Electronic structure modulation for low-power switching
© Raza; licensee Springer. 2013
Received: 26 July 2012
Accepted: 14 January 2013
Published: 13 February 2013
We report the transport characteristics of a novel transistor based on the electronic structure modulation of the channel. The gate voltage-controlled current modulation arises from the bandwidth manipulation of a midgap or a near-midgap state. We show that the transistor exhibits a gain and overcomes the 2.3 kBT/decade thermal limit in the inverse subthreshold slope where kB is the Boltzmann constant. The unique device physics also opens up many novel applications.
We present a novel concept for modulating the channel transport by all-electronic means. The working principle is based on the electronic structure modulation of a midgap or a near-midgap state due to an electric field by applying a gate voltage. Small bandwidths (BW) have large effective masses and hence poor transport characteristics due to strong scattering. This leads to the off state of the transistor. The on state has a large bandwidth and hence smaller effective mass, which gives the higher desired conduction. The proposed transistor, namely electronic structure modulation transistor (EMT), has also been analyzed as a possible replacement for metal oxide semiconductor field-effect transistor technology .
Conventional field-effect transistors (FET) rely on the band edge shift using an external gate voltage. Hence, FETs are limited by the 2.3 kBT/decade thermal limit in their subthreshold inverse slope , where kB is the Boltzmann constant and T is the temperature. With the scaling of the supply voltage, channel leakage current increases [2, 3], making the power dissipation a serious challenge. It is, therefore, desirable to reduce the off current with a low supply voltage by overcoming the subthreshold thermal limit, while retaining the gain and high speed device (pico-second) and circuit (nano-second) operation. Various devices have been under study as possible candidates to replace FETs in complementary metal-oxide semiconductor (CMOS) technology .
Concepts based on the modulation of various device parameters have been explored earlier. For example, velocity/mobility modulation transistors rely on the real-space transfer of carriers between two adjacent materials with different mobilities . Similarly, quantum modulation transistors are based on the constructive and destructive interference of the wavefunctions in the channel by electrically changing the T-shaped box dimensions . Furthermore, quantum effects in various planar heterostructures based on the modulation-doped field-effect transistor principle have been explored , where the field-effect is used to perturb the barrier for carriers flowing between the source and the drain electrodes. The localization of the state near the band edges due to disorder in the Anderson localization is also a relevant concept, which leads to a mobility edge , but this effect is also limited by the thermal limit. In this paper, our objective is to show that the proposed EMT may become a viable candidate for low-power applications within the device and circuit performance constraints under extreme scaling.
where α is a dimensionless parameter, called the modulation factor. BWo is the residual BW at zero gate voltage (Mag ≡ absolute magnitude) and Vg is the applied gate voltage. In Figure 1d, α = 0.47 and BWo = 0.12 eV. With increasing width of the nanoribbon, the residual bandwidth BWo, the modulation factor α, and the band gaps are expected to decrease . One could vary the device width, which will still result in qualitatively similar characteristics, as far as the conduction and valence band edges are well isolated from the near-midgap state.
Next, we consider the transport through the graphene nanoribbon by applying drain bias. In the limit of small drain bias, the channel transport is only dependent on the bandwidth of the near-midgap state. For zero bandwidth, no channel current flows through this state in the coherent limit, except for the dielectric leakage current and tunneling through the higher bands, which should be small given the conduction (valence) band is above (below) the localized state by about 1 eV. By applying a gate voltage to increase the bandwidth of the state, the channel current starts to flow. The operation of the EMT in this mode is equivalent to that of an n-MOS; hence, we refer to it as n-EMT. The equivalents of p-EMT can be realized by simply reversing the gate connections to induce an electric field in the reverse direction . This all-electronic scheme thus operates under complementary mode. We envision that such transistor action is more general and can be achieved in any dimension with a near-midgap state in the channel region, the bandwidth of which can be modulated by the external voltage and for which, one can make ohmic contacts with the midgap state. In the limit of high bias, this transport picture changes, which we discuss later. So far, to the best of our knowledge, an experimental observation of such a state in a zzGNR has not been made.
To understand the transport in the high-bias regime, we consider a gedanken one-dimensional device and start with the ansatz of Equation 1. For such a device, we use single-band tight-binding approximation , where the channel bandwidth is 4|to| and to is the nearest neighbor hopping parameter. For simplicity, we take five lattice points in the device region corresponding to a channel length and width of about 2 and 1 nm, respectively. The channel length can be decreased to about 1 nm as long as there is an unperturbed region in the middle with a near-midgap state, whereas the upper limit on the channel length can be bound by the scattering length, which can be in micrometer range for graphene. Similarly, the width can be varied as well which will result in a different gate voltage applied to achieve similar device characteristics. The Laplace’s potential due to the drain bias (Vd) is included as a linear voltage drop. The Hartree potential is ignored for simplicity, since it does not affect the device operating principle, although it may affect the quantitative results. The choice of a simple model allows us to study the device and the circuit characteristics in terms of the modulation factor α and the residual bandwidth BWo. The transport characteristics of EMTs with different channel materials can thus be conveniently casted into this model very efficiently.
where I is an identity matrix and UL is the Laplace’s potential drop. Self-energies and broadening functions are and Γs,d = i[Σs,d − Σs,d+], respectively. are the contact Fermi functions. μs,d are source/drain chemical potentials. μd is shifted due to drain bias as μd = μo − qVd and μs = μo, where μo is the equilibrium chemical potential.
Results and discussion
We further report the output characteristics in Figure 2b for Vg = 0.04, 0.08, 0.12, 0.16, and 0.2 V, which show a negative differential resistance (NDR) behavior that is crucial for the low-power inverter operation (Additional file 1). The current cut-off mechanism is similar to the Bloch condition through minibands in superlattices, giving rise to an NDR event, when the drain voltage exceeds the miniband width [15, 16]. The miniband in superlattices is formed by the overlap of quantized states through tunnel barriers, inherently leading to small miniband widths and large effective masses . The NDR events mediated by minibands have been reported in III-V heterostructures  and graphene superlattices . However, the peak-to-valley ratio in such structures is limited to about 1.1 to 1.2. In comparison, the NDR feature reported for near-midgap state in this work shows a peak-to-valley ratio of greater than 103, which is important for the low-power operation. The reason behind a higher ratio could be the formation of a midgap state band without any tunnel barriers, giving rise to higher conduction to yield a large peak current compared to the role of tunnel barriers in the miniband conduction, which lead to a small peak current due to quantum mechanical tunneling. On the other hand, the near-midgap state in this work is highly sensitive to the edge geometry. Therefore, achieving high material quality (with defect density less than parts per billion) is imperative for a proper operation of the proposed transistor. Moreover, the bandwidth of the near-midgap state is gate-voltage dependent; the Vd corresponding to peak and valley currents increases with increasing gate bias Vg due to a larger conduction window. Such peculiar drain voltage-dependent transport features are not exclusive for this device. In a three-terminal device, the electrostatics due to the drain bias introduces various non-trivial effects, e.g., pinch-off in FETs, etc.
To understand these device characteristics further, we report the drain bias dependence of the transmission window in Figure 2c for a gate voltage of 0.2 V. Without any drain bias, a wide transmission window is observed, which monotonically decreases with increasing bias (see Additional file 1 for further discussion). It is more interesting to look at the product of the transmission and the Fermi function difference of source/drain contacts T(E)[fs − fd]. With the increasing bias, since the Fermi function difference monotonically increases, the overall trend as shown in Figure 2d is observed. Using Equation 2, one can also relate these trends in Figure 2d to the negative differential resistance trends of Figure 2b.
In the reported device, the threshold voltage can be engineered by optimizing the side gate electrostatics to vary the modulation factor α. Yet, another way to change the threshold voltage is by engineering the work function of the side gate materials to create an intrinsic electric field, thereby changing the BWo. n-EMT device characteristics are shown in Figure 2. Similarly, by gate work-function and dielectric engineering, one can also achieve p-EMT characteristics by reversing the gate connections. Moreover, the optical phonon energy in graphene is about 200 meV. The choice of 0.2 V supply voltage allows us to ignore the electron–phonon inelastic scattering in these calculations.
We have reported an all-electronic transistor with low supply voltage based on the electronic structure modulation of a near-midgap state in the channel using an external gate voltage. The device physics, however, may lead to various applications of technological importance. We have shown that one can obtain gain and large on/off channel current ratio with few kBT supply voltage. We envision that the transistors based on the electronic structure modulation can open up a new class of post-CMOS logic devices. The concept is analyzed in zzGNR, provided the challenges related to the atomic control of the graphene nanoribbon edge quality and side gate electrostatics, and ohmic contacts with the near-midgap state can be overcome.
HR is an assistant professor in Electrical and Computer Engineering at the University of Iowa since May 2009. For two years, he was a postdoctoral associate at Cornell University. He received his BS on July 2001 from the University of Engineering and Technology Lahore Pakistan, MSc on December 2002, and Ph.D. on May 2007 from Purdue University. He has received “Magoon Award for Excellence in Teaching” from Purdue University. He is also the recipient of “Presidential Faculty Fellowship” and “Old Gold Fellowship” from the University of Iowa. His research group is focused on “anything that is small” for low-power post-CMOS transistor, spintronics, sensors, and solid-state energy harvesting applications from theoretical, experimental, and computational approaches using graphene, molecule, silicon, novel dielectrics, and carbon nanotube material systems. He has served as an editor of a 600-page book on Graphene Nanoelectronics published by Springer in 2012.
We acknowledge fruitful discussions with E. C. Kan and T. H. Hou about the experimental implementation of the transistor. We are grateful to T. Z. Raza for the computer codes of the tight-binding models. We are also thankful to S. Datta, D. R. Andersen, M. A. Alam, D. Stewart, K. Bernstein, and J. Welser for the useful discussion.
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