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Electrical gating and rectification in graphene three-terminal junctions
Applied Surface Science 291 (2014) 87–92
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Electrical gating and rectification in graphene three-terminal junctions B. Händel a,b , B. Hähnlein a , R. Göckeritz c , F. Schwierz b , J. Pezoldt a,∗ a FG Nanotechnologie, Institut für Mikro- und Nanotechnologien MacroNano® and Institut für Mikro- und Nanoelektronik, Postfach 100565, 98684 Ilmenau, Germany b FG Festkörperelektronik, Institut für Mikro- und Nanotechnologien MacroNano® and Institut für Mikro- und Nanoelektronik, Technische Universität Ilmenau, Postfach 100565, 98684 Ilmenau, Germany c FG Nanostrukturierte Materialien, Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, Von-Danckelmann-Platz 3, 06120 Halle (Saale), Germany
a r t i c l e
i n f o
Article history: Received 24 May 2013 Received in revised form 11 September 2013 Accepted 11 September 2013 Available online 20 September 2013
a b s t r a c t Graphene was grown on semiinsulating silicon carbide at 1800 ◦ C and atmospheric argon pressure. The all carbon T- and Y-shape three terminal junction devices were fabricated using electron beam lithography. All devices featured the negative rectification effect. The exact properties of the devices like the curvature of the output voltage response can be tuned by changing the branch width in the T- and Y-shape devices. Beside the rectification a switching behavior is demonstrated with the same three terminal junctions. © 2013 Elsevier B.V. All rights reserved.
Keywords: Three terminal junction Rectification Transistor Switch Graphene Silicon carbide
1. Introduction Graphene is a fascinating new material, whose properties are very promising for electronic applications [1–4]. Especially the mobility [5,6], the electrostatic control of the carrier type and their concentration [7] and the aggressive scalability of graphene devices [8] offer advantages for deep scaled and high frequency devices. Nevertheless, the unique band structure of large area graphene with no band gap and linear dispersion around the Dirac point [9,10] ask for new concepts to explore graphene’s unique capabilities in electronics [11–20]. One promising concept in this respect is the three-terminal junction devices (TTJ) [21]. These T- or Yshaped devices of sub-micrometer dimensions capitalize on certain characteristics of the nonlinear response regime of charge carrier transport appearing in two dimensional charge carrier gases confined laterally in channels [22,23]. In this devices geometrically [24,25] or electrically [26] controlled charge transport as well as a rectification [22,27] can be used for information processing and was demonstrated on III–V heterostructures. Recently, the rectification effect was also demonstrated on graphene [28–33]. As a first
∗ Corresponding author. Tel.: +49 3677 693412; fax: +49 3677 693356. E-mail address: [email protected] (J. Pezoldt). 0169-4332/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.apsusc.2013.09.066
application the frequency doubling was realized using graphene TTJ devices [30]. In contrast to III–V heterostructures where only on carrier type is available, the graphene TTJ allows the electrical tuning of the rectification effect from negative to positive rectification in dependence on the carrier type in the device, negative rectification in case of electron conduction and positive rectification if the holes are responsible for the charge transport [28,32]. Many attempts have been made to explain this effect based on the propagation of electromagnetic waves [34], ballistic [35,36] or diffuse–ballistic [37] transport. Here we show a dependence of the rectifying effect on the geometry of graphene TTJ which supports the theory set forth by Sadi et al. [37]. Yet actually the original idea behind the TTJ design was to obtain a new electronic switch [24] or an in-plane gated field effect transistor [38]. With TTJs made of carbon nanotubes a switching behavior [39] and differential current amplification [40] was actually observed. Here we report the observation of such behavior in TTJs made of graphene. 2. Experimental Epitaxial graphene was grown on Si-face semiinsulating silicon carbide. Before the growth process the SiC underwent a direct capping procedure during which it was annealed with a graphite cover
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at 1800 ◦ C for 3 min. Thereby a stepped surface morphology was obtained [41]. Consecutively graphitization was realized by heating the SiC at a rate of 10 K/min up to 1800 ◦ C in a graphite furnace in an argon atmosphere at normal pressure. Raman spectroscopy revealed the formation of 1–4 layers of graphene sheets with sizes up to several micrometers on the SiC surfaces [41]. Hall measurements of the grown graphene layers were carried out with an Accent Hall measurement system in van der Pauw geometry. A Hall mobility of 1000 cm2 /V/s and a sheet carrier concentration ns between 3 and 5 × 1012 cm−2 was obtained. Using a semiclassical model the electron mean free path le can be calculated as le = (h/2e) (ns /)1/2 , where h is the Planck constant and e is the elementary charge [42]. The estimated mean free path of the electrons is between 20 and 26 nm. These graphene sheets were structured into T- and Y-shaped structures with graphene contact pads at the end of each terminal with an electron beam lithography system Raith 150 using negative tone electron beam resist HSQ. Next the graphene was etched with oxygen plasma in an electron cyclotron resonance plasma etching apparatus. Afterwards the HSQ was removed by 10 min exposure to HF-vapor. The graphene contact pads, included in the design, had a size of 90 × 90 m and were situated at the end of each of the three terminals giving the ability for electric measurements. This solution is minimizing problems of contact resistance due to the metal graphene interface. Graphite tips were lowered onto these contact pads to allow for electronic measurements with a Keithley SCS 4200. The pads were framed by gold frames to ease finding them during measurement. The electrical measurements were conducted in two different modes. The rectifying behavior of the three terminal junctions was probed with so called push–pull-measurements. All three branches are contacted. A voltage VL is applied to one branch and another voltage VR to another. At the third branch the resulting voltage VC was measured with a high impedance voltmeter. Push–pull hereby implies that VL = −VR while VL was varied from 0 to 4 V. This setup is shown in Fig. 1. The switching ability of the TTJ was tested by connecting one branch to mass. At the other a voltage was varied from 0 to 8 V each for a different voltage applied to the third branch. In essence this third branch was hereby considered a gate and the other two branches of the channel as source and gate.
Fig. 2. The output voltage at the centre terminal VC in dependence on the push–pullvoltage for a T-shape device with a horizontal bar and a vertical bar width of 15 nm for both bars. The experimental output voltage is shown together with the fitted curve. The intersection of the dotted line gives the onset voltage Von .
3. Geometrical dependence of rectifying behavior A typical device used is shown in Fig. 1. It consists of a T-shape conductor with three ohmic contacts at the end of each branch. The fabricated T-shape devices have horizontal terminals bar widths H of 15 to 150 nm and a length from one terminal end to the other L of approximately 400 nm. The vertical terminal bar width W varied between 13 and 30 nm with respect to the horizontal terminal bar width. L1 is the length of the left terminal; L2 is the length of the right terminal and W is the width of the vertical terminal. The overall length of the horizontal bar L is L = L1 + L2 + W. The Y-shape devices had a symmetrical structure with an angel between the bars of 120◦ . The bar width varied between 15 and 100 nm and was equal in width for all bars within the limitations of electron beam lithography. Push–pull-measurements with the circuit setup shown in Fig. 1 of the T- and Y-shaped TTJ structures revealed the characteristic dependence of the electrical response VC at the central terminal in dependence on the applied push–pull voltage VPP . Even if the third branch is situated exactly half way from the other two branches, the observed potential is negative and not zero as expected from classical transport theory. This behavior has been observed and identified as rectification regardless of whether the horizontal channel is in the diffusive or linear-ballistic regime [22]. A typical dependence of the output voltage VC at the central branch is shown in Fig. 2 for a T-shape TTJ structure. The voltage response at the central contact consists of two parts. At low push–pull voltages VPP the voltage response follows the equation [22]: 2 VC = kVPP + O(V 4 ),
Fig. 1. A typical T-shape TTJ structure made of graphene is shown. The three branches end in larger graphene pads whose beginning can be discerned in this image. These endings are called terminal 1, 2 and 3. The setup of the measurement is sketched around the image of the T-structure. The small inset defines the definition of the device geometry.
(1)
where k is a fitting parameter representing the curvature of the voltage response at VPP = 0 V. The constant k is negative if the two dimensional electron gas consists of electrons and positive if the charge carriers are holes. This was recently demonstrated for grapheme TTJs [29–33]. For all fabricated structures regardless of the shape and the geometrical dimensions a down bending of the voltage response VC was observed which indicates electron transport. Different attempts have been made to explain the occurrence of this phenomenon [22,37,43]. The most plausible seems to be the one from Sadi et al. as, contrary to others it is not solely based on the assumption of ballistic transport alone [37]. Sadi et al. argue that the electrons in the three terminal junctions undergo a “quasi” ballistic transport. “Quasi” ballistic transport is hereby understood as instances where scattering on boundaries tends toward dominating
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scattering on defects and phonons. A mean free path of electrons le in our structures of 20 to 26 nm excludes the possibility of the rectifying behavior being exclusively a ballistic transport phenomenon in our devices. Therefore the electrons undergo around 20 instances of scattering when traveling through the device. Thus a large number of the electrons transgress the actual channel dimensions unlike a traditional conductor on micrometer or larger scale, wherein the majority of electrons never reaches the boundaries of the same. Keeping this in mind we can for one part think of the electron movement as a billiard ball like movement, bouncing on and of the channel edges on their transgression of the device. At the intersections of all three terminals some electrons will be scattered into the third branch on their movement from the end of branch one to the end of branch two. These electrons are the first contribution to the negative potential measured at the end of the central terminal. Next, slowdown of the electrons is mainly caused by scattering on defects at the boundary of the device channels. At the intersection these scattering instances are halved due to only a boundary on one side of the channel. Twice as many scattering instances on boundaries occur over the remaining length of the branches, as they feature boundaries on both sides. Consequently electrons will be accelerated more over the length of the intersection of the three branches then on the remaining length of the branches. This leads to the asymmetry in the distribution of velocity of the electrons across the channel length from the end of branch 1 to that of branch 2 as was observed by other groups [36]. The asymmetry of electron speed causes an asymmetry of the electron concentration along the channel length. This result in the second contribution to the negative potential observed at the end of the third branch. Finally, it has to be mentioned that recently, it was shown experimentally that the nonlinear response of the TTJ might not be limited to ballistic or diffuse-ballistic transport [44]. The second part of the voltage response is observed for high push–pull voltages VPP . In this region the voltage response can be approximated by the following formula [45]:
VC = − VPP + Von ,
(2)
where Von is the extrapolation of the linear dependence of VC to VC = 0 V. The appearance of the linear dependence at high voltages was explained for III–V materials by intervalley transfer [43,45,46]. As in the case of TTJs made of III–V heterostructures the values of the onset voltage of the graphene devices are too large to be explained only by the poor ballistics theory [45]. The onset voltages are determined for the graphene devices are between 0.5 and 1 V. The measured voltage response curve is non-symmetric. According to [22], the origins of asymmetry of the output voltage are asymmetric geometrical properties of the TTJ structures. It is caused by geometrical fluctuations stemming from inhomogeneities in the device geometry due to electron beam lithography effects and graphene thickness fluctuations. They are visible in Fig. 1, where different shades of gray, indicative for fluctuating graphene thickness deviations, as well as width fluctuations of the bar geometry and nonidentical left and the right contact areas near the active part of the TTJ structure can be noticed. Fig. 3 displays the dependence of the curvature k on the horizontal bar width H at constant length of the horizontal branch of the T-shape TTJ. The dependence of the curvature shows a minimum at approximately 40 nm. The absolute value of the curvature is a function of the relation between the diffusive and ballistic transport components of the charge carrier and their interplay with the TTJ device geometry. The strength of the rectification, i.e. the absolute value of the curvature, should increase with the intersection length to the remaining channel length between terminal 1 and terminal 2 due to nonzero vertical velocity component of the charge
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Fig. 3. Curvature dependence of T-shape TTJ devices as a function of the horizontal branch width H at constant length of the horizontal branch of 400 nm.
carriers. Therefore, if the horizontal bar width H is decreasing below the mean free path le of the charge carriers, here electrons, the scattering on the graphene stripe edges becomes dominant and the mean velocity and their component decreases leading to a lower scattering into the vertical branch and to lower absolute curvature values k. Additionally, the increased edge scattering equilibrates the velocity distribution along the horizontal bar reducing the second source for the negative curvature. For horizontal branch width H > le the curvature decreases again in agreement with the observation of T-shape TTJ devices fabricated on III–V materials [47]. So, it is believed that the origin of the observed dependence of the curvature k on the horizontal branch width H has similar origins. The dependence of the absolute curvature k on the bar width H of Y-shape TTJ, structures not shown here, does not show an extremal behavior within the range of the width of the fabricated structures and decrease with increasing branch width. For Y-shaped structures the electrons penetrate more in the central branch due to a stronger field enhanced vertical charge carrier component. This reduces the effect of the shrinking branch width and led to a weaker dependence of the curvature k on the branch width [48,49]. The effect of the vertical branch width W on the shape of the T-shape TTJ devices response is given in Fig. 4. For horizontal bar width H below the mean free path of the electrons le the absolute value of second derivative of the bell shape curve increases with increasing vertical bar width W, whereas for horizontal bar width H exceeding the mean free path of the electrons le the absolute value decreases with increasing the width of the vertical bar W. Again the observed behavior for the second case is equivalent
Fig. 4. Influence of the width of the vertical branch W on the curvature k of the voltage response curve for two different horizontal branch width H of T-shape TTJ devices.
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Fig. 5. Output characteristic of a Y-shaped TTJ device. The device is shown in the inset of the graph.
Fig. 6. Current gain in dependence on the drain-source UDS and the gate-source UGS voltage.
to the observations made in T-shape TTJ devices in III–V materials [48]. The other case W < le might have a similar explanation as given for the dependence of the curvature k on the horizontal branch width H if H is below the mean free path of the charge carriers le . It is worth to be mentioned here that the given interpretation of the geometrical dependent properties of the fabricated TTJ structures include the influence of surface charges.
electrons will flow through the gate. The following equation, where UGS is the gate voltage, ϕXG the local potential of the drain-source channel where it intersects the gate, xchannel-length the source-drain channel length and xG the position of the gate is, describes this condition:
4. Electrical gating in T- and Y-shaped junctions For the investigation of the switching ability of the fabricated Tand Y-shaped devices in the measurements terminal 1 was considered as source and terminal 2 as drain. Between these two a source-drain voltage was varied from 0 to 8 V. Source-drain and gate current were measured simultaneously. This procedure was repeated for different gate voltages from −4 to 4 V applied to terminal 3. The two decisive indicators for electrical gating are the ratio of source-drain current to gate current, the current gain, and the strength of control of the source-drain current by variation of the gate voltage, which manifests in the value of transconductance. Fig. 5 shows the source-drain current over the source-drain voltage for different gate voltages. If source-drain voltage is kept constant and only the gate voltage is changed, which represents in the graph a change along a vertical line, the source drain-voltage changes. At a drain source voltage of 3 V a maximum transconductance of 440 S/m was obtained for T-shape devices. For the fabricated Y-shape devices the maximum transconductance reached 300 S/m. For certain source-drain and gate voltages a current gain of up to 510 was obtained from the measured output characteristics of the T- and Y-shaped devices. The transconductance of the first 100 GHz graphene MOSFET demonstrated by IBM was 150 S/m [50]. The record transconductance achieved so far on epitaxial graphene MOSFET devices is 2000 S/m [51]. Typical values of the transconductance of graphene field effect transistors independent on the used graphene synthesis method are in the range between 100 and 600 S/m [52–55]. So, the obtained transconductance values of the Tand Y-shaped TTJ devices are comparable to standard graphene transistors. A current gain in these metallic structures is rather surprising considering that there is no gate insulator to limit the gate current. The key to understanding this effect is the source-drain and gate voltages at which the maximum current gain occurs. The local potential drops linear over the length of the channel between terminal 1 and 2. If the applied gate voltage is equal to this potential, no
UGS = ϕXG =
UDS xG , xchannel−length
(3)
Therefore, the T- or Y-structures behave like an electron valve gated by the voltage difference between terminal 1 and terminal 3. If the above condition (3) is fulfilled the valve is closed and the electrons only flow from source (terminal 1) to drain (terminal 2). In any other case the valve is opened and a gate current larger zero exists. Fig. 6 shows in the x–y-plane the source-drain current over the source-drain voltage. The z-axis is ascribed to the ratio of drain current ID over gate current IG . Peaks can be seen at the positions where the above condition is fulfilled. The peaks have a negative and positive part as the gate current changes direction as maximum current gain occurs when the same alternates its flow direction. The varying height of the maxima is due to gate voltage having been varied in 1 V steps. A finer tuning of the same should reveal maxima of equal height. The observed gating in T- and Y-shaped junctions is surprising, as there is no gap between gate and the channel between source-drain, as for example in side-gate field effect transistors [56–58]. Furthermore, the observed output characteristic shows not the classical transistor behavior. Nevertheless, the detected gating effect was also observed in Y-shaped junctions made of carbon nanotubes [39], but the observed current gain in the graphene based structures is considerably higher, than the current gain of 10 obtained for carbon nanotubes [39]. 5. Conclusion Dependence between the geometry of the TTJs and the rectifying behavior was observed. This supports the explanation of the rectifying behavior brought forth by Sadi et al. which suggests the presence of a “quasi” ballistic transport of the electrons in the TTJ. These insights about the rectifying behavior enable one to design TTJs more effectively. Next a switching behavior in TTJs made of graphene was demonstrated. The transconductance is much higher than the one obtained in other carbon nanotube TTJs exhibiting switching behavior. The observed current gain was explained by observing the voltages it occurs at. This may open the path to new electronic
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Electrical gating and rectification in graphene three-terminal junctions B. Händel a,b , B. Hähnlein a , R. Göckeritz c , F. Schwierz b , J. Pezoldt a,∗ a FG Nanotechnologie, Institut für Mikro- und Nanotechnologien MacroNano® and Institut für Mikro- und Nanoelektronik, Postfach 100565, 98684 Ilmenau, Germany b FG Festkörperelektronik, Institut für Mikro- und Nanotechnologien MacroNano® and Institut für Mikro- und Nanoelektronik, Technische Universität Ilmenau, Postfach 100565, 98684 Ilmenau, Germany c FG Nanostrukturierte Materialien, Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, Von-Danckelmann-Platz 3, 06120 Halle (Saale), Germany
a r t i c l e
i n f o
Article history: Received 24 May 2013 Received in revised form 11 September 2013 Accepted 11 September 2013 Available online 20 September 2013
a b s t r a c t Graphene was grown on semiinsulating silicon carbide at 1800 ◦ C and atmospheric argon pressure. The all carbon T- and Y-shape three terminal junction devices were fabricated using electron beam lithography. All devices featured the negative rectification effect. The exact properties of the devices like the curvature of the output voltage response can be tuned by changing the branch width in the T- and Y-shape devices. Beside the rectification a switching behavior is demonstrated with the same three terminal junctions. © 2013 Elsevier B.V. All rights reserved.
Keywords: Three terminal junction Rectification Transistor Switch Graphene Silicon carbide
1. Introduction Graphene is a fascinating new material, whose properties are very promising for electronic applications [1–4]. Especially the mobility [5,6], the electrostatic control of the carrier type and their concentration [7] and the aggressive scalability of graphene devices [8] offer advantages for deep scaled and high frequency devices. Nevertheless, the unique band structure of large area graphene with no band gap and linear dispersion around the Dirac point [9,10] ask for new concepts to explore graphene’s unique capabilities in electronics [11–20]. One promising concept in this respect is the three-terminal junction devices (TTJ) [21]. These T- or Yshaped devices of sub-micrometer dimensions capitalize on certain characteristics of the nonlinear response regime of charge carrier transport appearing in two dimensional charge carrier gases confined laterally in channels [22,23]. In this devices geometrically [24,25] or electrically [26] controlled charge transport as well as a rectification [22,27] can be used for information processing and was demonstrated on III–V heterostructures. Recently, the rectification effect was also demonstrated on graphene [28–33]. As a first
∗ Corresponding author. Tel.: +49 3677 693412; fax: +49 3677 693356. E-mail address: [email protected] (J. Pezoldt). 0169-4332/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.apsusc.2013.09.066
application the frequency doubling was realized using graphene TTJ devices [30]. In contrast to III–V heterostructures where only on carrier type is available, the graphene TTJ allows the electrical tuning of the rectification effect from negative to positive rectification in dependence on the carrier type in the device, negative rectification in case of electron conduction and positive rectification if the holes are responsible for the charge transport [28,32]. Many attempts have been made to explain this effect based on the propagation of electromagnetic waves [34], ballistic [35,36] or diffuse–ballistic [37] transport. Here we show a dependence of the rectifying effect on the geometry of graphene TTJ which supports the theory set forth by Sadi et al. [37]. Yet actually the original idea behind the TTJ design was to obtain a new electronic switch [24] or an in-plane gated field effect transistor [38]. With TTJs made of carbon nanotubes a switching behavior [39] and differential current amplification [40] was actually observed. Here we report the observation of such behavior in TTJs made of graphene. 2. Experimental Epitaxial graphene was grown on Si-face semiinsulating silicon carbide. Before the growth process the SiC underwent a direct capping procedure during which it was annealed with a graphite cover
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at 1800 ◦ C for 3 min. Thereby a stepped surface morphology was obtained [41]. Consecutively graphitization was realized by heating the SiC at a rate of 10 K/min up to 1800 ◦ C in a graphite furnace in an argon atmosphere at normal pressure. Raman spectroscopy revealed the formation of 1–4 layers of graphene sheets with sizes up to several micrometers on the SiC surfaces [41]. Hall measurements of the grown graphene layers were carried out with an Accent Hall measurement system in van der Pauw geometry. A Hall mobility of 1000 cm2 /V/s and a sheet carrier concentration ns between 3 and 5 × 1012 cm−2 was obtained. Using a semiclassical model the electron mean free path le can be calculated as le = (h/2e) (ns /)1/2 , where h is the Planck constant and e is the elementary charge [42]. The estimated mean free path of the electrons is between 20 and 26 nm. These graphene sheets were structured into T- and Y-shaped structures with graphene contact pads at the end of each terminal with an electron beam lithography system Raith 150 using negative tone electron beam resist HSQ. Next the graphene was etched with oxygen plasma in an electron cyclotron resonance plasma etching apparatus. Afterwards the HSQ was removed by 10 min exposure to HF-vapor. The graphene contact pads, included in the design, had a size of 90 × 90 m and were situated at the end of each of the three terminals giving the ability for electric measurements. This solution is minimizing problems of contact resistance due to the metal graphene interface. Graphite tips were lowered onto these contact pads to allow for electronic measurements with a Keithley SCS 4200. The pads were framed by gold frames to ease finding them during measurement. The electrical measurements were conducted in two different modes. The rectifying behavior of the three terminal junctions was probed with so called push–pull-measurements. All three branches are contacted. A voltage VL is applied to one branch and another voltage VR to another. At the third branch the resulting voltage VC was measured with a high impedance voltmeter. Push–pull hereby implies that VL = −VR while VL was varied from 0 to 4 V. This setup is shown in Fig. 1. The switching ability of the TTJ was tested by connecting one branch to mass. At the other a voltage was varied from 0 to 8 V each for a different voltage applied to the third branch. In essence this third branch was hereby considered a gate and the other two branches of the channel as source and gate.
Fig. 2. The output voltage at the centre terminal VC in dependence on the push–pullvoltage for a T-shape device with a horizontal bar and a vertical bar width of 15 nm for both bars. The experimental output voltage is shown together with the fitted curve. The intersection of the dotted line gives the onset voltage Von .
3. Geometrical dependence of rectifying behavior A typical device used is shown in Fig. 1. It consists of a T-shape conductor with three ohmic contacts at the end of each branch. The fabricated T-shape devices have horizontal terminals bar widths H of 15 to 150 nm and a length from one terminal end to the other L of approximately 400 nm. The vertical terminal bar width W varied between 13 and 30 nm with respect to the horizontal terminal bar width. L1 is the length of the left terminal; L2 is the length of the right terminal and W is the width of the vertical terminal. The overall length of the horizontal bar L is L = L1 + L2 + W. The Y-shape devices had a symmetrical structure with an angel between the bars of 120◦ . The bar width varied between 15 and 100 nm and was equal in width for all bars within the limitations of electron beam lithography. Push–pull-measurements with the circuit setup shown in Fig. 1 of the T- and Y-shaped TTJ structures revealed the characteristic dependence of the electrical response VC at the central terminal in dependence on the applied push–pull voltage VPP . Even if the third branch is situated exactly half way from the other two branches, the observed potential is negative and not zero as expected from classical transport theory. This behavior has been observed and identified as rectification regardless of whether the horizontal channel is in the diffusive or linear-ballistic regime [22]. A typical dependence of the output voltage VC at the central branch is shown in Fig. 2 for a T-shape TTJ structure. The voltage response at the central contact consists of two parts. At low push–pull voltages VPP the voltage response follows the equation [22]: 2 VC = kVPP + O(V 4 ),
Fig. 1. A typical T-shape TTJ structure made of graphene is shown. The three branches end in larger graphene pads whose beginning can be discerned in this image. These endings are called terminal 1, 2 and 3. The setup of the measurement is sketched around the image of the T-structure. The small inset defines the definition of the device geometry.
(1)
where k is a fitting parameter representing the curvature of the voltage response at VPP = 0 V. The constant k is negative if the two dimensional electron gas consists of electrons and positive if the charge carriers are holes. This was recently demonstrated for grapheme TTJs [29–33]. For all fabricated structures regardless of the shape and the geometrical dimensions a down bending of the voltage response VC was observed which indicates electron transport. Different attempts have been made to explain the occurrence of this phenomenon [22,37,43]. The most plausible seems to be the one from Sadi et al. as, contrary to others it is not solely based on the assumption of ballistic transport alone [37]. Sadi et al. argue that the electrons in the three terminal junctions undergo a “quasi” ballistic transport. “Quasi” ballistic transport is hereby understood as instances where scattering on boundaries tends toward dominating
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scattering on defects and phonons. A mean free path of electrons le in our structures of 20 to 26 nm excludes the possibility of the rectifying behavior being exclusively a ballistic transport phenomenon in our devices. Therefore the electrons undergo around 20 instances of scattering when traveling through the device. Thus a large number of the electrons transgress the actual channel dimensions unlike a traditional conductor on micrometer or larger scale, wherein the majority of electrons never reaches the boundaries of the same. Keeping this in mind we can for one part think of the electron movement as a billiard ball like movement, bouncing on and of the channel edges on their transgression of the device. At the intersections of all three terminals some electrons will be scattered into the third branch on their movement from the end of branch one to the end of branch two. These electrons are the first contribution to the negative potential measured at the end of the central terminal. Next, slowdown of the electrons is mainly caused by scattering on defects at the boundary of the device channels. At the intersection these scattering instances are halved due to only a boundary on one side of the channel. Twice as many scattering instances on boundaries occur over the remaining length of the branches, as they feature boundaries on both sides. Consequently electrons will be accelerated more over the length of the intersection of the three branches then on the remaining length of the branches. This leads to the asymmetry in the distribution of velocity of the electrons across the channel length from the end of branch 1 to that of branch 2 as was observed by other groups [36]. The asymmetry of electron speed causes an asymmetry of the electron concentration along the channel length. This result in the second contribution to the negative potential observed at the end of the third branch. Finally, it has to be mentioned that recently, it was shown experimentally that the nonlinear response of the TTJ might not be limited to ballistic or diffuse-ballistic transport [44]. The second part of the voltage response is observed for high push–pull voltages VPP . In this region the voltage response can be approximated by the following formula [45]:
VC = − VPP + Von ,
(2)
where Von is the extrapolation of the linear dependence of VC to VC = 0 V. The appearance of the linear dependence at high voltages was explained for III–V materials by intervalley transfer [43,45,46]. As in the case of TTJs made of III–V heterostructures the values of the onset voltage of the graphene devices are too large to be explained only by the poor ballistics theory [45]. The onset voltages are determined for the graphene devices are between 0.5 and 1 V. The measured voltage response curve is non-symmetric. According to [22], the origins of asymmetry of the output voltage are asymmetric geometrical properties of the TTJ structures. It is caused by geometrical fluctuations stemming from inhomogeneities in the device geometry due to electron beam lithography effects and graphene thickness fluctuations. They are visible in Fig. 1, where different shades of gray, indicative for fluctuating graphene thickness deviations, as well as width fluctuations of the bar geometry and nonidentical left and the right contact areas near the active part of the TTJ structure can be noticed. Fig. 3 displays the dependence of the curvature k on the horizontal bar width H at constant length of the horizontal branch of the T-shape TTJ. The dependence of the curvature shows a minimum at approximately 40 nm. The absolute value of the curvature is a function of the relation between the diffusive and ballistic transport components of the charge carrier and their interplay with the TTJ device geometry. The strength of the rectification, i.e. the absolute value of the curvature, should increase with the intersection length to the remaining channel length between terminal 1 and terminal 2 due to nonzero vertical velocity component of the charge
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Fig. 3. Curvature dependence of T-shape TTJ devices as a function of the horizontal branch width H at constant length of the horizontal branch of 400 nm.
carriers. Therefore, if the horizontal bar width H is decreasing below the mean free path le of the charge carriers, here electrons, the scattering on the graphene stripe edges becomes dominant and the mean velocity and their component decreases leading to a lower scattering into the vertical branch and to lower absolute curvature values k. Additionally, the increased edge scattering equilibrates the velocity distribution along the horizontal bar reducing the second source for the negative curvature. For horizontal branch width H > le the curvature decreases again in agreement with the observation of T-shape TTJ devices fabricated on III–V materials [47]. So, it is believed that the origin of the observed dependence of the curvature k on the horizontal branch width H has similar origins. The dependence of the absolute curvature k on the bar width H of Y-shape TTJ, structures not shown here, does not show an extremal behavior within the range of the width of the fabricated structures and decrease with increasing branch width. For Y-shaped structures the electrons penetrate more in the central branch due to a stronger field enhanced vertical charge carrier component. This reduces the effect of the shrinking branch width and led to a weaker dependence of the curvature k on the branch width [48,49]. The effect of the vertical branch width W on the shape of the T-shape TTJ devices response is given in Fig. 4. For horizontal bar width H below the mean free path of the electrons le the absolute value of second derivative of the bell shape curve increases with increasing vertical bar width W, whereas for horizontal bar width H exceeding the mean free path of the electrons le the absolute value decreases with increasing the width of the vertical bar W. Again the observed behavior for the second case is equivalent
Fig. 4. Influence of the width of the vertical branch W on the curvature k of the voltage response curve for two different horizontal branch width H of T-shape TTJ devices.
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Fig. 5. Output characteristic of a Y-shaped TTJ device. The device is shown in the inset of the graph.
Fig. 6. Current gain in dependence on the drain-source UDS and the gate-source UGS voltage.
to the observations made in T-shape TTJ devices in III–V materials [48]. The other case W < le might have a similar explanation as given for the dependence of the curvature k on the horizontal branch width H if H is below the mean free path of the charge carriers le . It is worth to be mentioned here that the given interpretation of the geometrical dependent properties of the fabricated TTJ structures include the influence of surface charges.
electrons will flow through the gate. The following equation, where UGS is the gate voltage, ϕXG the local potential of the drain-source channel where it intersects the gate, xchannel-length the source-drain channel length and xG the position of the gate is, describes this condition:
4. Electrical gating in T- and Y-shaped junctions For the investigation of the switching ability of the fabricated Tand Y-shaped devices in the measurements terminal 1 was considered as source and terminal 2 as drain. Between these two a source-drain voltage was varied from 0 to 8 V. Source-drain and gate current were measured simultaneously. This procedure was repeated for different gate voltages from −4 to 4 V applied to terminal 3. The two decisive indicators for electrical gating are the ratio of source-drain current to gate current, the current gain, and the strength of control of the source-drain current by variation of the gate voltage, which manifests in the value of transconductance. Fig. 5 shows the source-drain current over the source-drain voltage for different gate voltages. If source-drain voltage is kept constant and only the gate voltage is changed, which represents in the graph a change along a vertical line, the source drain-voltage changes. At a drain source voltage of 3 V a maximum transconductance of 440 S/m was obtained for T-shape devices. For the fabricated Y-shape devices the maximum transconductance reached 300 S/m. For certain source-drain and gate voltages a current gain of up to 510 was obtained from the measured output characteristics of the T- and Y-shaped devices. The transconductance of the first 100 GHz graphene MOSFET demonstrated by IBM was 150 S/m [50]. The record transconductance achieved so far on epitaxial graphene MOSFET devices is 2000 S/m [51]. Typical values of the transconductance of graphene field effect transistors independent on the used graphene synthesis method are in the range between 100 and 600 S/m [52–55]. So, the obtained transconductance values of the Tand Y-shaped TTJ devices are comparable to standard graphene transistors. A current gain in these metallic structures is rather surprising considering that there is no gate insulator to limit the gate current. The key to understanding this effect is the source-drain and gate voltages at which the maximum current gain occurs. The local potential drops linear over the length of the channel between terminal 1 and 2. If the applied gate voltage is equal to this potential, no
UGS = ϕXG =
UDS xG , xchannel−length
(3)
Therefore, the T- or Y-structures behave like an electron valve gated by the voltage difference between terminal 1 and terminal 3. If the above condition (3) is fulfilled the valve is closed and the electrons only flow from source (terminal 1) to drain (terminal 2). In any other case the valve is opened and a gate current larger zero exists. Fig. 6 shows in the x–y-plane the source-drain current over the source-drain voltage. The z-axis is ascribed to the ratio of drain current ID over gate current IG . Peaks can be seen at the positions where the above condition is fulfilled. The peaks have a negative and positive part as the gate current changes direction as maximum current gain occurs when the same alternates its flow direction. The varying height of the maxima is due to gate voltage having been varied in 1 V steps. A finer tuning of the same should reveal maxima of equal height. The observed gating in T- and Y-shaped junctions is surprising, as there is no gap between gate and the channel between source-drain, as for example in side-gate field effect transistors [56–58]. Furthermore, the observed output characteristic shows not the classical transistor behavior. Nevertheless, the detected gating effect was also observed in Y-shaped junctions made of carbon nanotubes [39], but the observed current gain in the graphene based structures is considerably higher, than the current gain of 10 obtained for carbon nanotubes [39]. 5. Conclusion Dependence between the geometry of the TTJs and the rectifying behavior was observed. This supports the explanation of the rectifying behavior brought forth by Sadi et al. which suggests the presence of a “quasi” ballistic transport of the electrons in the TTJ. These insights about the rectifying behavior enable one to design TTJs more effectively. Next a switching behavior in TTJs made of graphene was demonstrated. The transconductance is much higher than the one obtained in other carbon nanotube TTJs exhibiting switching behavior. The observed current gain was explained by observing the voltages it occurs at. This may open the path to new electronic
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