IEEE ELECTRON DEVICE LETTERS, VOL. 29, NO. 9, SEPTEMBER 2008
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The Gate Leakage Current in Graphene Field-Effect Transistor Ling-Feng Mao, Xi-Jun Li, Zi-Ou Wang, and Jin-Yan Wang
Abstract—The unique band structure of graphene makes the gate leakage current in a graphene field-effect transistor (FET) different from that in silicon FET. Theoretical investigation in this letter demonstrates that the Fowler–Nordheim tunneling current (TC) in a graphene FET is different from that in a silicon FET. Numerical calculations show that a higher oxide electric field results in larger TC in a graphene FET than that in a silicon FET. This implies that, to ensure a workable graphene FET, a thicker gate oxide is needed to limit the gate leakage current compared to that for a silicon FET. Index Terms—Dielectric films, graphene electronics, metal–oxide–semiconductor field-effect transistors (MOSFETs), tunneling.
II. M ETHODS In an electronic device with two terminals—left and right—for electron injection and collection, according to the 1-D Landauer formula [12], the electric current that originates from the left (right) and goes to the right (left) can be written as JL (E, V, T ) = q v(E)T (E, V )N (E) × fL (E − EF , T )dE JR (E, V, T ) = q v(E)T (E, V )N (E) × fR (E − (EF − qV ), T ) dE
I. I NTRODUCTION
A
S THE conventional silicon technology is approaching its scaling limit, both the academics and industry have been searching for new materials with properties that can be better controlled by the electric field to replace silicon. The most notable candidates of such materials are organic conductors [1] and carbon nanotubes [2]. However, a hexagonal lattice of a single layer of carbon atoms called graphene has recently emerged as both a unique system for fundamental studies of condensed matter and quantum physics [3] and a fascinating building block for future devices in the era of postsilicon [4]–[8]. The unique properties of graphene come from its energy band structure characterized by the zero gap width and a linear energy–momentum relation [8]. Several groups have succeeded in controlling both carrier types and density in graphene using a single back gate [9]–[11]. To explore graphene for such a gate-controlled electronic device, gate leakage current is an important factor of quality for the device reliability. However, there is little discussion on this factor for graphene devices utilizing gate-controlled configurations. In this letter, we investigate the gate leakage current in a graphene field-effect transistor (FET).
Manuscript received March 16, 2008. This work was supported by the National Natural Science Foundation of China under Grant 60606016. The review of this letter was arranged by Editor A. Nathan. L.-F. Mao and Z.-O. Wang are with the School of Electronics and Information Engineering, Soochow University, Suzhou 215021, China (e-mail:
[email protected]). X.-J. Li is with the Department of Materials Science and Engineering, National University of Singapore, Singapore 117574. J.-Y. Wang is with the Institute of Microelectronics, Peking University, Beijing 100871, China. Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LED.2008.2001475
(1)
(2)
where E is the electron energy, v(E) is the velocity of electrons moving in the tunneling direction, N (E) is the 1-D density of states, f is the Fermi–Dirac distribution, T is the temperature, EF is the electrochemical potential, q is the electron charge, and T (E, V ) is the transmission coefficient. The tunneling current (TC) through the ultrathin oxide in a FET can be written as J(E, V, T ) = JL (EF , V, T ) − JR (EF , V, T ).
(3)
As we know, the conductivity in a graphene FET linearly increases with the gate voltage for both polarities, and its Hall coefficient changes sign at Vg ≈ 0. The gate voltage induces a surface charge (electron or hole) density [8], which satisfies n=
ε0 εV Cox V = qS tox q
(4)
where Cox is the capacitance of the gate oxide; S and tox are the area and thickness of the gate oxide, respectively; V is the oxide voltage; and ε0 and ε are the permittivities of free space and the gate oxide, respectively. The area carrier (electron or hole) density can be written as n = αV
(5)
where α ≈ 7.3 × 1010 cm−2 · V−1 has been obtained from the experiment and agrees well with the theoretically estimated value for the surface charge density induced by the electric field effect. For a graphene FET, its Landauer current from substrate to gate can thus be simplified as (6) JL (E, V, T ) = q T (E, V )n(E, V )v(E)dE
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IEEE ELECTRON DEVICE LETTERS, VOL. 29, NO. 9, SEPTEMBER 2008
where n(E, V ) is the 1-D electron density along the tunneling direction as a function of both the electron energy in the conduction band and the gate voltage, and v(E) is the electron velocity perpendicular to the graphene sheet. If the device works at a temperature that is not high enough, for simplicity, we can assume that all the tunneling electrons are located at the bottom of the 2-D conduction band, because the change in the redistribution caused by electron thermal energy along the tunneling direction can be neglected. It is well known that there are two types of electron motion for an electron in a FET. One is the thermal velocity, which is the random motion velocity of an electron due to the thermal energy. Usually, it is quite high (higher than 1 × 107 cm/s) but does not contribute to the net current flow in a FET. The other is the drift velocity, which is the motion velocity of an electron driven by an electric field. The drift velocity is usually smaller in magnitude than the thermal velocity but contributes to the net current. Although this drift velocity dominates the net device current, it has little to do with the TC. It will not be considered, because only gate leakage current is studied in this letter. Meanwhile, it needs to be noted that there is no group velocity of electrons moving in the tunneling direction in a graphene FET. Thermal velocity can be used as the electron velocity in Landauer’s formula. Thus ε0 εV JL = T (E, V ) tox d
kB T m∗
(7)
where m∗ is the effective electron mass in graphene along the tunneling direction, and d is the thickness of the single-layer graphene. The Landauer current from right to left for a graphene FET can be obtained as 4qm∗g π JR = (2π)3
∞
∞ fR (E − EF + qV, T )dE
T (Ez , V )dEz 0
Ez
(8) where m∗g is the effective electron mass in the gate along the tunneling direction, and Ez is the electron energy along the tunneling direction. As we know, if the Fermi–Dirac distribution is assumed, for the TC through the oxide in a silicon FET, (3) can be rewritten as [13] ∞ J= 0
qm∗Si kB T T (Ez , V ) 2π 2 3
1 + exp(Ef −L − Ez ) × ln 1 + exp(Ef −R − Ez − qV )
dEz
(9)
where m∗g is the effective electron mass in the silicon along the tunneling direction, and Ef −L and Ef −R are the Fermi levels of the left and right contacts, respectively. The transmission coefficient can be calculated using the transfer matrix method. Thus, the TC through the gate oxide in both graphene and silicon FET under ballistic transport can be calculated.
Fig. 1.
TC and oxide field curve of a graphene FET.
III. R ESULT AND D ISCUSSION In the calculations, an effective electron mass 0.5 m0 of SiO2 , effective electron mass 0.26 m0 of Si, longitudinal effective electron mass 1.0 m0 of the graphene sheet, temperature of 300 K, relative dielectric constant of the gate oxide of 3.9, oxide thickness of 13 nm, thickness of single-layer graphene of 0.34 nm, barrier height of silicon FET of 3.0 eV, and barrier height of graphene FET of 2.5 eV have been used. We choose a thin gate oxide, because the leakage current becomes serious with shrinking oxide thickness and the thick gate oxide could not meet the issue of gate leakage current. Fig. 1 shows the current–voltage curve of a graphene FET. The transfer matrix method has been used to calculate the transmission coefficient, and the TC has been calculated using (3). From this figure, it is clear that log(J/F ) is linearly dependent on the inverse of the electric field oxide across the gate oxide. Such a linear relationship can be analytical explained in the following. According to Wentzel–Kramer–Brillouin (WKB) [14] approximation, for Fowler–Nordheim tunneling, we can obtain the following analytical formula for the transmission coefficient: 4 2m∗ox φ3/2 (10) TFN (E, V ) = exp − 3 2 q tV ox where m∗ox is the effective electron mass in the gate oxide, is the reduced Planck’s constant, and Φ is the potential barrier height. Usually, the TC from substrate to gate is of orders of magnitude higher than that from gate to substrate for Fowler–Nodheim tunneling when the gate voltage is positive; (7) can be simplified as 4 2m∗ox φ3/2 ε0 εV kB T exp − . (11) J= tox d m∗ 3 2 q tV ox As graphene is a 2-D crystal, it has a 2-D band structure. Free electron mass can be as adopted as the mass along the tunneling direction perpendicular to the graphene sheet. Equation (11) can be rewritten as √ C J = B1 F T exp − (12) F
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At last, the quantum capacitance (basically, the effect of the increase in the Fermi energy in the graphene layer) could be of significant value for thin dielectric layers. One can introduce a step function θ(V ), and let quantum capacitance CQ = θ(V )Cox . Thus, the TC calculated from (7) or (12) after considering the reduction in the barrier height caused by the Fermi energy in the graphene layer multiplies the factor of θ(V ) that can be used to estimate the TC in the case of quantum capacitance. IV. C ONCLUSION
Fig. 2. Comparison of the TC between a graphene and silicon FET. The TC and the temperature curve of a graphene FET are also given.
where F is the oxide electric field, and ε0 ε kB B1 = d m∗ ∗ 1/2 4 (2mox ) C= φ3/2 . 3 q
(13) (14)
For a silicon FET with the same WKB approximation, its Fowler–Nordheim tunneling can be written as [15] C 2 JFN = BF exp − (15) F q 3 m∗Si (16) B= 16π 2 m∗ox φ 4 (2m∗ox )1/2 3/2 φ . C= (17) 3 q It can easily be seen from (12)–(17) that, for a graphene FET, the term at the side from the exponential term of Fowler–Nordheim TC linearly depends on both the square root of temperature and the electric field across the gate oxide, whereas it only linearly depends on the square of the electric field but independent of the temperature in a silicon FET. The exponential term in the analytical expressions remains the same for both the graphene and silicon FETs. Now, we turn to the difference in the TC of a graphene FET and a silicon FET. Fig. 2 depicts the calculated results for each. For comparison, the results obtained by using (9) and the parameters of the graphene FET have also been given in this figure. This figure clearly illustrates that the TC in a graphene FET will be much larger than that in a silicon FET in the regime of high electric field. This figure also shows that a higher oxide electric field leads to a larger TC in a graphene FET than that in a silicon FET. This implies that, for practical applications, a thicker gate oxide is needed to reduce the leakage current in a graphenelike FET. This requirement will limit the ultimate device size for graphene electronics. Fig. 2 also shows the TC as a function of the square root of temperature. It clearly illustrates that TC linearly depends on the square root of temperature, which is described by (7) and (12). Such a dependent relation also implies a possibility of controlling the gate leakage current in a graphene FET via changing working temperature.
In conclusion, the Fowler–Nordheim TC of a graphene FET linearly depends on both the electric field oxide across the gate oxide and the square root of temperature. Numerical calculations demonstrate that a higher gate electric field leads to a larger TC in a graphene FET than that in a silicon FET. The difference between the gate leakage current in a graphene FET and that in a silicon FET originates from the low dimensionality of graphene that is 2-D. It is worth noting that graphene electronics may suffer from an integration scale limit due to the gate material thickness required to reduce the TC. On the other hand, by controlling device-working temperature, this thickness requirement may partially be erased. R EFERENCES [1] C. D. Dimitrakopoulos and D. J. Mascaro, “Organic thin-film transistors: A review of recent advances,” IBM J. Res. Develop., vol. 45, no. 1, pp. 11–27, Jan. 2001. [2] R. H. Baughman, A. A. Zakhidov, and W. A. de Heer, “Carbon nanotubes: The route toward applications,” Science, vol. 297, no. 5582, pp. 787–792, Aug. 2002. [3] A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater., vol. 6, no. 3, pp. 183–191, Mar. 2007. [4] C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science, vol. 312, no. 5777, pp. 1191–1196, May 2006. [5] A. Rycerz, J. Tworzydlo, and C. W. J. Beenakker, “Valley filter and valley valve in graphene,” Nature. Phys., vol. 3, no. 3, pp. 172–175, Mar. 2007. [6] M. C. Lemme, T. J. Echtermeyer, M. Baus, and H. Kurz, “A graphene field effect device,” IEEE Electron Device Lett., vol. 28, no. 4, pp. 282– 284, Apr. 2007. [7] B. Huard, J. A. Sulpizio, N. Stander, K. Todd, B. Yang, and D. Goldhaber-Gordon, “Transport measurements across a tunable potential barrier in graphene,” Phys. Rev. Lett., vol. 98, no. 23, pp. 236 803.1– 236 803.4, Jun. 2007. [8] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Crigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science, vol. 306, no. 5696, pp. 666–669, Oct. 2004. [9] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Gregorieva, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature, vol. 438, no. 7065, pp. 197–200, Nov. 2005. [10] Y. B. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum hall effect and berrys phase in graphene,” Nature, vol. 438, no. 7065, pp. 201–204, Nov. 2005. [11] H. B. Heersche, P. Jarillo-Herrero, J. B. Oostinga, L. M. K. Vandersypen, and A. F. Morpurgo, “Bipolar supercurrent in graphene,” Nature, vol. 446, no. 7131, pp. 56–59, Mar. 2007. [12] P. F. Bagwell and T. P. Orlando, “Landauers conductance formula and its generalization to finite voltages,” Phys. Rev. B, vol. 40, no. 4, pp. 1456– 1464, Jul. 1989. [13] Y. Ando and T. Itoh, “Calculation of transmission tunneling current across arbitrary potential barriers,” J. Appl. Phys., vol. 61, no. 4, pp. 1497–1502, Feb. 1987. [14] D. Bohm, Quantum Theory. Englewood Cliffs, NJ: Prentice-Hall, 1951. [15] Z. A. Weinberg, “On tunneling in metal–oxide–silicon structure,” J. Appl. Phys., vol. 53, no. 7, pp. 5052–5056, Jul. 1982.
