Space-charge limited and ultrafast dynamics in graphene-based nano-gaps

Graphene is considered to be one of the key materials in future ultrafast optoelectronic circuits and applications,^1–3 particularly because of its rather low electrical resistivity and its high optical-to-electrical conversion efficiency due to an ultrafast hot charge carrier multiplication.^1,2,4 Recent work by Yoshioka et al. showed that this fast response can be utilized to transduce optical signals into electronic circuits with a bandwidth of up to 200 GHz.⁵ Certain applications, such as ultrafast photoswitches in on-chip THz-circuits, however, require devices with a high resistivity to allow amplification schemes with a high gain. Over the last decades, defect-rich semiconducting materials, such as oxygen-implanted silicon and low temperature grown gallium arsenide (LT-GaAs), have been identified to be most suitable for this application.^6–9 The corresponding photoconductive switches can be operated by ultrafast near-infrared pulses, which excite charge carriers into both the defect states and the conduction band of the semiconductors. Typically, the overall carrier lifetime of the defect-rich materials limits the achievable switching time of such photoconductive switches.¹⁰ The reported minimal switching time has been stagnating at slightly below 0.5 ps since the late 1980s.^7,9,11 A promising way to overcome the carrier-lifetime-limitation in photoconductive switches is to exploit femtosecond photoemission processes in nanoscale gaps,¹² where electrons are ejected out of a solid by the impact of an ultrafast laser pulse.^13–15 In metal structures, however, the necessary strong optical pulses can limit the device lifetime due to laser-ablation processes.¹⁶ In graphene, the ablation is reduced because of the very good mechanical and thermal stability of the material,^5,14,17–19 but nano-gaps filled with a dielectric material still seem to be the geometry of choice to allow an ultrafast and stable operation of the photoswitches.

In this work, we discuss resistive nano-gaps formed in graphene by utilizing the focused beam of a helium ion microscope (HIM). The graphene layers are split into two sections by the direct impact of the helium ions as well as the ion-induced swelling of the underlying sapphire at the positions of the ion-impact.^20–22 The resulting two-terminal devices exhibit a resistivity exceeding 2 kΩ m compared to 0.1 Ω m for the pristine graphene. We demonstrate that the devices can be operated as photoswitches with a sub- to picosecond temporal optoelectronic response. This fast response occurs at all investigated temperatures, and it can be explained by the dynamics of hot electrons in the two graphene-sections.^2,5,19,23 We observe an additional space-charge limited current^24,25 contributing to the optoelectronic response with a timescale on the order of tens of picoseconds. The process arises from the HIM-defined nano-gaps, and it does not occur in pristine graphene.²³ The optoelectronic response, which is stable over several months, suggests a potential of such resistive graphene nano-gap devices in photoconductive switches and ultrafast circuitries. In turn, our results extend the library of space-charge limited nanodiodes in two-dimensional materials toward ultrafast photocurrent switches.^24,26–28

Figure 1(a) shows the HIM image of a graphene sheet on sapphire. The graphene was produced by chemical vapor deposition (CVD) in a horizontal hot-wall furnace using a Cu foil as a catalyst and methane as a carbon source. The detailed CVD process employed to achieve single-layer graphene with a high structural quality can be found here.²⁹ The transferred graphene is electrically contacted by two Ti/Au-pads (2/25 nm), which act as the electronic source and drain contacts. In the center of the two-terminal device, the graphene is split into two parts by a line-exposure of the HIM with a fluence of helium ions in the range of 60–100 ions/ $Å$².³⁰ For such fluences, substrate-swelling is expected in sapphire, which originates from the formation of helium bubbles below the lithographic line.^20–22 Indeed, a line cut of an atomic force microscope (AFM) image confirms that the nano-gap has a height of 42 nm and a width of 84 nm [Fig. 1(b)]. Particularly, the width exceeds the focus of the HIM at least by one to two orders of magnitude.³¹ Moreover, since the ion-induced swelling is expected to start at fluences larger than 0.2 ions/ $Å$²,³⁰ we infer that the nano-gap is filled by a bulge of defect-rich sapphire. For comparison, the HIM image of Fig. 1(a) is taken at an average fluence of ∼0.008 ions/ $Å$².

Measuring the current–voltage characteristics of the devices before and after the HIM patterning, we see a transition from a linear behavior with a resistance of ∼10 kΩ before (not shown) to a highly non-Ohmic behavior with a differential resistance larger than 200 MΩ at zero bias; cf. Fig. 1(c) with data for a bath temperature of T_bath = 80 K. At this low temperature, the overall characteristics can be well described by a power law of V_bias^ζ, with ζ = (5.8 ± 0.9) [red line in Fig. 1(c)], and V_bias the applied bias voltage. The large bias-exponent points toward the dominance of a space-charge limited current in the so-called trap-filled limit for the current across the nano-gaps without illumination.²⁴ The overall temperature dependence of the current–voltage characteristics is discussed below.

For the photocurrent studies, we use a pulsed laser at a center photon energy of E_photon = 1.1 eV with a pulse duration of ∼14 fs and a repetition rate of 80 MHz, as in Ref. 12. The light is focused onto the sample with a Gaussian laser spot of ∼3 μm. With a maximum intensity of 24 mW at the position of the graphene, the peak vacuum field can be as high as ∼1 V/nm. In principle, the latter value is sufficient to observe electronic photoemission processes across the nano-gaps.^16, Figure 1(d) shows a spatial map of the time-integrated photocurrent I_photo for the circuit as depicted in Fig. 1(a) measured again at T_bath = 80 K. The maximum photocurrent occurs at the position of the nano-gap, which demonstrates that the optoelectronic response at the 84 nm-wide nano-gap is larger than the one at the position of the graphene and at the graphene–metal interfaces.³² Analyzing I_photo vs the applied bias V_bias at the position of the nano-gaps, we find a quadratic relation for all measured laser intensities at this temperature [I_photo ∼ V_bias^ξ, with ξ = 2, Fig. 1(e)]. The quadratic bias-exponent suggests that a space-charge limited current is also responsible for I_photo, however, in the so-called trap-free limit.²⁴ In this interpretation, the laser excitation gives rise to extra charge carriers, which fill the traps within the nano-gap. Such a photoexcited filling explains the difference of the bias exponents for I_dark and I_photo.²⁴ 

To further explore the optoelectronic dynamics, we plot the time-integrated photocurrent intensity I_photo vs the laser power P_laser measured at the position of the nano-gap. We observe that I_photo ∼ P_laser^γ, with γ = (0.73 ± 0.01) [Fig. 1(f)]. The measured power-exponent is consistent with the theoretically expected value of 3/4 for a space-charge limited photocurrent.³³ However, we note that the experimentally determined exponent is temperature-dependent, as will be discussed below. Generally, the observed exponent γ < 1 rules out multiphoton-absorption processes, which would exhibit an exponent larger than one.¹⁶ Moreover, we observe that the laser-polarization has only a minor impact on the photocurrent across the nano-gaps, as is exemplarily indicated by presenting data for an horizontal and vertical polarization in Fig. 1(f). This observation further supports the interpretation that a photoemission process only plays a minor role to describe the time-integrated photocurrent across the presented nano-gaps.¹⁴ 

In a next step, we explore the underlying optoelectronic time-scales within the graphene-based nano-gaps. To this end, the nano-gaps are integrated into coplanar stripline circuits as photoswitches [inset of Fig. 2(a)]. At a distance of approximately 300 μm to them, an additional oxygen-implanted silicon-based photoswitch is placed (named Auston-switch in the inset).⁶ After the excitation with the 14-fs pump pulse, the ultrafast photocurrent components across the nano-gaps give rise to a short electric field pulse in the coplanar striplines, which travels along them as a THz-transient with the dispersion and absorption given by the effective dielectric function of the coplanar striplines.^12,34 When the THz-transient passes the Auston-switch, we hit the latter with a time-delayed probe laser pulse (E_photon = 1.59 eV, pulse duration ∼100 fs). In turn, we detect a sampling current I_sampling across the Auston-switch, which linearly depends on the field strength of the THz-transient originating from the optoelectronic response within the nano-gaps. In this manner, we can reveal the photocurrent across the graphene-based nano-gaps with a minimal time-resolution of ∼0.5 ps given by the overall carrier lifetime in the silicon-based Auston-switch.⁹ 

The main graph of Fig. 2(a) shows I_sampling for a graphene-based nano-gap measured at T_bath = 80 K for V_bias = +6 and −6 V. In both cases, we can distinguish a fast signal followed by a slow one, the latter of which persists up to several picoseconds. The fast signal exhibits a timescale of τ_fast = (1.1 ± 0.1) ps. Hereby, it is not limited by the time-resolution of the silicon-based photoswitch nor the utilized probe laser. Moreover, it shows no dependence on V_bias, but it increases with laser intensity in a monotonous to linear fashion [Fig. 2(b) and supplementary material]. We explain this fast decay by a displacement current, i.e., hot charge carriers displaced according to the local potential landscape.³² In principle, the underlying timescale of hot-charge carriers can be as fast as femtoseconds in pristine graphene.² In the current case, however, the measured timescale is consistent with the bandwidth limitation of the utilized stripline circuit.^23,34

In contrast to the fast signal, the slow signal switches sign with the polarity of V_bias [Fig. 2(a)], and it exhibits an exponential decay with a timescale of τ_slow = (26 ± 12) ps. To demonstrate the bias-dependence, we numerically integrate the signal within the colored area of Fig. 2(a) for different V_bias. Figure 2(c) reveals a quadratic relationship of this slow signal vs V_bias. In turn, the slow decay of the time-resolved I_sampling is also consistent with the aforementioned space-charge limited current in the trap-free limit,²⁴ as is the time-integrated signal I_photo at the position of the nano-gaps [cf. Fig. 1(e)].

Generally, there are several physical processes, which can give rise to non-linear current–voltage characteristics in the case of both a laser irradiation and without. The most relevant ones for this work are the class of space-charge limited²⁴ currents as bulk conduction processes within the bulge. The space-charge limited conduction of charge carriers, e.g., in an insulator, is described by the so-called Mott–Gurney (MG) law, and it is the solid-state counterpart to the Child–Langmuir law, which is valid in vacuum. In the presence of one shallow trap level, the corresponding MG-current shows a quadratic bias-dependence.²⁴ This model can be extended by considering an exponential distribution of shallow traps with a characteristic trap-depth of E_trap. The corresponding Mark–Helfrich (MH) law suggests a temperature-dependent bias-exponent of ζ^MH = 1 + E_trap/(k_BT_bath), with I α V_bias^ζMH, below a critical temperature T_c = E_trap/k_B and a quadratic dependence above, with k_B the Boltzmann constant.²⁴ When the current–voltage curves are interpolated to higher biases for varying T_bath < T_c, the curves are supposed to intersect at a crossover voltage V_c, from which one can estimate the defect density n_traps as n_traps = 2·V_Cε₀ε_r/(e·d²), with e the electron charge, ε_r (ε₀) the relative (vacuum) permittivity of the material, and d the distance of charge transport.³⁵ 

We observe that the dark current across the nano-gaps is consistent with the MH-model. Namely, we measure steeper non-linear current–voltage characteristics for lower bath temperatures, i.e., I_dark α V_bias^ζ with ζ ∼ 6 for 80 K and ζ ∼ 3 for 310 K [Fig. 3(a)]. The red line in Fig. 3(a) depicts a fit according to the MH-model as ζ = 1 + E_trap/(k_BT_bath), which suggests a characteristic energy of the exponentially distributed trap states of E_trap = (40 ± 6) meV and a corresponding critical temperature of T_c = (465 ± 70) K. Interpolating the IV curves to higher V_bias for each T_bath, we, indeed, observe a crossover voltage [vertical line in Fig. 3(b)], which turns out to be V_C = 13.2 V. The value translates into an effective defect density of n_traps = (2.1 ± 0.2) × 10¹⁸ cm⁻³ (for d = 84 nm the AFM-width of the nano-gaps and ε_r = 9–11 as for sapphire),³⁶ which is consistent with the utilized high fluences of the helium beam to write the nano-gaps.

In the case of the time-integrated photocurrent I_photo, we find a power-exponent of γ = 0.73 ± 0.01 at 80 K close to the theoretically expected value of 3/4 for a space-charge limited photocurrent (i.e., I_photo ∝ P_laser^γ).³³ However, the exponent γ increases with the bath temperature [Fig. 3(c)]. Fitting the temperature-dependence of γ linearly (red line), the power-exponent γ reaches a virtual value of ∼2/3 at zero temperature, which suggests underlying defect-dominated dynamics as is consistent with a defect-rich bulge within the nano-gap.²⁵ Moreover, the exponent γ reaches a value close to unity at a virtual temperature of about 450 K. The latter value is surprisingly consistent with the already mentioned critical temperature T_c = (465 ± 70) K derived from the IV-characteristics without illumination. We interpret the consistency by a transition from a space-charge limited current in the trap-filled limit for the dark current to an effectively trap-free scenario for the photocurrent. In other words, the illumination effectively fills the traps of the nano-gaps, which tentatively explains the deduced photocurrent being linear in laser power at 450 K.²⁴ We note, however, that the dynamic filling depends on the relaxation and thermalization processes of the optically excited charge carriers as well as on the specific distribution of the trap levels in energy and space within the nano-gap, and it is not sufficiently described by the MG- and MH-models, which assume homogeneous bulk materials. The short-coming of both models can be seen by the bias-exponent ξ of the photocurrent (i.e., I_photo ∝ V_bias^ξ), which reaches values below 2 for increasing T_bath [Fig. 3(d)], which is neither described by the MG- nor MH-models.²⁴ In a naïve way, ξ seems to decrease to a linear bias dependence at high temperature, which we expect for an effectively trap-free, photon-assisted transport of charge carriers. Future theory studies on optically excited charge carriers in a space-charge limited setting might be able to describe this transition in more detail. At presence, we are able to give a timescale of the underlying photocurrent dynamics, since the slower contribution of the time-resolved photocurrent dynamics shows a consistent bias-exponent ξ [e.g., Fig. 2(c)]. The corresponding time constant τ_slow = (26 ± 12) ps exceeds the expected transit time of charge carriers across the nano-gaps, which is on the order of femtoseconds for $v Fermi graphene$ ∼ 10⁶ ms⁻¹ and d = 84 nm. It also surpasses the femtosecond to sub-picosecond electron–electron and electron–phonon scattering times in graphene.^2,5,23 Instead, the timescale is consistent with a dominating cooling pathway of the optically excited electron- and then lattice-temperature to the sapphire substrate. Namely, assuming that the time constant is proportional to the ratio of the interfacial thermal conductance G of graphene vs the underlying substrate to its heat capacitance C = 1.5 × 10⁻¹⁵ J K⁻¹μm⁻²,³⁷ we obtain an interfacial thermal conductance on the order of G = (5.8 ± 2.4) × 10³ W/(K cm²). The value is consistent with similar graphene-sapphire devices without a nano-gap.³⁷ Therefore, it is very likely that the laser heats the nano-gaps as well as the adjacent graphene sections to temperatures significantly above T_bath such that an effectively trap-free transport describes the slow photocurrent response, while the relaxation of the hot carriers gives rise to the fast response in form of an ultrafast displacement current. We finally note that the slow timescale τ_slow might also point toward an additional charge depletion process, as soon as the laser is switched-off. However, our methodologies cannot distinguish between both processes. In any case, without illumination, the traps seem to dominate the non-linear current–voltage characteristics.

In conclusion, we show that it is possible to generate resistive nano-gaps in graphene-based two-terminal circuits by utilizing a focused helium ion beam. We demonstrate that the devices show space-charge limited current–voltage characteristics, which can be well described by the Mark–Helfrich law. In the case of photon irradiation, we observe an ultrafast response of the nano-gaps on the order of sub- to picosecond and a slower optoelectronic response, which is consistent with the space-charge limited scenario in combination with a photon-assisted trap filling.