An analytical approach to calculate effective channel - Semantic Scholar
Microelectronics Reliability 53 (2013) 540–543
Contents lists available at SciVerse ScienceDirect
Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel
An analytical approach to calculate effective channel length in graphene nanoribbon field effect transistors M. Ghadiry a,⇑, M. Nadi b, M. Bahadorian c, Asrulnizam ABD Manaf a, H. Karimi d, Hatef Sadeghi e a
School of Electrical and Electronic Engineering, Engineering Campus, Universiti Sains Malaysia, Penang, Malaysia Department of Computer Engineering, Ashtian Branch, IAU, Ashtian, Iran c Institute of Advanced Photonics Science, Nanotechnology Research Alliance, Universiti Teknologi Malaysia, 81310 Johor Bahru, Malaysia d Japan International Institute of Technology (MJIIT), Universiti Teknologi Malaysia, 54100 Kuala Lumpur, Malaysia e Lancaster Quantum Technology Center, Physics Department, Lancaster University, LA1 4YB Lancaster, UK b
a r t i c l e
i n f o
Article history: Received 12 September 2012 Received in revised form 23 November 2012 Accepted 3 December 2012 Available online 14 February 2013
a b s t r a c t A compact analytical approach for calculation of effective channel length in graphene nanoribbon field effect transistor (GNRFET) is presented in this paper. The modelling is begun by applying Gauss’s law and solving Poisson’s equation. We include the effect of quantum capacitance and GNR’s intrinsic carrier concentration in our model. Based on the model the effects of several parameters such as drain-source voltage, channel length, and oxide thickness are studied on the length of effective channel in GNRFETs. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction By using graphene nanoribbon it seems to be possible to make devices with channels that are extremely thin and will allow FETs to be scaled to shorter channel lengths and higher speeds without encountering the adverse short-channel effects restricting the performance of the existing devices. As a result, high performance logic circuits such as high speed full adders could be realized [1,2]. Recently, experimental and theoretical studies such as [3–6] have shown it is possible to fabricate GNR transistors. As a result, many researchers have been attracted to this field and provided several models for GNR’s properties [7–12]. Nevertheless, there is lack of research in modelling the behaviour of GNRFET near the drain junction and the breakdown mechanism. The effective channel length is one of the most important parameters of MOSFETs showing the portion of the channel contributing to the properties of MOSFETs such as I–V characteristic. In order to calculate the effective channel length, the width of the drain region, where impact ionization and carrier velocity saturation occur, has to be computed. It controls the lateral drain breakdown [13,14], substrate current, hot-electron generation [15,16], and drain current at the drain region [17,18]. Although several models are available for saturation region of silicon-based MOSFETs such as [19,20,16,21,22], there is still plenty of room for research in modelling of this region for car-
bon-based FETs. In order to gain insights into reliability issues of these devices, close analysis of this region is necessary. In addition, these kinds of models open the way to explore the possibility of designing power transistors using carbon. Since at the moment the fabrication technology is still at its very first steps, analytical modelling seems to be a useful tool in the case of examining saturation region. In this paper,s simple and compact analytical models for surface potential, lateral electric field and effective channel length are proposed and the behaviour of a general top-gate GNRFET in the saturation region is studied.
2. Effective channel length model for GNRFET with top gate A schematic cross-section of top-gated GNRFET is shown in Fig. 1, where tox is the oxide thickness of top gate with dielectric constant of ox; tg, W and L are the GNR’s thickness, width and the channel length respectively. The channel is divided into two sections. Section 1 is defined between drain and saturation point and Section 2 between saturation point and source junction. We begin with applying Gauss’s law in the Section 1, shown in Fig. 1.
q
Z 0
x
Z
tg
ðn þ NÞ dx dt ¼
0
Z
x
ox nox
dx
þ
tg
g n0
dt
0
0
Z
Z
tg
g nx
dt
ð1Þ
0
⇑ Corresponding author. Tel.: +60 127427906. E-mail address: [email protected] (M. Ghadiry). 0026-2714/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.microrel.2012.12.002
where q is the charge magnitude, g and ox are the graphene and oxide dielectric constants, n is the intrinsic carrier concentration,
M. Ghadiry et al. / Microelectronics Reliability 53 (2013) 540–543
ðC q þ C g ÞC ox C q þ C g þ C ox
C tg ¼
541
ð8Þ
it is found that
ðC g þ C q ÞV g þ /ch ð0Þ /ch ðV g Þ ¼ V g 1 C q þ C g þ C ox
where Cq is the quantum capacitance of the channel which is given by
Fig. 1. Schematic cross section of a top-gated GNRFET.
@n @E
C q ¼ q2 N is doping concentration, nox,n0, and nx are the oxide, saturation, and lateral electric fields. Taking derivation over 1 yields
@ 2 /1 ðxÞ V g V bi /1 ðxÞ qðN þ nÞ þ ¼ @x2 g k2
ð2Þ
where /1(x) is the surface potential of GNR at any point along the x direction inside thep Section 1 (0 < x < DL), and Vg is the gate voltage. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The parameter k ¼ g tg t ox =ox is relevant length scale for potential variation [13]. Built-in voltage Vbi in GNR with a bandgap Eg = htF/ 3w is written as
htF N V bi ¼ V T ln n 6qW
ð3Þ
6
where tF 10 m/s is the Fermi Velocity, and VT = KT/q is thermal voltage. Since quantum capacitance is significant in nanoscale transistors, we need to include the effect of that in the surface potential model. As the enclosed charge in the Gauss’s surface is given by
Q ¼q
Z 0
tg
Z
x
ðn þ NÞ dt dx;
ð4Þ
qn can be replaced by qðn þ NÞ ¼ tQg x. Therefore the surface potential is written as
ð5Þ
n2D ¼
Z
Q ¼ ðC g þ C q ÞðV fb þ /ch V sub Þtg
ð6Þ
where C g ¼ tg is the GNR capacitance, Cq is the quantum capacitance of the channel, Vfb is flat band voltage,Vsub is the substrate voltage and /ch is the self consistent potential in the central region of the channel. The concentration of the holes has been neglected here. In order to calculate /ch, one can use total capacitance seen from the top gate. According to Fig. 2, Ctg is given by
where C ox ¼ toxox is the gate capacitance. Using
DOSðf ðE EFd Þ f ðE EFs ÞÞ dE
ð7Þ
ð11Þ
0
where for i = s,d,
f ðE EFi Þ ¼
1 EE F 1 þ exp KT i
ð12Þ
We approximate EFs = EF and EFd = EF qVds and the density of states DOS(E) is given by [23]
DOS ¼
2me
ð13Þ
ph2 1EE ; n2D is written as
Replacing fi ¼
n2D ¼
Z
þ1
1þexp
Fi KT
DOSðfd fs Þ dE
ð14Þ
0 1 finally, n ¼ n2D btg þt c, with tint being the interlayer distance of int graphene is written and limited as
n¼
Z gd Z gs 1 2me fd dE fs dE 2 t g þ t int ph 0 0
C q ¼ q2
g
@/ C tg ¼ C ox 1 ch @V g
þ1
where gi ¼
The charge Q can also be calculated from
ð10Þ
where E is the energy. The /ch(0) can be approximated to / 2L where /(x) is the surface potential at any point along the channel. This term will be addressed later in this paper. The two-dimensional carrier concentration n2D is written as
0
@ 2 /ðxÞ V g V bi /ðxÞ Q þ ¼ @x2 g t g x k2
ð9Þ
EF i ðEg =2Þ . KT
ð15Þ
Finally, quantum capacitance is given as
1 2me ðf fs Þ t g þ t int ph2 d
ð16Þ
Now we can proceed to calculate surface potential and lateral electric field analytically. Boundary conditions for /1(x) are defined as /1(0) = V0 + Vbi, /1(DL) = Vbi + Vds, n1(0) = n0, where V0, Vds, DL, and n0 are the saturation voltage at the onset of saturation region, drain voltage, length of saturation region and saturation surface electric field respectively. Solving the differential equation and taking
A¼
Q V g V bi tg g x k2
ð17Þ
yields
h x i x 1 þ ðV 0 þ V bi Þ cosh þ kn0 /1 ðxÞ ¼ k2 A cosh k k x sinh k
ð18Þ
Since n(x) = @/(x)/@x, surface electric field distribution n1(x) is expressed as
x x V 0 þ V bi sinh n0 cosh n1 ðxÞ ¼ kA þ k k k
Fig. 2. Equivalent circuit of device electrostatics.
ð19Þ
In order to model the surface potential /2(x) between source and saturation point (Section 2), we apply Gauss’s law at the region 2 with boundary conditions of n2(0) = n0 and /2(0) = V0 + Vbi.
542
M. Ghadiry et al. / Microelectronics Reliability 53 (2013) 540–543
Assuming that DL < L/2, /(L/2) = /2(L/2 DL). As a result /ch(0) is expressed as
L 2 DL 2 k A0 k2 A0 þ ðV 0 þ V bi Þ 2k L 2DL L 2 DL cosh þ kn0 sinh 2k 2k
/ch ð0Þ ¼ cosh
ð20Þ
where A0 at Vg = 0 V is defined as
A0 ¼
Q
g t g x
þ
V bi
ð21Þ
k2
To calculate DL, Eq. (18) can be numerically solved at x = DL. As a result, the effective channel length LE = L D L is given as
LE ¼ L
sinh DkL V 0 k V ds k2 A cosh DkL 1 ðV 0 þ V bi Þ cosh DkL
ð22Þ
Fig. 4. The length of velocity saturation region (LVSR) with different channel lengths and drain voltages.
According to [24,22,13] the electric field at Section 2 can be assumed to be linear. As a result it is concluded that @ 2/(LE)/@x2 = n0/LE. In addition, it is assumed that /2(0) = V0 + Vbi = n0LE + Vbi. Therefore using Poison’s equation again, n0 is given as
n0 ¼
LE k2 L2E k2
Q V g 2V bi þ g t g x k2
ð23Þ
3. Simulation results and discussion In this section, the results calculated from presented equation is presented. Table 1, shows the values employed in calculation. Fig. 3 shows the electric field distribution in the lateral direction with the different drain voltages. The values for Vds have been chosen based on saturation voltage V0, which has been almost 0.23 V in this simulation. This figure indicates that the potential distribution along the nanoribbon surface is similar to the profile of an abrupt junction at the edges of p-base/drift region and drain/drift region junctions [25]. In other word, it shows the electric field profile follows an exponential form depending on the distance from the source. Fig. 4 depicts the effect of the drain voltage and channel length on the length of saturation region. The higher the drain voltage and longer channel are, the longer the DL is. In addition, the Figure shows that the ratio of DL/L increases as L decreases. In Fig. 5 it is shown that increasing oxide thickness causes decrease in LE. In addition, the term d(LE)/d(Vds) increases as tox increases. To put in nutshell, using empirical equation extracted from presented charts and equations, the dependence of effective channel length on controllable parameters can be simply given as
Fig. 5. The effective channel length vs. oxide thickness variations at different drain voltages.
LE /
L:N:ðC q þ C gnr Þ V ds :tox :Ld
ð24Þ
4. Conclusion A simple model for the effective channel length of graphene nanoribbon FET is presented in this paper. The model is developed by solving Poison’s equation in two different sections of the channel. Quantum capacitance and intrinsic carrier concentration were modelled for the studied device and included in the surface potential model. Using the proposed models, we plotted the profile of quantum capacitance, lateral electric field, effective channel length, and length of velocity saturation l = region. In addition, the effects of device parameters such as drain-source voltage, channel length, and oxide thickness were examined on the length of effective channel. We showed that in the presented GNRFET, the effective channel can occupy up to half of the channel and decreases as Vds, and tox increases.
Appendix A
Fig. 3. Lateral electric field at different drain voltages.
Typical simulation parameters used in this paper is given in Table 1.
M. Ghadiry et al. / Microelectronics Reliability 53 (2013) 540–543 Table 1 Typical value for parameters used in simulation. Some values such as channel length and GNR’s width may treat as variable. Parameter name
Symbol and value
Doping concentration Carrier concentration Fermi velocity Carbon–carbon distance Graphene layer’s interlayer distance GNR’s thickness GNR width GNR dielectric constant Charge magnitude Temperature Bolzman’s constant Planck’s constant Thermal voltage SiO2 dielectric constant Vacuum permittivity Oxide thickness Channel length Gate voltage
tF = 106 m/s a = 0.14 109 m tint = 0.34 109 m tg = 0.39 nm W = 5 nm g = 3.5 e0 F/m q = 1.6 1019 C 300 K K = 1.38 1023 J/K h = 6.6 1034 J s VT = KBT/q V ox = 3.9 e0 F/m 0 = 8.85 1012 F/m tox = 2 nm L = 20 nm Vg = 0.1 V
References [1] Ghadiry MH, AbdManaf A, Ahmadi MT, Sadeghi H, Nadisenejani M. Design and analysis of a new carbon nanotube full adder cell. J Nanomater 2011;2011:1–6. [2] Ghadiry M, Nadi M, A’Ain A. Dlpa: discrepant low pdp 8-bit adder. Circ, Syst, Signal Process 2012:1–14. http://dx.doi.org/10.1007/s00034-012-9438-6. [3] Wang Y, Huang B-C, Zhang M, Woo JC. Op timizing the fabrication process for high performance graphene field effect transistors. Microelectr Reliab 2012;52(8):1602–5. ICMAT 2011 – Reliability and variability of semiconductor devices and ICs. doi:10.1016/j.microrel.2011.09.036. .. [4] Moslemi MR, Sheikhi MH, Saghafi K, Moravvej-Farshi MK. El ectronic properties of a dual-gated gnr-fet under uniaxial tensile strain. Microelectr Reliab 2012;52(11):2579–84. http://dx.doi.org/10.1016/j.microrel.2012.05.009. . [5] Lemme MC, Echtermeyer TJ, Baus M, Kurz H. A graphene field-effect device. Electr Dev Lett, IEEE 2007;28(4):282–4.
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[6] Berger C, Song Z, Li X, Wu X, Brown N, Naud C, et al. Electronic confinement and coherence in patterned epitaxial graphene. Science 2006;312(5777):1191–6. [7] E.V. Castro, K.S. Novoselov, S.V. Morozov, N.M.R. Peres, J.M.B. Lopes dos Santos, Johan Nilsson, F. Guinea, A.K. Geim, A.H. Castro Neto, Electronic properties of a biased graphene bilayer, J. Phys.: Condens. Matter 22 (17). [8] Dorgan VE, Bae M-H, Pop E. Mobility and saturation velocity in graphene on sio2. Appl Phys Lett 2010;97(8):082112–21. [9] Ghadiry M, Manaf ABA, Nadi M, Rahmani M, Ahmadi M. Ionization coefficient of monolayer graphene nanoribbon. Microelectr Reliab 2012;52(7):1396–400. [10] Geim AK, Novoselov KS. The rise of graphene. Nat Mater 2007;6(3):183–91. [11] Ghadiry MH, Ahmadi MNS M, Manaf AA. A model for length of saturation velocity region in double-gate graphene nanoribbon transistors. Microelectr Reliab J 2011;51:2143–6. [12] Giovannetti G, Khomayakov PA, Brocks G, Karpan VM, van den Brink J, Kelly PJ. Doping graphene with metal contacts. Phys Rev Lett 2008;101(2):026803. 1–4. [13] Wong H. Drain breakdown in submicron mosfets: a review. Microelectr Reliab 2000;40(1):3–15. [14] Wong H. A physically-based mos transistor avalanche breakdown model. IEEE Trans Electr Dev 1995;42(12):2197–202. [15] Arora ND, Sharma MS. Mosfet substrate current model for circuit simulation. IEEE Trans Electr Dev 1991;38(6):1392–8. [16] Fang FF, Fowler AB. Hot electron effects and saturation velocities in silicon inversion layers. J Appl Phys 1970;41(4):1825–31. [17] Gildenblat G, Xin L, Wu W, Hailing W, Jha A, van Langevelde R, et al. Psp: an advanced surface-potential-based mosfet model for circuit simulation. IEEE Trans Electr Dev 2006;53(9):1979–93. [18] Baum G, Beneking H. Drift velocity saturation in mos transistors. IEEE Trans Electron Dev 1970;17(6):481–2. [19] Dang LM. Drain-voltage dependence of igfet turn-on voltage. Solid-State Electr 1977;20(10):825–30. [20] El-Mansy YA, Boothroyd AR. A simple two-dimensional model for igfet operation in the saturation region. IEEE Trans Electr Dev 1977;24(3):254–62. [21] Kim I-J, Matsumoto S, Sakai T, Yachi T. Analytical approach to breakdown voltages in thin-film soi power mosfets. Solid-State Electr 1996;39(1):95–100. [22] Krizaj D, Charitat G, Amon S. A new analytical model for determination of breakdown voltage of resurf structures. Solid-State Electr 1996;39(9):1353–8. [23] Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, et al. Electric field effect in atomically thin carbon films. Science 2004;306(5696): 666–9. [24] Banna ME, Nokali ME. A pseudo-two-dimensional analysis of short channel mosfets. Solid-State Electr 1988;31(2):269–74. [25] Yang W, Cheng X, Yu Y, Song Z, Shen D. A novel analytical model for the breakdown voltage of thin-film soi power mosfets. Solid-State Electr 2005;49(1):43–8.
Contents lists available at SciVerse ScienceDirect
Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel
An analytical approach to calculate effective channel length in graphene nanoribbon field effect transistors M. Ghadiry a,⇑, M. Nadi b, M. Bahadorian c, Asrulnizam ABD Manaf a, H. Karimi d, Hatef Sadeghi e a
School of Electrical and Electronic Engineering, Engineering Campus, Universiti Sains Malaysia, Penang, Malaysia Department of Computer Engineering, Ashtian Branch, IAU, Ashtian, Iran c Institute of Advanced Photonics Science, Nanotechnology Research Alliance, Universiti Teknologi Malaysia, 81310 Johor Bahru, Malaysia d Japan International Institute of Technology (MJIIT), Universiti Teknologi Malaysia, 54100 Kuala Lumpur, Malaysia e Lancaster Quantum Technology Center, Physics Department, Lancaster University, LA1 4YB Lancaster, UK b
a r t i c l e
i n f o
Article history: Received 12 September 2012 Received in revised form 23 November 2012 Accepted 3 December 2012 Available online 14 February 2013
a b s t r a c t A compact analytical approach for calculation of effective channel length in graphene nanoribbon field effect transistor (GNRFET) is presented in this paper. The modelling is begun by applying Gauss’s law and solving Poisson’s equation. We include the effect of quantum capacitance and GNR’s intrinsic carrier concentration in our model. Based on the model the effects of several parameters such as drain-source voltage, channel length, and oxide thickness are studied on the length of effective channel in GNRFETs. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction By using graphene nanoribbon it seems to be possible to make devices with channels that are extremely thin and will allow FETs to be scaled to shorter channel lengths and higher speeds without encountering the adverse short-channel effects restricting the performance of the existing devices. As a result, high performance logic circuits such as high speed full adders could be realized [1,2]. Recently, experimental and theoretical studies such as [3–6] have shown it is possible to fabricate GNR transistors. As a result, many researchers have been attracted to this field and provided several models for GNR’s properties [7–12]. Nevertheless, there is lack of research in modelling the behaviour of GNRFET near the drain junction and the breakdown mechanism. The effective channel length is one of the most important parameters of MOSFETs showing the portion of the channel contributing to the properties of MOSFETs such as I–V characteristic. In order to calculate the effective channel length, the width of the drain region, where impact ionization and carrier velocity saturation occur, has to be computed. It controls the lateral drain breakdown [13,14], substrate current, hot-electron generation [15,16], and drain current at the drain region [17,18]. Although several models are available for saturation region of silicon-based MOSFETs such as [19,20,16,21,22], there is still plenty of room for research in modelling of this region for car-
bon-based FETs. In order to gain insights into reliability issues of these devices, close analysis of this region is necessary. In addition, these kinds of models open the way to explore the possibility of designing power transistors using carbon. Since at the moment the fabrication technology is still at its very first steps, analytical modelling seems to be a useful tool in the case of examining saturation region. In this paper,s simple and compact analytical models for surface potential, lateral electric field and effective channel length are proposed and the behaviour of a general top-gate GNRFET in the saturation region is studied.
2. Effective channel length model for GNRFET with top gate A schematic cross-section of top-gated GNRFET is shown in Fig. 1, where tox is the oxide thickness of top gate with dielectric constant of ox; tg, W and L are the GNR’s thickness, width and the channel length respectively. The channel is divided into two sections. Section 1 is defined between drain and saturation point and Section 2 between saturation point and source junction. We begin with applying Gauss’s law in the Section 1, shown in Fig. 1.
q
Z 0
x
Z
tg
ðn þ NÞ dx dt ¼
0
Z
x
ox nox
dx
þ
tg
g n0
dt
0
0
Z
Z
tg
g nx
dt
ð1Þ
0
⇑ Corresponding author. Tel.: +60 127427906. E-mail address: [email protected] (M. Ghadiry). 0026-2714/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.microrel.2012.12.002
where q is the charge magnitude, g and ox are the graphene and oxide dielectric constants, n is the intrinsic carrier concentration,
M. Ghadiry et al. / Microelectronics Reliability 53 (2013) 540–543
ðC q þ C g ÞC ox C q þ C g þ C ox
C tg ¼
541
ð8Þ
it is found that
ðC g þ C q ÞV g þ /ch ð0Þ /ch ðV g Þ ¼ V g 1 C q þ C g þ C ox
where Cq is the quantum capacitance of the channel which is given by
Fig. 1. Schematic cross section of a top-gated GNRFET.
@n @E
C q ¼ q2 N is doping concentration, nox,n0, and nx are the oxide, saturation, and lateral electric fields. Taking derivation over 1 yields
@ 2 /1 ðxÞ V g V bi /1 ðxÞ qðN þ nÞ þ ¼ @x2 g k2
ð2Þ
where /1(x) is the surface potential of GNR at any point along the x direction inside thep Section 1 (0 < x < DL), and Vg is the gate voltage. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The parameter k ¼ g tg t ox =ox is relevant length scale for potential variation [13]. Built-in voltage Vbi in GNR with a bandgap Eg = htF/ 3w is written as
htF N V bi ¼ V T ln n 6qW
ð3Þ
6
where tF 10 m/s is the Fermi Velocity, and VT = KT/q is thermal voltage. Since quantum capacitance is significant in nanoscale transistors, we need to include the effect of that in the surface potential model. As the enclosed charge in the Gauss’s surface is given by
Q ¼q
Z 0
tg
Z
x
ðn þ NÞ dt dx;
ð4Þ
qn can be replaced by qðn þ NÞ ¼ tQg x. Therefore the surface potential is written as
ð5Þ
n2D ¼
Z
Q ¼ ðC g þ C q ÞðV fb þ /ch V sub Þtg
ð6Þ
where C g ¼ tg is the GNR capacitance, Cq is the quantum capacitance of the channel, Vfb is flat band voltage,Vsub is the substrate voltage and /ch is the self consistent potential in the central region of the channel. The concentration of the holes has been neglected here. In order to calculate /ch, one can use total capacitance seen from the top gate. According to Fig. 2, Ctg is given by
where C ox ¼ toxox is the gate capacitance. Using
DOSðf ðE EFd Þ f ðE EFs ÞÞ dE
ð7Þ
ð11Þ
0
where for i = s,d,
f ðE EFi Þ ¼
1 EE F 1 þ exp KT i
ð12Þ
We approximate EFs = EF and EFd = EF qVds and the density of states DOS(E) is given by [23]
DOS ¼
2me
ð13Þ
ph2 1EE ; n2D is written as
Replacing fi ¼
n2D ¼
Z
þ1
1þexp
Fi KT
DOSðfd fs Þ dE
ð14Þ
0 1 finally, n ¼ n2D btg þt c, with tint being the interlayer distance of int graphene is written and limited as
n¼
Z gd Z gs 1 2me fd dE fs dE 2 t g þ t int ph 0 0
C q ¼ q2
g
@/ C tg ¼ C ox 1 ch @V g
þ1
where gi ¼
The charge Q can also be calculated from
ð10Þ
where E is the energy. The /ch(0) can be approximated to / 2L where /(x) is the surface potential at any point along the channel. This term will be addressed later in this paper. The two-dimensional carrier concentration n2D is written as
0
@ 2 /ðxÞ V g V bi /ðxÞ Q þ ¼ @x2 g t g x k2
ð9Þ
EF i ðEg =2Þ . KT
ð15Þ
Finally, quantum capacitance is given as
1 2me ðf fs Þ t g þ t int ph2 d
ð16Þ
Now we can proceed to calculate surface potential and lateral electric field analytically. Boundary conditions for /1(x) are defined as /1(0) = V0 + Vbi, /1(DL) = Vbi + Vds, n1(0) = n0, where V0, Vds, DL, and n0 are the saturation voltage at the onset of saturation region, drain voltage, length of saturation region and saturation surface electric field respectively. Solving the differential equation and taking
A¼
Q V g V bi tg g x k2
ð17Þ
yields
h x i x 1 þ ðV 0 þ V bi Þ cosh þ kn0 /1 ðxÞ ¼ k2 A cosh k k x sinh k
ð18Þ
Since n(x) = @/(x)/@x, surface electric field distribution n1(x) is expressed as
x x V 0 þ V bi sinh n0 cosh n1 ðxÞ ¼ kA þ k k k
Fig. 2. Equivalent circuit of device electrostatics.
ð19Þ
In order to model the surface potential /2(x) between source and saturation point (Section 2), we apply Gauss’s law at the region 2 with boundary conditions of n2(0) = n0 and /2(0) = V0 + Vbi.
542
M. Ghadiry et al. / Microelectronics Reliability 53 (2013) 540–543
Assuming that DL < L/2, /(L/2) = /2(L/2 DL). As a result /ch(0) is expressed as
L 2 DL 2 k A0 k2 A0 þ ðV 0 þ V bi Þ 2k L 2DL L 2 DL cosh þ kn0 sinh 2k 2k
/ch ð0Þ ¼ cosh
ð20Þ
where A0 at Vg = 0 V is defined as
A0 ¼
Q
g t g x
þ
V bi
ð21Þ
k2
To calculate DL, Eq. (18) can be numerically solved at x = DL. As a result, the effective channel length LE = L D L is given as
LE ¼ L
sinh DkL V 0 k V ds k2 A cosh DkL 1 ðV 0 þ V bi Þ cosh DkL
ð22Þ
Fig. 4. The length of velocity saturation region (LVSR) with different channel lengths and drain voltages.
According to [24,22,13] the electric field at Section 2 can be assumed to be linear. As a result it is concluded that @ 2/(LE)/@x2 = n0/LE. In addition, it is assumed that /2(0) = V0 + Vbi = n0LE + Vbi. Therefore using Poison’s equation again, n0 is given as
n0 ¼
LE k2 L2E k2
Q V g 2V bi þ g t g x k2
ð23Þ
3. Simulation results and discussion In this section, the results calculated from presented equation is presented. Table 1, shows the values employed in calculation. Fig. 3 shows the electric field distribution in the lateral direction with the different drain voltages. The values for Vds have been chosen based on saturation voltage V0, which has been almost 0.23 V in this simulation. This figure indicates that the potential distribution along the nanoribbon surface is similar to the profile of an abrupt junction at the edges of p-base/drift region and drain/drift region junctions [25]. In other word, it shows the electric field profile follows an exponential form depending on the distance from the source. Fig. 4 depicts the effect of the drain voltage and channel length on the length of saturation region. The higher the drain voltage and longer channel are, the longer the DL is. In addition, the Figure shows that the ratio of DL/L increases as L decreases. In Fig. 5 it is shown that increasing oxide thickness causes decrease in LE. In addition, the term d(LE)/d(Vds) increases as tox increases. To put in nutshell, using empirical equation extracted from presented charts and equations, the dependence of effective channel length on controllable parameters can be simply given as
Fig. 5. The effective channel length vs. oxide thickness variations at different drain voltages.
LE /
L:N:ðC q þ C gnr Þ V ds :tox :Ld
ð24Þ
4. Conclusion A simple model for the effective channel length of graphene nanoribbon FET is presented in this paper. The model is developed by solving Poison’s equation in two different sections of the channel. Quantum capacitance and intrinsic carrier concentration were modelled for the studied device and included in the surface potential model. Using the proposed models, we plotted the profile of quantum capacitance, lateral electric field, effective channel length, and length of velocity saturation l = region. In addition, the effects of device parameters such as drain-source voltage, channel length, and oxide thickness were examined on the length of effective channel. We showed that in the presented GNRFET, the effective channel can occupy up to half of the channel and decreases as Vds, and tox increases.
Appendix A
Fig. 3. Lateral electric field at different drain voltages.
Typical simulation parameters used in this paper is given in Table 1.
M. Ghadiry et al. / Microelectronics Reliability 53 (2013) 540–543 Table 1 Typical value for parameters used in simulation. Some values such as channel length and GNR’s width may treat as variable. Parameter name
Symbol and value
Doping concentration Carrier concentration Fermi velocity Carbon–carbon distance Graphene layer’s interlayer distance GNR’s thickness GNR width GNR dielectric constant Charge magnitude Temperature Bolzman’s constant Planck’s constant Thermal voltage SiO2 dielectric constant Vacuum permittivity Oxide thickness Channel length Gate voltage
tF = 106 m/s a = 0.14 109 m tint = 0.34 109 m tg = 0.39 nm W = 5 nm g = 3.5 e0 F/m q = 1.6 1019 C 300 K K = 1.38 1023 J/K h = 6.6 1034 J s VT = KBT/q V ox = 3.9 e0 F/m 0 = 8.85 1012 F/m tox = 2 nm L = 20 nm Vg = 0.1 V
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