charged impurities and acoustic phonons. Note that the phonon contribution does not depend on V_{G}. We have

The ratio $\rho i\rho aph$ 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 (*n*_{i}) and to the corresponding lengthening of the scattering time (*τ*_{i}). Graphene is then doped by the uncompensated charges (i.e., *n* increases) and partly by the electric field produced by V_{G}. 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, $\rho i\rho aph$ falls as temperature rises, and the fall is significantly more pronounced at lower ΔV. As shown in Fig. 6(b), *ρ*_{i} is significantly greater than *ρ*_{aph} (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, *ρ*_{aph} reaches about 10% of *ρ*_{i}. 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 *n*_{i}. The relevant data for the quantity $\sigma i\sigma 0=Anni$, 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 *n*_{i}. 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,

At the same time, we have a relationship^{19,20}

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 SiO_{2} 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.

## CONCLUSIONS

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/SiO_{2} 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 *τ*_{i} strongly depended on V_{G}. The V_{G} induced polarization reduced *n*_{i}, thus increasing *τ*_{i} 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 SiO_{2} back gated devices.

## SUPPLEMENTARY MATERIAL

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

## ACKNOWLEDGMENTS

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

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

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