Effects of Electrode Layer Band Structure on the Performance of

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Amritesh Rai, Nano Kyounghwan Letters is Kim, published Xuehao by Subscriber the access American provided Chemical bySociety. the University 1155

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Interlayer Current (µA/µm2)

2.5

Page 1quadlayer Nano of 34 Letters 2.0 1.5 1.0

1 2 0.5Paragon Plus Environmen ACS pentalayer 3 0.0 0.25 0.5 0.75 1 4 0 Interlayer Voltage (V)

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Effects of Electrode Layer Band Structure on the Performance of Multi-Layer Graphene-hBNGraphene Interlayer Tunnel Field Effect Transistors Sangwoo Kang*‡, Nitin Prasad‡, Hema C. P. Movva, Amritesh Rai, Kyounghwan Kim, Xuehao Mou, Takashi Taniguchia), Kenji Watanabea), Leonard F. Register, Emanuel Tutuc, Sanjay K. Banerjee Microelectronics Research Center, Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas, 78758, USA a) National Institute for Material Science, 1-1Namiki, Tsukuba, Ibaraki, 305-0044, Japan KEYWORDS Interlayer Tunnel FET (ITFET), resonant tunneling, negative differential resistance (NDR), multi-layer graphene, band structure, stacking order

ABSTRACT Interlayer tunnel field-effect transistors based on graphene and hexagonal boron nitride (hBN) have recently attracted much interest for their potential as beyond-CMOS devices. Using a recently developed method for fabricating rotationally aligned two-dimensional heterostructures, we show experimental results for devices with varying thicknesses and stacking order of the graphene electrode layers, and also model the current-voltage behavior. We show

ACS Paragon Plus Environment

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that an increase in the graphene layer thickness results in narrower resonance. However, due to a simultaneous increase in the number of sub-bands and decrease of sub-band separation with an increase in thickness, the negative differential resistance peaks becomes less prominent and do not appear for certain conditions at room temperature. Also, we show that due to the unique band structure of odd number of layer Bernal-stacked graphene, the number of closely spaced resonance conditions increase, causing interference between neighboring resonance peaks. Although this can be avoided with even number layer graphene, we find that in this case, the bandgap opening present at high biases tend to broaden the resonance peaks.


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Advancements in the fabrication techniques for heterostructures based on two-dimensional materials has facilitated the study of such structures for various applications.1 One such application is the tunnel field effect transistors (TFETs) which has attracted interest in recent years as a possible beyond-CMOS device candidate.2,3 Especially, interlayer tunnel FETs (ITFETs), whose operation is based on band alignment and resultant resonant tunneling between two electrode layers through a tunnel barrier, offer the possibility of obtaining sharp negative differential resistance (NDR), and has received much attention.4-17 Such devices have been demonstrated for double monolayer graphene4,5 and double bilayer graphene6,7 functioning as the electrode layer, with hexagonal boron nitride (hBN) used as the interlayer tunnel dielectric. These devices exhibited NDR characteristics which are gate tunable. Also, there have been theoretical predictions of such phenomenon occurring even between distinct number of graphene layers,8 and also reports of possible phonon-mediated resonant tunneling.9 Attempts have been made to model such behavior, which has helped in better understanding the physics behind such devices.10-16 Various applications that utilize this NDR characteristic have also been demonstrated.5,7,17 However, the NDR observed in these devices were only comparable in terms of peak-to-valley current ratio (PVCR) to that which have been obtained in the past, for instance, in Ge pn-junction Esaki diodes,18 and inferior to that obtained for InGaAs/AlAs/InAs resonant tunneling diodes,19 although the voltage at which the NDR peaks occur was somewhat scaled down. Notwithstanding theoretical work on what kind of materials and device dimensions would be advantageous for obtaining higher PVCR in these two-dimensional heterostructures,20,21 corresponding experimental efforts into identifying performance-limiting factors for ITFETs have been limited. Here, we present the results of an exploration of various combinations of different graphene layer thickness and hBN, with the aim of providing a list of lessons learned


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which will act as a guide to what kind of materials or band structures should be used in order to achieve improved performance in ITFETs; i.e., enhanced PVCR for lower-power operation and larger output swing in logic gates, and narrower resonant peaks for lower-voltage and lowerpower operation. We explored double bilayer, trilayer, quadlayer, and both Bernal(ABA)-stacked and rhombohedral(ABC)-stacked pentalayer graphene as the electrode layer, with varying hBN tunnel barrier thicknesses. We find that the differing band structures for the multi-layer graphene layers with varying thicknesses cause significant changes in the interlayer current-voltage characteristics, and that certain band structures are more preferable than others in obtaining higher PVCR. Also, we find that with extremely thin interlayer hBN thicknesses of two atomic layers and even number of layer graphene, the resonance peaks can split into smaller sub-peaks at high biases due to the bandgap opening effect, and cause the overall peaks to broaden. The ITFETs, whose structural schematic and bias conditions are shown in Fig. 1(a), are fabricated using a process similar to a recently developed dry transfer method that can ensure the rotational alignment of the two graphene electrode layers.22 A large graphene flake (Fig. 1(b)) whose thickness and stacking order is determined by a combination of optical contrast and Raman spectroscopy23-26 - is first sectioned into separate parts with clearly identifiable straight edges by means of electron beam lithography (EBL), as shown in Fig. 1(c). It has been reported that the stacking order of graphene does not change with conventional processing used for device fabrication.25 Nevertheless, we confirmed through Raman mapping that no change in the stacking order for multi-layer graphene occurred during the transfer or device fabrication process. Using a polymer-coated glass stamp, the top hBN is first picked up, with which the top graphene, interlayer hBN, bottom graphene, and bottom hBN flakes are picked up sequentially and selectively. The position and rotation of the samples on the stage or the stamp itself are


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unaltered throughout the transfer process. This process flow ensures that the two graphene layers are rotationally aligned in real space, and as a result, that the two K-points of the top and bottom layer graphene are rotationally aligned in k-space,22 which allows momentum conserving tunneling to occur. The whole stack is then landed onto a fresh Si-SiO2 substrate as shown in Fig. 1(d), vacuum annealed, and then, the active region is patterned using EBL and plasma etching. The metal contacts are also defined through EBL and with subsequent thermal evaporation of Cr and Au and lift-off, which result is shown in Fig. 1(e). This process flow results in independent one dimensional contacts to the edges of the top and bottom graphene.27 For this particular device shown in Fig. 1, we used a graphene flake that had regions of bilayer, Bernal-stacked trilayer and rhombohedral-stacked pentalayer graphene. The flake was sectioned in such a way, that when transferred on top of each other, the final device would have regions of double bilayer, bilayer-to-trilayer, double trilayer, trilayer-to-pentalayer, and double pentalayer, using the same top, interlayer and substrate hBN dielectric. In the discussions below, we only deal with devices that have the same number of layers of graphene on either side of the tunnel barrier. This experimental setup was intended to minimize the microscopic device-to-device fluctuations in terms of the rotational alignment, variations in the interlayer hBN, and surrounding environment. This ensured that these factors that can influence the tunneling characteristics were limited, and we were mostly observing the effects of the graphene layer thickness. In addition, we fabricated and characterized double bilayer devices with two and six atomic layers of interlayer hBN in a single batch, and a double quadlayer device with two atomic layers of interlayer hBN. The devices were characterized under high vacuum, and all measurements were conducted at room temperature. A simple schematic of the band structures for the graphene electrode layers used in this study are given in Fig. 1(f).28


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Fig. 2(a) shows the area normalized interlayer current (IINT) versus top layer bias (VTL) characteristics at room temperature of the three devices with differing graphene layer thickness fabricated at the same time. Only step-like features could be observed in the IINT-VTL characteristics for these samples, unlike previous results where multiple NDR peaks were observed for the case of double bilayer device.6,7 This could be attributed to a slight rotational misalignment between the two graphene layers as discussed below regarding the analysis of simulation results for pentalayer graphene ITFET. It should be noted here that there is still the potential of misalignment when the sample or stamp is unintentionally shifted during the transfer process. In such cases, the straight edges of the graphene flakes are used to visually identify the rotational alignment. Under an optical microscope with a 10X objective, a misalignment of ~2 ° can be readily identified, which may be considered the maximum rotational misalignment. Nevertheless, in Fig. 2(b) for the differential conductance (dIINT/dVTL) versus VTL plot, it can be noted that the step-like features of (a) show up as clear conductance peaks which correspond to conditions where the bands of the graphene layers would be aligned and resonant tunneling would occur. It is clear that an increase in the number of graphene layers results in a sharper peak (narrower width and higher intensity), as shown in the inset of Fig. 2(b), where the width of the first resonance peak for each device extracted through a Lorentzian fit, is plotted against graphene layer thickness. The width Γ decreases from 295 mV for bilayer to 55 mV for pentalayer. This is an expected result from the increase in the density of states (DOS) for thicker graphene.29 The increase in the DOS can be noted by the relative insensitivity of the characteristics with back gate bias (VBG) for trilayer graphene compared to pentalayer graphene (Figs. 2(c,d)). When the VBG is varied from -40 V to 40 V, the shift in the peak position for trilayer is 80 mV while the pentalayer characteristic is unchanged. This means that the DOS is


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large enough for pentalayer graphene that the quantum capacitance (CQ in inset of Fig. 2(d)) takes on a more dominant role in determining the characteristics of the device rather than the external capacitances (CBG and CINT). Nevertheless, even with a sharper peak with increase in the number of graphene layers, it is apparent that there is no NDR region in the characteristics for double trilayer and double pentalayer devices. It is highly probable that this is due to an increase in the number of sub-bands and decrease in the sub-band spacing for multi-layer graphene which gives rise to an increase in closely spaced resonance conditions. It has recently been suggested that an increase in the number of sub-bands and decrease in sub-band spacing can result in a less prominent NDR peak.30 This becomes more apparent when we compare the characteristics of double pentalayer graphene with differing stacking order, namely, Bernal-stacking versus rhombohedral-stacking, as is shown in Fig. 3(a). Since Bernal-stacked pentalayer graphene has a Dirac cone band like monolayer graphene in addition to the four parabolic sub-bands, there are much more conditions when resonances can occur, as was shown for tunneling between monolayer graphene (Dirac cone) and bilayer graphene (two parabolic bands) separated by hBN.8 It was shown that there will be more closely spaced resonance conditions than compared to devices with the same thickness for the two graphene electrode layer, due to the alignment of the Dirac cone of the monolayer graphene and the two parabolic bands of the bilayer graphene making intersections at various biases. In other words, there will be many more resonance peaks for Bernal-stacked double pentalayer graphene due to additional resonance conditions established between the Dirac cone sub-band of one layer to the parabolic sub-bands of the other layer. Hence, the resonance peaks will be more closely spaced for Bernal-stacking, and therefore, interference effects for it will be more pronounced when compared to rhombohedral-stacking. This is evident in the


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dIINT/dVTL plot of Fig. 3(a) where the valleys of the red curve (rhombohedral-stacking) are seen to be lower in intensity compared to the blue curve (Bernal-stacking), due to the slightly larger spacing between the first and second resonance peak. In our Bernal to rhombohedral-stacking comparison, we see that in the range of VTL = -0.7 ~ 0.7 V, Bernal-stacking shows 10 distinguishable peaks while the rhombohedral case shows only 5 such peaks. It should be noted here that the two device did not have the same interlayer hBN thickness, with the Bernal-stacked device having 3 atomic layers while the rhombohedral-stacked device had 5. As shown in Fig. 3(b), when two double bilayer graphene devices are fabricated in a single batch with different hBN thicknesses, we consistently observe that thinner hBN thickness results in resonance peaks that are more spaced apart with a lower valley in the differential conductance on either side of the first resonance peak. This is a result of the increase in the interlayer capacitance, as discussed by Kim et al.22 In other words, if the Bernal-stacked device had been fabricated with the same hBN thickness as that of the rhombohedral-stacked device, the resonance peaks would have been more closely spaced and the differential conductance valleys would have been even higher. In Fig. 4, we compare the results of a double bilayer graphene with that of a double quadlayer ITFET. Though the devices were fabricated separately, they had the same interlayer hBN thickness. The band structure of even number of layer graphene does not exhibit a Dirac cone. Therefore, additional resonance conditions that arise from the tunneling between the linear and parabolic bands are absent. As a result, we expect fewer resonance conditions and improved NDR characteristics. As can be noted from the figures this is indeed the case. Further, the double quadlayer device exhibits a significant NDR comparable to that of double bilayer device. As expected, the double quadlayer device has an enhanced NDR resonance due to its higher DOS. This is evident from Fig. 4(b) where the normalized peak intensity of the first resonance is larger


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than that of the double bilayer device. However, due to the smaller spacing from the first subband to the second sub-band for quad-layer graphene (0.6 γ1) compared to bilayer graphene (γ1), the secondary resonance peaks occur closer to the first peak, which, in turn, acts to increase the intensity of the valleys adjacent to the first peak, and prohibits the device from having a larger PVCR. In other words, the second resonance interferes with the first resonance. To corroborate the measurements with theory, we compute the tunneling currents using a Bardeen transfer Hamiltonian approach, following Feenstra et al.10 and de la Barrera et al.14 The tunneling current is obtained by carrying a probability weighted sum of the transition rates ( )

between all the states (, ) between the top layer and the bottom layer, i.e.,

8

= − !" # − ! ℏ

,

where and are Fermi distribution functions of the top and the bottom layers, respectively.

is the coherence length of an electron, is the total overlap area between the top and bottom

layers. The transition rates are computed by estimating the overlap integral of the wavefunctions in the top and the bottom layers (% and % , respectively) i.e.,

*% ℏ *% = ) *+ ψ∗− ψ∗0 2( */ */

where the integral is carried out over an area of

. A finite coherence length of the electron

< √ accounts for further resonance broadening observed in the measurements. is further simplified following the formalism of Feenstra et al.10


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To compute the tunneling currents, band structure for Bernal-stacked bilayer and quadlayer graphene are obtained using a tight-binding formalism. In the presence of transverse electric fields, bilayer graphene opens up a band-gap31,32 and the quadlayer Bernal-stacked graphene undergoes a band distortion without bandgap opening.32 For simplicity, the trigonal warping is neglected in both cases. Further, the screening effect to estimate the on-site potential for each layer in quadlayer graphene, in the presence of transverse electric fields, is not self-consistent. Finally, the case of mono/tri/pentalayer graphene, the band structure is approximated using a combination of linear and parabolic bands with no bandgap openings or distortion in the presence of transverse electric fields.28 The charging of the top and bottom layers are taken into account while estimating the electrostatic potentials (3 and 3 , respectively) in both layers needed to compute the tunneling currents. This is done by solving the following coupled non-linear equations, 45 65 − 3 + 489 3 − 3 + : 3 , 6 = 0, 489 3 − 3 + : 3 , 6 = 0. Here, 45 489 is the bottom gate (interlayer) capacitance, : 3 , 6 and : 3 , 6 are the charges accumulated on the top and the bottom layers of the ITFET, respectively, due to the finite DOS, determined by assuming a Fermi distribution of carriers. Using this model, we have calculated the dIINT/dVTL characteristics of a device with double Bernal-stacked pentalayer graphene structure. In Fig. 5(a), individual components from the tunneling between like-bands (middle curve) and between that of the unlike-bands (bottom curve) are plotted with the measurement result (top curve). It can be noted from the plot that the


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positions of the resonance peaks match very well with that obtained for the measurements. It is clear that the broad NDR regions occurring on either side of the first resonance peak at roughly VTL = ± 0.12 ~ 0.4 V which results from the like-band tunneling, is dampened out by the unlikeband tunneling which happens to occur in the same range of VTL. This confirms our previous argument that odd number of layer Bernal-stacked graphene is not suited for obtaining high PVCR NDR characteristics. One discrepancy between measurement and simulation is the peak width resulting from the unlike-band tunneling. This could be explained by a discrepancy in the assumed rotational alignment. In our simulations, we assumed perfect rotational alignment between the top and bottom graphene layers. The tunneling current between a Dirac cone and the parabolic bands in the twisted structure displays a much richer behavior analogous to the study done by Lane et al.8 Therefore, we surmise that an unintentional slight rotational misalignment during fabrication might have been the cause for the sharper peaks resulting from unlike-band tunneling relative to the simulations. This is also in line with the less prominent NDR for double bilayer device of Figs. 2(a,b) compared to previously reported results,6,7 that was transferred and fabricated simultaneously with the pentalayer device. Calculations were also carried out for the double bilayer and double quadlayer case which results are shown in Fig. 5(b,c) overlaid onto the measurement results. It can be noted from the figures that the overall peak positions obtained through our model agree well with measurements. The broadening of the peaks observed for the secondary peaks at high VTL is also evident in the simulations. From the model results, we note that the broadening of the peaks is actually a result of there being multiple sub-peaks. This splitting of the secondary peak is because of a bandgap opening at the Dirac point due to the presence of transverse electric fields.31,32 However, the bandgap opening around 6 = 0 due to the applied fields is close to


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zero, and cannot account for the flatness in the resonance around 6 = 0. We surmise that this may be a signature of a non-resonant tunneling mechanism9 or because of a contact issue. The performance of ITFETs in circuits is critical for beyond-CMOS logic applications. While the task of determining the numbers and the widths of the resonance peaks is complicated by the bandstructures of multi-layer graphene, the strongest resonance peak, occurring with fully aligned bands of graphene multi-layers and usually smaller gate/interlayer voltage combinations, is the main focus in low-power circuit applications. Our semi-quantitative compact model33 shares the same components as indicated in the quantum simulations, with IINT capturing only the main resonance peak by a linear leakage resistance in parallel with a Lorentzian-broadened resonance peak. The details of the SPICE circuit simulations are presented elsewhere.33 These simulations show that, in the presence of the increasing leakage currents with increasing VTL, a clock amplitude of 5~6 times the half-width of the resonance peaks is optimal for the performance of ITFETs in logic circuits. In terms of the gON/gOFF ratio, namely the ratio between the slope of the resonance current peak and the slope of the background current, for which the PVCR is a good metric, it should be at least 5 for the simple gates such as inverters, and above 20 for complex gates such as NAND and NOR gates. The 27.5 mV half-width of the resonance peak for graphene pentalayers, as illustrated in Fig. 2(b), would allow energy consumption on the order of aJ per ITFET per clock-cycle, assuming nanoscale devices and GHz clock frequency. Consequently, with sharper resonance peaks, reducing the background leakage current should be the next step for improved gON/gOFF ratio and, in turn, low-power operations. In summary, we have explored various combinations of graphene and interlayer hBN thicknesses in order to experimentally study the effects of electrode layer band structure on the characteristics of ITFETs. Although an increase in the number of graphene layer does bring


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about an enhanced resonance, due to an increase in the number of sub-bands and a decrease in the spacing between the sub-bands, the sharper resonance is dampened by closely spaced adjacent resonance peaks, inhibiting the device from attaining higher PVCR values. Especially, when using an odd number of layer graphene with Bernal-stacking, which has a Dirac cone band in addition to the parabolic bands, the number of resonance conditions radically increase causing further interference with the main resonance peaks. This is not the case for even number of layer graphene, which does not have a Dirac cone regardless of the stacking order, and therefore we are able to observe NDR. However, in this case also, the sharper resonance brought about by the increase in DOS is countered by the decrease in the sub-band spacing between parabolic subbands, limiting the PVCR. Also, for even number of layer graphene, bandgap opening effects, which tend to split and broaden out the resonance peaks at high fields should also be taken into consideration. Moreover, we find that the increase in the DOS with increase in number of graphene layers, increases the influence of the quantum capacitance on the device characteristics and decreases that of the external gate capacitances. In others words, the gate tunability is somewhat diminished. We suggest that the use of a material with an optimized DOS, minimal number of sub-bands or largely spaced sub-bands, and no change in the band structure as a function of applied bias would be advantageous in terms of implementing ITFETs with improved performance (higher PVCR) at low operating voltages. While the various TMD materials do have closely spaced sub-bands, especially for multi-layers,34 we do not think that they would influence ITFET performance due to the high DOS at the band edge, and as a result, the subbands not being able to be populated during normal operation.34-36 We consider various TMDs to be favorable in terms of ITFET implementation. Remaining challenges are obtaining large enough TMD flakes to which the above fabrication method can be applied, obtaining high


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quality flakes which eliminates the band tail states,20 and finally, establishing low resistance, stable contacts to the TMD layers.


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(b) r‐penta B‐tri (c)

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bi

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Figure 1. (a) Schematic of the device structure and biasing scheme. The two graphene electrode layers are separated by a thin interlayer hBN layer, and encapsulated with a thick hBN on the top and bottom. The heavily n-type doped silicon substrate is used as the back gate. The thickness of the top and bottom graphene layers is varied from 2 to 5 atomic layers. (b) As-exfoliated graphene flake showing regions of bilayer, trilayer and pentalayer. (c) The graphene flake is divided into two parts by means of electron beam lithography and O2 plasma etching. (d) The flakes are transferred to make an hBN-Gr-hBN-Gr-hBN heterostructure on top of a Si-SiO2 substrate, while maintaining the relative rotational alignment of the two graphene layers. (e) The stack is partitioned into separate parts to make devices with various combinations of electrode layers. (f) Band structure for the electrode layer used in the devices for this study.28 The top numbers indicate the number of graphene layers. Only odd number of layer graphene with Bernal(ABA)-stacking exhibit a Dirac cone sub-band.


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(c)

150

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bilayer trilayer pentalayer


(a)


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0

0.15

0.3

‐0.3

VTL (V)

‐0.15

0

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VTL (V)

Figure 2. (a) Normalized current-voltage (I-V) characteristics at room temperature for double bilayer, double trilayer, and double pentalayer ITFETs sharing the same 5 atomic layer thick interlayer hBN. Inset in (a) shows an equivalent circuit model of the device. (b) The differential conductance (dIINT/dVTL) versus top layer bias (VTL) of the same devices. The conductance peaks correspond to the various resonant tunneling conditions that are established when VTL is varied. The inset of (b) shows that with an increase in the number of graphene layers, the peak width, extracted by the width Γ of a Lorentzian fit on the first resonance peak, decrease. dIINT/dVTL as a function of VTL for varying VBG for (c) double trilayer and (d) double pentalayer. The insensitivity of the characteristics to VBG in the pentalayer case compared to trilayer is due


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to the larger DOS, and hence larger quantum capacitance (CQ), of pentalayer graphene compared to trilayer.


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8 0.4

d I IN T/d V TL (μ S /μ m 2 )

rh o m b o h e d ra l B e rn a l

2.0

d I IN T/d V TL (n S /μ m 2 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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6

60

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40

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‐20

V BG = 0 V

‐4

‐40 ‐1

‐0.5

V TL (V)

0

V TL (V)

0.5

1

Figure 3. Differential conductance comparison between (a) Bernal-stacked pentalayer graphene with 3 layer hBN (blue) and rhombohedral-stacked pentalayer graphene with 5 layer hBN (red), and (b) bilayer graphene ITFETs with differing interlayer hBN thickness. The colors of the curve and axis numbers are matched. There are many more conductance peaks for the Bernal-stacked case compared to the rhombohedral-stacked one. This is because odd number of layer Bernalstacked graphene has a Dirac cone sub-band which allow for more resonant tunneling conditions to be established with the parabolic bands of the other layer.8 The first conductance valleys of the Bernal-stacked ITFET is higher than that of the rhombohedral-stacked one due to the more closely spaced secondary resonance, even though the Bernal-stacked pentalayer device had a thinner hBN which tends to result in steeper valleys as shown in (b).


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Nano Letters

(a)

3.0 bilayer Bi‐layer quadlayer Quad‐layer

IINT (μA/μm2)

2.0 1.0 0.0 ‐1.0 ‐2.0

VBG = 0 V

‐3.0 ‐1

(b) dIINT/dVTL (μS/μm2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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‐0.5

0

0.5

1

0.5

1

VTL (V) 8.0 6.0 4.0 2.0 0.0 ‐2.0 ‐4.0 ‐6.0 ‐1

‐0.5

0

VTL (V)

Figure 4. (a) I-V characteristics at room temperature of double bilayer (blue) and double quadlayer (red) ITFETs with the same hBN thickness of 2 atomic layers but made separately. Clear NDR peaks occurring at regular intervals can be noted. (b) The differential conductance for the same devices. The first resonance peak of the double quadlayer device shows slightly higher differential conductance but the first resonance valley is less pronounced compared to the double bilayer device. This is a result of the secondary peaks occurring at a slightly lower bias for double quadlayers compared to double bilayers due to the smaller separation of sub-bands. The smaller separation of peaks means that the resonance of one interferes with the other and makes them less prominent.


19

(3)

(e)

(d)

0.4

0.2

(1)

0.0

dIINT/dVTL (μS/μm2)

(1) (2)

(a)

(b)

8 6 4 2 0 ‐2 ‐4

VBG = 0 V

‐6 ‐1

(f)

0.0

(2)

(c)

0.0


dIINT/dVTL (arb. units)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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‐0.5

0

0.5

1

VTL (V) 6 4 2 0 ‐2 ‐4

VBG = 0 V

‐6 ‐0.7

‐0.35

0

0.35

VTL (V)

0.7

(3)

‐1

‐0.5

0

0.5

1

VTL (V)

Figure 5. (a) Measurement result for a double Bernal-stacked pentalayer device. Simulation result for dIINT/dVTL of the same structure caused by (b) tunneling between like-bands and by (c) tunneling between the unlike-bands. Peak positions observed in the measurement result match well with the simulation results. Also, it can be noted that the NDR region observed for like-band tunneling is dampened out by the more closely spaced unlike-band tunneling. (d) Band diagrams for cases (1-3) in (a), where solid lines represent filled bands and dotted lines empty ones. Unlike-band tunneling occurs when the electrons in the Dirac-band have states in the parabolic band of the other layer where it can tunnel into without loss or gain of momentum (green dots). Comparison of measurement (symbols) and simulation (dashed line) for (e) double bilayer and (f) double quadlayer devices. The peak positions and peak broadening at high VTL are well reproduced through simulation.


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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AUTHOR INFORMATION Corresponding Author *[email protected] Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. ‡These authors contributed equally. Funding Sources This work was supported in part by SWAN, a funded center of NRI, a Semiconductor Research Corporation program (SRC), sponsored by NERC and NIST, and also in part by NSF NNCI program.


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Insert Table of Contents Graphic and Synopsis Here

Interlayer Current (µA/µm2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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2.5

quadlayer 2.0 1.5 1.0 0.5

pentalayer 0.0 0

0.25

0.5

0.75

1

Interlayer Voltage (V)


28

(a)

VTL

(b) r‐penta

(c)

Nano Letters B‐tri Page 30 of 34

hBN

VBG

1 2 3 4 5 (f)6 7 8 9 10 11

T‐Gr hBN B‐Gr (d) hBN SiO2 n+ Si

2

bi

(e)

T‐Gr

B‐Gr hBN

3

4

5

5


ABA

ABC

(a) Page150 31 of 34 100

(c) 1.2 Nano Letters bilayer trilayer pentalayer

‐40 V


40 V

80 mV 0

VTL (V)

0.15

VBG = 0 V

0.3

40 V


Peak Width (mV)


IINT (pA/μm2)

1 2 0.8 50 3 4 0 5 VBG CBG CQ,T VTL 6 0.4 ‐50 7 8 CINT CQ,B IINT 9 ‐100 10 11 ‐150 0.0 12 ‐0.3 ‐0.15 ‐0.3 ‐0.15 0 0.15 0.3 13 VTL (V) 14 400 (b) (d) 2.0 15 2.0 300 ‐40 V 16 bilayer 200 17 1.6 trilayer 1.6 100 18 0 pentalayer 19 1.2 0 1 2 3 4 5 6 20 1.2 # of Graphene Layers 21 0.8 Γ 22 0.8 23 0.4 24 25 0.4 0.0 26 VBG = ‐40 V 27 ACS Paragon Plus Environment ‐0.4 0.0 28 ‐0.3 ‐0.15 0 0.15 0.3 ‐0.3 ‐0.15 29 VTL (V) 30

VBG = 0 V

0

VTL (V)

0.15

0.3

1 2 1.6 3 4 1.2 5 6 0.8 7 8 0.4 9 10 11 0.0 ‐0.7 12 13

rhombohedral VBG = 0 V Bernal

0.5 0.4 0.3 0.2 0.1

2 layers of hBN 6 layers of hBN

8

0

VTL (V)

0.35

0.7

80

6

60

4

40

2

20

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0

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‐0.5

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ACS Paragon 0 Plus Environment ‐4

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dIINT/dVTL (pS/μm2)


2.0

Nano(b) Letters 10


(a)

(a) Page 33 3 of 34

bilayer Bi‐layer quadlayer Quad‐layer

2


IINT (μA/μm2)

1 2 3 4 5 6 7 8 9 10 11 12 (b) 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Nano Letters

1 0 ‐1 ‐2

VBG = 0 V

‐3 ‐1

‐0.5

0

0.5

1

VTL (V) 8 6 4 2 0 ‐2 ‐4 ‐6 ACS Paragon Plus Environment ‐1 ‐0.5 0 0.5

VTL (V)

1

(3)

(d)

Nano Letters

(e) dIINT/dVTL (μS/μm2)

0.4

0.2

0.0

(1)

(b)

8

Page 34 of 34

6 4 2 0 ‐2 ‐4

VBG = 0 V

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(f)

0.0

(2)

(c)

0.0



dIINT/dVTL (arb. units)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

(1) (2)

(a)

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(3)

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1

Effects of Electrode Layer Band Structure on the Performance of

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