Direct mapping of local Seebeck coefficient in 2D material nanostructures via scanning thermal gate microscopy

We first measure the gate dependent local thermoelectric signal of a graphene rectangle (see figure 2). The charge neutrality point of the sample is at a back gate voltage of around V_g = 28 V, as inferred from the gate dependent measurement of the two-terminal channel resistance (see figure 2(c)). Thus, the graphene is strongly p-doped at $V_\mathrm{g} = 0$ V, which is due to the interaction of the SiO₂ substrate surface charges with the graphene as reported previously [24, 25]. An STGM thermovoltage map with an excess tip apex-temperature of $T_\mathrm{tip} \approx 53$ K (where the tip-apex temperature is defined with respect to the microscope stage and when the tip is in contact with the sample surface) is recorded at three back gate voltages: for a p-doped graphene channel at −15V, close to the Dirac point at 28V and for an n-doped graphene channel at 45V (figure 2(a), left to right). In the p-doped regime, a thermovoltage on the order of 30 µV is generated across the device when the hot tip heats up the areas around the electrical contacts, where the polarity of the induced thermovoltage is negative (positive) when the source (drain) is locally heated up. As before, the source is defined as the contact kept on ground during the measurement. Only very little to no thermovoltage drop is observed when the tip is scanned over the graphene channel. At low carrier concentrations around the Dirac point, a seemingly chaotic distribution of thermovoltage signal of changing polarity appears in the graphene channel. In the n-doped regime, the signal along the graphene strip is again reduced. A larger voltage build up (compared to that in the channel) is observed when the source and drain contacts are heated up, with opposite polarity compared to the p-doped case.

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The chaotic signal observed close to the Dirac point is attributed to charge puddles inside the graphene channel, which locally vary the energy dependent position of the charge neutrality point and are commonly observed for graphene on SiO₂ substrates (see figure 2(b)) [19, 26]. This leads to sharp local variations in S(x) which in turn influences the thermovoltage signal. We note that the signal from the charge puddles dominates the map at the charge neutrality point. This is due to the high sensitivity of the magnitude and sign of the Seebeck coefficient on the carrier density around the Dirac point [17, 19], resulting in large differences in S(x) over the short distances associated with charge puddles.

It is important to note, that a change in the thermal spreading resistance of the sample and the tip-sample interface thermal resistance, which is dominated by the thermal conductivity of the measured surface, strongly influences the tip-apex temperature. This is particularly relevant when the hot tip is in contact with the Au electrodes since the thermal conductivity of Au is two orders of magnitude larger than that of SiO₂/graphene on SiO₂. This leads to a reduction of the excess tip-apex temperature from $T_\mathrm{tip} \approx 53$ K to $T_\mathrm{tip} \approx 1.5$ K when the tip is in contact with Au compared to SiO₂/graphene [27].

In the following we discuss the origin of the observed thermovoltage signals. When the hot tip heats up the left or right electrical contact, the measured thermovoltage drop corresponds to the sum of the thermoelectric voltage build-up of the graphene under the Au contacts and the graphene channel. This signal will later be used to estimate the (global) Seebeck coefficient of the graphene channel with respect to the graphene underneath the contact [5].

The origin of thermoelectric signals is the preference of charge carriers to diffuse from hot regions in a material towards colder regions. The driving force for this process is proportional to the temperature difference ΔT and the entropy the system gains per diffusing carrier. The latter is highest if a carrier diffuses towards regions of higher DOS. This, depending on the source/drain configuration and the majority carriers in the sample, results in the build up of a positive or negative thermovoltage under open circuit conditions.

Graphene underneath the gold contacts is shielded from the electric field created by a gate voltage, such that its carrier concentration is not affected when a $V_\mathrm{g}$ is applied. For a gate voltage $V_\mathrm{g} = 45$ V, where the graphene channel is highly n-doped (see figure 2(a) right) the DOS at the Fermi energy is larger in the channel than in the graphene underneath the gold contact (see figure 2(b) right), meaning electrons will diffuse from the contact area into the graphene channel. This results in a positive (negative) thermovoltage build up at the drain (source) contact. While the DOS in the channel is still larger in the channel than in the graphene underneath the gold contact for a p-doped sample (see figure 2(b) left), the majority carriers are now holes, resulting in a negative (positive) thermovoltage build up at the drain (source) contact.

In the following, we apply STGM to a second type of graphene device, a junction formed between SLG and BLG (similar to the device and measurement shown in figure 1(b)). Shown in figure 3, the sample consists of a graphene flake that has both a single-layer and a bilayer section and is contacted by four gold contacts. The two contacts labelled source (S) and drain (D) are used for measuring the thermovoltage drop, while the other contacts are left floating (F).

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In figure 3, in accordance with the previous measurement of the graphene rectangle, a thermovoltage signal on the contacts is observed, which switches polarity as the gate voltage is swept over the charge neutrality point. The gate dependent thermovoltage maps highlight the impact of the contacts on the graphene sample.

The two passive, floating contacts show bi-polar signals, reversing polarity as the gate voltage is increased. This is because the floating contacts locally dope the graphene and pin the carrier density, an effect that has previously been observed in scanning photocurrent microscopy measurements [28]. In addition, the Fermi level pinning from the contacts leads to a strongly non-homogenous change of the carrier density within the sample. Rather than occurring simultaneously over the whole sample area, the change is gradual, starting from the SLG/BLG junction and wandering out towards the contacts as can be seen for higher gate voltages (right side of figure 3 and further images in the SI). Furthermore, STGM allows us to image the gate screening due to a second layer of graphene on top of the SLG channel. While a multitude of charge puddles appear around the charge neutrality point for gate voltages $V_\mathrm{g} = 10/12.5$ V on the SLG side, the BLG area of the sample shows hardly any. This is attributed to the electric field created by surface dipoles in the oxide being screened by the bottom layer and consequently not affecting charge carriers in the top layer, as previously indirectly observed [29, 30]. In addition, this observation provides evidence that the two graphene layers are thermally largely decoupled [31]. Assuming charge puddles are present in the bottom layer as they are in the SLG area, any temperature differential caused by the tip should cause a thermovoltage signal, which is not observed in our experiment.

Furthermore, we observe thermovoltage signals along straight lines (one example is highlighted by a pink dashed line in the bottom left panel of figure 3(a)) which are not visible in the height topography AFM maps (see SI). This signal could be attributed to local strain in the graphene layer or wrinkles, induced by the deposition of the contacts or the annealing step (see experimental section). Strain has been shown to alter the band structure [32] and Seebeck coefficient in graphene as well as displaying a Peltier effect [33]. Notably, at all gate voltages a strong thermovoltage signal is observed when heating up the interface between SLG and BLG as reported in scanning photocurrent measurements [10, 11]. The signal is changing from negative to positive as the gate voltage is increased. Both the wrinkle and the SLG/BLG junction signal follow a non-linear gate dependence.

The change in the sign at the SLG/BLG interface with changing majority carrier concentration (here both the SLG and BLG area of the sample is assumed to be p-doped (n-doped) for low (high) gate voltages respectively) implies that there is a change in the Seebeck coefficient difference $\Delta S = S_\mathrm{SLG}-S_\mathrm{BLG}$.

As discussed above, diffusion of carriers is driven by a gain in entropy, and carriers diffuse from regions of low DOS into regions of high DOS. Due to the parabolic dispersion relation in BLG compared to the linear one in SLG around the charge neutrality point, the DOS in BLG is higher than in SLG [34, 35]. This leads to a positive (negative) thermovoltage build-up for electron (hole) doped graphene when measuring the voltage at the drain contact with respect to the source [11]. This development is shown in figure 4(a) where line traces along the green dotted line in figure 3(a) through the junction at different gate voltages are displayed. Figure 4(b), top, shows close ups of the position dependent thermovoltage signal for a high p-doped ($V_\mathrm{g} = -10$ V) and n-doped ($V_\mathrm{g} = 25$ V) sample. For p-doped (n-doped) graphene, the signal dips (peaks) at the interface between SLG and BLG. Such a dip (peak) observed for the highly p-doped (n-doped) sample, respectively, is predicted by the model presented in figure 1 and can be attributed to a step-wise change in the Seebeck coefficient between the SLG and BLG areas.

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However, at gate voltages close to the charge neutrality point of the sample ($V_\mathrm{g} = 10/12.5$ V) the thermovoltage exhibits a bipolar signal not compatible with a Seebeck coefficient changing from $S_\mathrm{1}$ to $S_\mathrm{2}$. A similar bipolar signal at $V_\mathrm{g} = 10$ V is found around wrinkles as shown for an exemplary linecut in figure 4(c). Rather than indicating a transition from one Seebeck coefficient value to another, a bipolar signal implies a drop/peak of the Seebeck coefficient (see SI for simulations similar to the ones shown in figure 1). For the wrinkles, the sudden spatial change in the Seebeck coefficient can be explained by local strain, which is the origin of the fold or wrinkle formation in the first place. It has been predicted that local strain in graphene can alter its Seebeck coefficient with the magnitude of this effect being largest around the charge neutrality point [36]. As a result, wrinkles in graphene exhibit a bipolar and prominent signal around the CNP. At higher doping, the change in the Seebeck coefficient is not as large and only leads to a small ΔS. While this describes the signal at the wrinkles, the nature of the bipolar thermovoltage displayed at the SLG/BLG junction is less clear. The bipolar thermovoltage could be a combination of the signal from charge puddles and the junction or be attributed to local symmetry breaking and disorder at the edges of the BLG which locally influences the Seebeck coefficient.

As explained above, the magnitude of the local thermovoltage signal measured by STGM is given by a convolution of a temperature difference with a given spatial Seebeck coefficient distribution. Thus it is possible to extract the length scales the Seebeck coefficient is changing on associated with the different origins of the signals. We note that the spatial distribution in the thermovoltage signals is directly influenced by two factors: the shape and height of the temperature profile distribution created by the STGM tip and the local changes of the Seebeck coefficient. In order to determine the width of the temperature distribution created by the STGM tip, we fit the step in the thermovoltage signal at the graphene flake edge with a sigmoidal Boltzmann curve $f(x) = y_1+(y_1-y_2)/(1+e^{\frac{x-x_0}{\Delta x}})$ (see orange dashed line in the top left panel in figure 3(a) and (SI) [37]. This fit gives Δx ≈ 60nm, suggesting a lateral resolution of 2Δx = 120nm. From this fit we can also extract a width of the Gaussian temperature distribution of $l_\mathrm{dT} \approx 1$ µm (see SI).

By comparing $l_\mathrm{dT}$ to the length scales observed in the experiment for the thermovoltage signal, $l_\mathrm{V} \geq 3 \, \mu$m (see figure 4(b) and (c)), it becomes clear that the width of the signal can not be explained by limited spatial resolution. Since the thermovoltage signal measured is a convolution of S and dT, the signal width is approximately given by the sum of $l_\mathrm{dT}$ and $l_\mathrm{S}$ (the length scale associated with changes in the Seebeck coefficient), making it possible to extract $l_\mathrm{S}$. The signal width can then be used to quantify the band bending at the transition between SLG to BLG (similarly to the depletion region width in a semiconductor p-n junction that is determined by doping levels and applied voltage). Applying this to the line traces shown in figure 4(a) which show $l_\mathrm{V} \approx 3$ µm, we can extract the length scale of the Seebeck coefficient transition when moving from SLG to BLG, giving $l_\mathrm{S} \approx$ 2 µm for both a p-doped and an n-doped sample. We note that our results suggest that $l_\mathrm{S}$ depends on the carrier density: by lowering the carrier density we find that $l_\mathrm{S}$ decreases to $l_\mathrm{S} \approx$ 1 µm at $V_\mathrm{g} = 5$ V (see figure 4(a) and (SI). It is worth to mention that, although the length scales on which the Seebeck coefficient changes around the SLG/BLG junction and a wrinkle are similar, the underlying mechanism for this length scale is different (see figure 4(b) top and (c)). While $l_\mathrm{S}$ around the junction depends on charge carrier density, $l_\mathrm{S}$ for wrinkles in graphene is expected to depend more on the elastic properties of graphene. Thus, the above analysis suggests that even for an atomically sharp step between two 2D materials, the Seebeck coefficient changes gradually on a length scale of micrometers.

Following equation (1), the local thermovoltage is a convolution of the local variation of the Seebeck coefficient and the temperature gradient induced by the STGM tip. Therefore, it is possible to extract the local Seebeck coefficient variations from a measurement of the local thermovoltage by deconvolution. We test this idea by making the following assumptions: 1) we assume a Gaussian distribution of the temperature profile induced by the heated tip, with σ ≈ 100nm as extracted from the sigmoidal Boltzmann curve fit (see SI). 2) we assume a constant thermal spreading resistance and therefore a constant tip apex- temperature. The error making this assumption is low if only the graphene channel is investigated and contact areas are neglected. 3) the thermovoltage is measured between two sample contacts in the x-direction and therefore contributions in the y-direction can be neglected (here we rotate the thermovoltage maps before performing the deconvolution).

From equation (1) we know that $V_{\rm th}$ is the convolution of S and the position dependent derivative of the temperature profile caused by the hot tip $T^\prime(x_{\rm T})$, i.e. $V_{\rm th}(x_{\rm T}) = (S*T^\prime)(x_{\rm T})$. Using the temperature distribution $T(x_{\rm T})$, and thereby $T^\prime(x_{\rm T})$, we can then deconvolute equation (1) and obtain an estimate of the Seebeck coefficient distribution along the sample, S(x) (see SI). Using a regularized filter in combination with a constraint least-square restoration algorithm to perform the 1D deconvolution, a quasi two-dimensional temperature distribution is line-by-line deconvoluted with the thermovoltage map to produce the Seebeck map (see methods and SI).

As the tip is moved over the sample, the centre of its two-dimensional temperature distribution moves as well. Then, at every point of the thermovoltage map, a complete deconvolution of each line in the temperature distribution with the thermovoltage image gives a point in the deconvolution image.

The resulting deconvolution is shown in figure 5(a). First, we note that the signal on and around the contacts is not accurate as we assumed a constant excess temperature of the tip-sample contact (of $T_\mathrm{tip}\approx$ 53K) for the deconvolution. This is not given on the Au contacts where the tip-sample contact temperature drops substantially as discussed earlier. Therefore, the areas of the Au contacts, which effectively show the difference in Seebeck coefficients between gold and graphene [38], do not necessarily have the right magnitude (see discussion below). The graphene channel in between the contacts however, satisfies assumption 2 and displays the expected change in the Seebeck coefficient from positive to negative with increasing gate voltage [17]. In addition, large local variations of the magnitude and sign of the local Seebeck coefficient are observed around the Dirac point (see figure 5). These give rise to the thermovoltage signal shown in figure 2(a).

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In order to evaluate the deconvolution method presented in this work, we can compare its result to the Seebeck coefficient of the graphene channel obtained via two additional methods: 1) using the gate dependent conductance data of the device and the Mott formula to estimate $S(V_g)$ [17] and 2) estimating the temperature drop ΔT over the whole sample when the STGM tip heats up one electrical contact, and use $S = -V_{\rm th}/\Delta T$. For case 1, the Mott formula is given by

$$\begin{equation} S = -\frac{\pi^2k_B^2\mathrm{T}}{3 |e|}\frac{1}{G}\frac{dG}{dV_g}\frac{dV_g}{dE}\big|_{E = E_F} ~, \end{equation} \tag{ 2 }$$

where e is the electron charge, G the electrical conductance, $k_\mathrm{B}$ the Boltzman constant and T the operating temperature (see SI). The gate dependent Seebeck coefficient of the rectangular sample can then be calculated using the gatetrace from figure 2(c) with the results of this calculation shown in figure 5(b). The calculated Seebeck coefficient overall first increases, changes sign when crossing the charge neutrality point, further decreases, and subsequently increases towards zero again. We observe maximum and minimum values of the Seebeck coefficient around S = 30-40 µV/K and S = −40 to −50 µV/K, respectively.

Then, for method 2), we divide the average thermovoltage value on the contacts in figure 2(a) by the assumed temperature difference between the contacts (≈ 1.5K as discussed) giving S = 5.92 ± 3.2 µV/K and S = -4.19 ± 1.49 µV/K for the Seebeck coefficient at low and high gate voltage respectively. The distribution of values of $V_{\rm th}$ on the contacts is plotted in a bar graph and fitted with a multi peaked Gaussian to determine the most likely event with the error then given by the width of the Gaussian (see SI).

Lastly, the Seebeck coefficient obtained via the deconvolution is extracted from the graphene channel in the maps of the spatial distribution of the Seebeck coefficient in figure 5(a). Again, the values of S in the channel are found by fitting the distribution with a multi peaked Gaussian to determine the most likely event with the error then given by the width of the Gaussian (see SI). This gives S = 44.94 ± 5.43 µV/K and S = -42.91 ± 9.4 µV/K for the p-doped and n-doped channel respectively. The result of the three different methods to obtain the Seebeck coefficient in the sample are shown in figure 5(b). The values found for all methods are qualitatively on the same order of magnitude, considering the assumptions necessary and are in accordance with previously reported global thermovoltage measurements in graphene. In addition, all methods find the same trend of a reversal in sign when changing majority carriers as expected for graphene [17]. Method 2), that is the Seebeck coefficient estimated via the thermovoltage drop, quantitatively differs most likely due to the large error in determining the temperature difference across the device from heating up one contact. While this makes it difficult to quantify the Seebeck coefficient precisely, this is a well-known obstacle in every thermoelectric measurement and all estimates for the Seebeck coefficient presented here are consistent.

The deconvolution can also be applied to the more complex thermovoltage maps of the SLG/BLG sample (figure 5(c)). Here, artefacts introduced by the changing tip-sample thermal resistance are clearly visible in the contact area. However, trends predicted by theory like the change in the Seebeck coefficient of the SLG area $S_\mathrm{SLG}$ with respect to the BLG area $S_\mathrm{BLG}$ are visible, with $S_\mathrm{SLG} \lt S_\mathrm{BLG}$ at $V_\mathrm{g} = -10$ V and $S_\mathrm{SLG} > S_\mathrm{BLG}$ at $V_\mathrm{g} = 25$ V [11]. In addition, again, a high local variation of $S_\mathrm{SLG}$ with changing polarity can be seen around the charge neutrality point, caused by charge puddles. While the calculated Seebeck coefficients around 500 µV/K may seem high for a graphene channel, we note that these are local variations due to local effects rather than representations of the global Seebeck coefficient.