JOURNAL OF APPLIED PHYSICS 101, 073709 共2007兲

Oscillation of gate leakage current in double-gate metal-oxide-semiconductor field-effect transistors V. Nam Doa兲 and P. Dollfus Institut d’Electronique Fondamentale, Bâtiment 220—UMR8622, CNRS, Université Paris Sud, 91405 Orsay, France

共Received 3 November 2006; accepted 2 February 2007; published online 13 April 2007兲 Using the nonequilibrium Green’s function method, gate current characteristics are investigated for nanometer-scaled double-gate metal-oxide-semiconductor field-effect transistor. The mode-space approximation is, at the first stage of the calculation, used to obtain self-consistently the potential profile and the charge distribution in the structure. This solution is then used to solve the two-dimensional transport equation to extract the desired quantities. In addition to the dependence of the gate-leakage current on the gate bias and on the oxide thickness, our calculation shows the oscillation behavior of the leakage current versus the drain voltage. It is explained as the result of the strong quantization of electronic states inside the device, giving a resonant-like character to the tunneling of charges from source and drain contacts to the gates. This effect is strongly dependent on the gate length. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2716874兴 I. INTRODUCTION

Scaling down the size of field-effect transistors 共FETs兲, one has to face with arising of quantum effects that may strongly affect their operation and performance. Since charges can move from source to drain by tunneling through the channel, the off-current value and the subthreshold swing may be dramatically degraded.1 Additionally, because of the quantum mechanical confinement of the electron states in the channel, the carrier density is low at the Si/ SiO2 interface and reaches its maximum inside the Si film, contrary to the classical picture,2,3 which results in a reduction of gate capacitance. As a consequence of ultra-thin gate oxide requirements, electrons can tunnel to the gate, creating the so-called gate-leakage current that is considered as one of the most severe challenge to take up for next complementary metaloxide-semiconductor 共CMOS兲 generations, especially regarding the power consumption. To date, a lot of experimental4,5 and theoretical works have focused on this problem. On the theoretical point of view, both semiclassical6 and quantum mechanical approaches have been developed 共see Refs. 2, 7,8, and 9, and references therein兲. Except for the semiclassical approach of Ref. 6 coupled with two-dimensional 共2D兲 Monte Carlo simulation of transistor, most of them are based on the one-dimensional description of tunneling from the quasibounded states in the inversion layer of MOS capacitors, which is accurate but not enough to understand the gate current behavior in the full range of transistor operating conditions. In this article we present a new interesting feature of the gate leakage current in ultra-thin double-gate metal-oxidesemiconductor field-effect transistors 共DG MOSFETs, depicted in Fig. 1兲: the oscillation versus the drain bias. As explained later, this phenomenon is a consequence of the fact that electrons tunnel out-of/into quantized electronic states inside the device through the gate oxide layer. Besides, we

systematically investigate the dependence of this current on the gate voltage, the oxide thickness, and the gate length. This study is based on the nonequilibrium Green’s function 共NEGF兲 formalism,10,11 which is considered as one of the most powerful tools to treat the carrier transport in nanoscaled FETs since quantum mechanical effects can be rigorously included. In small devices, where the transport may be considered as essentially ballistic, the NEGF method simply leads to a current formula similar to that of the Landauer formalism.11,12 However, it still costs expensive computational time and memory for the exact treatment of 2D problems. Therefore, efficient approximations have been developed as the so-called mode-space approach.13,14 As proved elsewhere, e.g., in Refs. 13 and 15, this approximation works well in DG-MOSFET with body thickness less than 5 nm. Unfortunately, it does not allow to investigate the oxideleakage effects because of difficulty of treating the boundary condition for the wave function. To overcome this obstacle we propose an efficient procedure to extract the essential information on the gate-leakage current as follows: First, in the mode-space approximation, a self-consistent procedure is used to find out the potential profile and the carrier distribution in the device under a given transport condition. Second, the resulting self-consistent potential is used to solve exactly the quantum transport equation, i.e., to determine every 2D Green’s function, from which desired quantities are extracted. Practically, this procedure is simple but it guarantees

FIG. 1. 共Color online兲 Schematic of the DG MOSFETs.

a兲

Electronic mail: [email protected]

0021-8979/2007/101共7兲/073709/6/$23.00

101, 073709-1

© 2007 American Institute of Physics

Downloaded 13 Apr 2007 to 129.175.97.14. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

073709-2

J. Appl. Phys. 101, 073709 共2007兲

V. N. Do and P. Dollfus

a potential profile very close to the exact one and it strongly reduces computational requirements compared to a full 2D treatment. The paper is organized as follows: Sec. II is devoted to the description of the model and its treatment using the NEGF approach. The gate-leakage current is reformulated in an appropriate form for the aim of numerical calculation. Section III presents the numerical results and their discussions. The last section is the conclusion. II. FORMALISM

As usually done, we consider a model describing the transport of electrons in a DG MOSFET 共see Fig. 1兲 in the ballistic regime using this standard effective mass Hamiltonian13,14 H=兺 ␯

−

再冋 冉 冊 冉 冊 冉 冊冎 −

ប2 ⳵ 1 ⳵ ប2 ⳵ 1 ⳵ − + Ec共x, y兲 2 ⳵ x mx␯ ⳵ x 2 ⳵ y m␯y ⳵ y

ប2 ⳵ 1 ⳵ 2 ⳵ z mz␯ ⳵ z

,

册

共1兲

where ␯ 共running from 1 to 6兲 is the valley index of the ␯ silicon electronic structure; mx/y/z are the effective masses along the three direction in the valley ␯; Ec共x , y兲 is the bottom of conduction band, which includes the discontinuity of the conduction band between the oxide and the silicon, and the potential U共x , y兲 created by all charges in the device. For simplicity, we now ignore the band index ␯. The three-dimensional Hamiltonian before is actually decoupled into two independent parts, the two first terms and the last one, wherein the eigenstates of the latter are simply the plane wave and the associated eigenvalues 具z兩k典 =

1

冑L z

eikz ; ⑀k =

ប 2k 2 . 2mz

共2兲

The spirit of the mode-space approximation is to completely decouple the Hamiltonian by choosing an appropriate representation basis for the two first terms, for example 兵兩i典 丢 兩␣i典其, where i denotes the position xi along the OX direction in the real space 共real-space representation兲 and 具y 兩 ␣i典 = ␾␣i 共y兲 is the eigenvector of the second term in Eq. 共1兲 with x = xi, associated with the eigenvalue E␣i ⬅ E␣共xi兲,

冋

−

冉 冊

册

ប2 ⳵ 1 ⳵ + Ec共xi, y兲 ␾␣i 共y兲 = E␣i ␾␣i 共y兲. 2 ⳵ y my ⳵ y

共3兲

So that, if we neglect the i dependence of ␾␣i 共y兲, in the basis 兵兩i典 丢 兩␣i典 丢 兩k典其 the Hamiltonian Eq. 共1兲 is completely decoupled and the transport of carriers can be described by a onedimensional 共1D兲 effective Hamiltonian Heff = −

冉 冊

ប2 ⳵ 1 ⳵ + 关⑀k + E␣共x兲兴. 2 ⳵ x mx ⳵ x

共4兲

The quantum transport equation is, in principle, written in terms of three independent Green functions, for example the retarded 共Gr兲, advanced 共Ga兲, and lesser 共G⬍兲 functions. By denoting T as the matrix form of the first term of Heff, the retarded Green function matrix is given by

关G␣r 共E, k兲兴 = 关共E − ⑀k − E␣兲I − 关T兴 − 关⌺rS共E, k兲兴 r − 关⌺D 共E, k兲兴兴−1 ,

共5兲

r where 关⌺S共D兲 兴 is the retarded self-energy matrix describing the coupling between the source 共drain兲 and the reservoir. These self-energy matrices can be calculated exactly r 关⌺S共D兲 共E, k兲兴i, i⬘ = t2x gS共D兲共E, k兲␦k, k⬘␦i, i⬘␦i, 1共N兲 ,

共6兲

tx = ប2 / 2mx␯a2

where is the hopping parameter between the device and the contact; gS共D兲共E , k兲 is the reservoir-surface Green function, which satisfies this normal algebra equation 2 − ␭S共D兲共E, k兲gS共D兲 + 1 = 0, t2x gS共D兲

共7兲

with the convenient sign for the root to guarantee the Green function to be finite. In this equation ␭S共D兲共E , k兲 = E − ⑀共k兲 + 2tx − E␣共x1共N兲兲.16 The advanced Green function and the advanced selfenergies are Hermitian adjoints of the corresponding retarded matrices: Ga = 关Gr兴+ and ⌺a = 关⌺r兴+. The lesser Green function is thus expressed as G␣⬍共E, k兲 = =

兺

a G␣r 共E, k兲⌺⬍ c 共E, k兲G␣共E, k兲

共8兲

兺

⌫c␣共E, k兲兩G␣r 共E, k兲兩 2 f c共E兲,

共9兲

c=S, D

c=S, D

␣ r a = i关⌺S共D兲 − ⌺S共D兲 兴 and f c is the Fermi function aswhere ⌫S共D兲 sociated with the contact c. To determine the conduction band profile Ec共x , y兲, one needs to calculate the electron density which is given by this formula

关ne兴i, j =

冑 冕

1 ab ⫻

2mzkBT 兺 ␲2ប2 c=S, D

dE兩␾␣i共j兲兩2关Ac␣共E兲兴iF− 1 2

冉 冊

␮c − E , k BT

共10兲

E = E − ⑀ k; 关Ac␣共E兲兴i where we have introduced ␣ r 2 = 关⌫c 共E兲兴ic兩关G␣共E兲兴i,ic兩 ; a and b are the grid spacings along the OX and OY directions. We then put the electron density into the Poisson equation to take out the potential profile U共x , y兲. Once the the conduction band is self-consistently calculated, we use it to solve Eq. 共1兲 in the full real space, i.e., considering every Green function as a four-index matrix as for instance 关G␣r 共E, k兲兴i, i⬘ → 关Gr共E, k兲兴i, j; i⬘, j⬘ ,

共11兲

and Eq. 共5兲 is replaced by r 共E, k兲 关Gr共E, k兲兴 = 关共E − ⑀k兲I − H − ⌺rS共E, k兲 − ⌺D

− ⌺Tr共E, k兲 − ⌺Br共E, k兲兴−1 ,

共12兲

r where 关⌺T共B兲 共E , k兲兴 is the self-energy matrix describing the coupling between the top 共bottom兲 gate and the device. These coupling self-energy matrices are generally determined through the surface Green functions, which, in Eq. 共6兲, becomes a two-index matrix satisfying this matrix equation

Downloaded 13 Apr 2007 to 129.175.97.14. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

073709-3

J. Appl. Phys. 101, 073709 共2007兲

V. N. Do and P. Dollfus

关gT−1共E, k兲兴i; i⬘ = 共E − ⑀k − 2ty − UG i 兲␦i; i⬘ − 关Tx兴i; i⬘ − t2y 关gT共E, k兲兴i; i⬘ ,

共13兲

where ty = ប2 / 2m␯y b2 is the hopping parameter between the top gate and the device; and 关Tx兴 is the matrix describing the kinetic energy on the top-gate surface. We distinguish the two cases of metal and polysilicon gate by setting UG = 0 for the former case and UG i = Ui,1共Ny兲 for the latter one. To extract the gate-leakage current, we use the Landauer formula by noting that the current is due to the contribution of all carriers from the source and the drain tunneling through the oxide layer to the gate. Accordingly, IG is characterized by the transmission coefficients TS/G and TD/G and it can be expressed as IG = IS/G + ID/S, where IS共D兲/G = e

冑

2mzkBT ␲ 3ប 4

− F− 1 2

冉

冕

␮G − E k BT

冋 冉

dETS共D兲/G共E兲 F− 1

冊册

2

,

␮S共D兲 − E k BT

冊

共14兲

and Ny

iG

2

G r 2 ⌫S共D兲 兺 1, j ⌫i, 1兩G1, j; i, 1兩 . j=1 i=i

TS共D兲/G = 兺

FIG. 2. 共Color online兲 Gate leakage current density in the 关6/2/0.5兴 device at various gate voltages, from 0.1 to 0.6 V 共upward兲. Temperature: 300 K.

共15兲

G1

Here, for simplicity, we just keep the diagonal elements of the tunneling rates and iG1, iG2 denote the coordinates of the gate ends. III. RESULTS AND DISCUSSION

Using the earlier simulation procedure we analyze the gate leakage current in DG-MOSFETs. For short, the simulated structures are denoted 关Lg / TSi / Tox兴, where Lg, TSi, and Tox, given in nanometers, are the gate length, the silicon body thickness, and the gate oxide thickness, respectively. Source/channel and channel/drain junctions are assumed to be abrupt with source 共S兲 and drain 共D兲 regions uniformly doped to 1020 cm−3 and not overlapped by the gate. We have considered the leakage current in case of poly-silicon gate and we have realized that the gate current always oscillates as a function of drain bias in the regime of weak inversion. However, when increasing the gate voltage the oscillation is suppressed and, as a general behavior, the current tends to decrease when increasing the drain bias. Although the gate has to work as a conductor rather than a semiconductor, in order to highlight the phenomenon we essentially present below the results for the case of a work function corresponding to a weakly doped gate, wherein the phenomenon can be clearly described. However, at the end of the article we make a brief comparison of the gate current in the two cases of weakly and heavily doped gates for the same operating state of the device. In Fig. 2 the gate leakage current in 关6/2/0.5兴 is plotted versus the drain bias for various gate voltages. Obviously, pronounced oscillations occur at small VGS and suppress at high VGS. Before going to explain such features of the gate leakage current we plot in Fig. 3 the distribution of this current along the gate length. Note that, for instance when

VDS = 0 V, in the two cases of low and high gate voltage, the shape of this current distribution is different. This is because the gate surface electronic states contribute differently to the leakage current. When VGS is low, obviously only the lowest level of such states is important, thus the distribution of IG takes the simple form shown in Fig. 3共a兲. However, when VGS is high, IG get the multihump form as shown in Fig. 3共b兲 since several states contribute to the current. When increasing the drain bias, the area limited by the curve strongly varies on a large portion of the gate surface. It is obviously due to the drain-induced drop of potential in the channel. Particularly, one can realize the oscillation of such area in Fig. 3共a兲 while it decreases monotonically in Fig. 3共b兲. This behavior is in agreement with the results shown in Fig. 1. We can define a particular value of energy, denoted as Eoff, such that electrons with E ⬍ Eoff hardly tunnel to the gate. In the case of weakly doped gate, i.e., the gate Fermi level E f is below the bottom of conduction band Ec in the gate, Eoff can

FIG. 3. 共Color online兲 Distribution of leakage-current along the gate at various drain biases, VDS = 0.0 V 共circles兲, 0.08 V 共squares兲, 0.14 V 共diamonds兲, 0.2 V 共up-triangles兲, and 0.3 V 共down-triangles兲. 共a兲 Weak inversion regime 共VGS = 0.1 V兲 and 共b兲 strong inversion regime 共VGS = 0.6 V兲.

Downloaded 13 Apr 2007 to 129.175.97.14. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

073709-4

V. N. Do and P. Dollfus

FIG. 4. 共Color online兲 Leakage-current spectral density in the 关6/2/0.5兴 device at two typical values of gate voltage, 共a兲 0.1 and 共b兲 0.6 V. Temperature: 300 K.

be well identified with Ec, but in the case of heavily doped gate Eoff may be roughly defined in the inferior vicinity of E f 共Eoff ⬍ E f 兲, so that the number of unoccupied states 1 − f G at Eoff is small. Additionally, one should notice the fact that Eoff does not depend on VDS but on VGS only. i In the local picture of the gate leakage current IG , where i denotes the position along the gate, its oscillation versus VDS is straightforward. The maxima correspond to the cases of alignment between Eoff and one of the quantized levels in the channel and the minima occur when Eoff is in between two such levels. This picture resembles that in a double barrier structure 共DBS兲 when increasing the bias.16 Comparing these two pictures, Eoff apparently plays a role similar to the emitter conduction band in the DBS, i.e., the tunneling cutoff energy. The total gate leakage current can be separated as IG = Iground + Iosci, where Iground and Iosci are due to the contribution of local currents that are insensitive and sensitive to VDS, respectively. When the gate voltage is high enough, the channel potential barrier is low so that at the source-end and on a large part of the channel the quantized energy levels are quite flat and weakly sensitive to VDS. The contributions to Iosci only come from the drain-end of the channel, which results in an oscillatory part Iosci much smaller than Iground. In contrast, when the gate voltage is low, the channel potential barrier is high but short channel effects make it sensitive to VDS on the whole channel length. As a consequence, almost the full gate area provides contributions to Iosci. The magnitudes of Iground and Iosci become comparable and the gate leakage current significantly oscillates as a function of VDS. To strengthen the mechanism analyzed earlier we plot in Fig. 4 the spectral density of this current in the two typical cases of weak 共VGS = 0.1 V兲 and strong 共VGS = 0.6 V兲 inversion regime at various drain voltages. Figure 4共a兲 shows significant peaks and their harmonious changing versus the drain bias. The evolution of IG共E兲 vs VDS concretely seems to be: the peaks for VDS = 0.14 V, denoted as 共0.14兲␣ for short, move to 共0.20兲␣, and 共0.20兲␣+1 become 共0.30兲␣; the value of Eoff in this case is about 0.26 eV, i.e., close to the main peak

J. Appl. Phys. 101, 073709 共2007兲

FIG. 5. 共Color online兲 Gate length effect on the leakage current plotted in the strong inversion regime 共VDS = 0.1 V兲, Lg = 3 nm 共lozenges兲, Lg = 6 nm 共squares兲, and Lg = 12 nm 共circles兲. Temperature: 300 K.

of IG共E兲, which explains why IG regularly oscillates. Moreover, this global picture can be deeper understood if we note that the position of the peak 共0.14兲1 is mostly the same as that of 共0.30兲1 共these two VDS values correspond to the current maxima while the minimum is for VDS = 0.20 V, see Fig. 1兲. The main contribution to IG virtually comes from the gate portion close to the source where it is weakly sensitive to the drain bias. The other part of the channel is sensitive to VDS and gives the oscillatory contribution to IG. In contrast, when VGS = 0.6 V, there exist very thin peaks localized at high energy relative to Eoff ⯝ −0.23 eV 关see Fig. 3共b兲兴. Obviously, such peaks of the leakage current spectral density simply contribute to the magnitude of the total current but do not cause significant oscillations. From Fig. 3, the tendency of decreasing leakage current as a function of VDS is straightforward to understand. The fact is that the gate leakage current is due to the contribution of two particle fluxes from the source and from the drain. When VDS = 0 V, these two flux components are equal in their magnitude but they are opposite. Thus there is, of course, no drain current but IG is maximal. When increasing VDS, since the potential in the channel is dropped in the drain-end of the channel, the flux from the drain decreases, and may even cause a negative contribution to IG. Meanwhile the flux contributing to IG from the source mostly does not change 共since there is no potential drop in the source-end of the channel兲 in spite of increasing total particle flux. Consequently, it results in the decreasing of the leakage current as VDS increases. The effect of gate length on the current oscillation for VGS = 0.1 V is shown in Fig. 5. We consider the gate length of 3, 6, and 12 nm, all other things being equal 共Tox = 0.5 nm, TSi = 2 nm兲. The result shows that the amplitude of the oscillations are strongly affected by the gate length as it influences the potential barrier in the channel. For a gate length of 12 nm, the short channel effects can be neglected and the channel potential is influenced by VDS only at the drain-end of the channel that becomes the only position in the channel giving rise to oscillations of the current. As a consequence,

Downloaded 13 Apr 2007 to 129.175.97.14. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

073709-5

J. Appl. Phys. 101, 073709 共2007兲

V. N. Do and P. Dollfus

FIG. 6. 共Color online兲 共a兲 Gate leakage current density vs gate voltages in the 关6 / 2 / Tox兴 devices 关Tox = 0.5, . . . , 1.0共nm兲兴 with drain voltage VDS = 0.0 V. Temperature: 300 K. 共b兲 Dependence of gate leakage current on the oxide thickness for VDS = 0 V and VGS = 0.6 V.

the oscillation amplitude is smaller and the current peak spacings are greater than those previously obtained for Lg = 6 nm. In contrast, for Lg = 3 nm, the enhanced short channel effects increase the amplitude of oscillations and they seem to mix some current peaks. Furthermore, the short channel effects are so strong here that the electron injection in the channel increases under the effect of VDS, which enhances the gate current and the amplitude of oscillation. Our calculation also reemphasizes the conclusions about the dependence of the leakage current on the gate voltage, as well as the exponential law of IG as a function of Tox in the important case of VDS = 0 V, as shown in Fig. 6. Such results were already derived using semiclassical models6 or quantum mechanical 1D models2,7,9 and importantly they are confirmed by experimental data.5 However, according to our results here one can conclude that using semiclassical and/or quantum 1D models is not appropriate to fully describe the elegant properties of the gate leakage current in the ultrathin and ultrashort MOSFET structures. We now plot in Fig. 7 the gate leakage current in the two cases of weakly and heavily doped gate. In the former case, in order the device to be in the same operating state as that of the latter one, a higher gate voltage is to be applied and the gate current magnitude is also greater. Besides, the negative contribution of the particle fluxes to the gate current is also revealed in Fig. 7. In the latter case 共high gate doping兲, there exists a range of high drain voltage where the gate current is negative. Physically, at high VDS, electrons are accelerated by the electric field and almost all of them then move to the drain. Consequently, the difference between local Fermi level in the channel and that of the gate E f decreases. The gate leakage current, therefore, decreases as VDS increasing as previously discussed in the article. A flux of electrons can even tunnel from the gate through the gate oxide layer to the channel in the drain-end of the channel. One also notices that the current peak spacings in the two cases are not the same since the relative position of Eoff to the quantized energy levels in the two cases is different. We should mention that

FIG. 7. 共Color online兲 Leakage current in two cases of weakly and heavily doped polysilicon gate. The devices 共关6/2/0.5兴兲 are in the same operating state in the weak inversion regime. Temperature: 300 K.

the model used here neglects the accumulation and depletion effects in the polysilicon gate. However, these effects are currently on the way of studying and we do not expect them to qualitatively change the results obtained here. IV. CONCLUSION

In conclusion, we have presented some important investigations of the gate leakage current in double-gate MOSFET structures using the NEGF technique. Once again, this technique proves its power to describe quantum mechanically the properties of an open system, which is extremely important to understand the operation and performance of nanoscaled devices. The problem of device-reservoirs coupling is exactly treated in the Green’s function approach in terms of the self-energies while it is a big obstacle in other methods. Our quantum mechanical treatment shows for the first time the oscillation of the gate leakage current versus the drain bias. This interesting feature is then explained using a picture which is similar to the resonant tunneling through double barrier structures. The effects of gate length as well as the gate doping regimes are also discussed. Additionally, our results reconfirm the available conclusions about the gate bias and oxide thickness dependences of the gate leakage current, which were previously deduced using both quantum and semiclassical 1D models, in qualitative agreement with the experimental data. ACKNOWLEDGMENTS

This work has been partially done with the support of the European Community under Contract No. IST-506844 共NoE SINANO兲. The authors would like to thank V. L. Nguyen, from the Institute of Physics and Electrics, Hanoi, Vietnam, for helpful discussions. 1

J. Wang and M. Lundstrom, Tech. Dig. - Int. Electron Devices Meet. 2002, 707 共2002兲. 2 E. Cassan, P. Dollfus, S. Galdin, and P. Hesto, IEEE Trans. Electron Devices 48, 715 共2001兲. 3 L. Ge, F. Gamiz, O. G. Workman, and S. Veeraraghavan, IEEE Trans.

Downloaded 13 Apr 2007 to 129.175.97.14. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

073709-6

Electron Devices 53, 753 共2006兲. H. S. Momose, S.-I. Nakamura, T. Ohguro, T. Yoshitomi, E. Morifuji, T. Morimoto, Y. Katsumata, and H. Iwai, IEEE Trans. Electron Devices 45, 691 共1998兲. 5 K. J. Yang and C. Hu, IEEE Trans. Electron Devices 46, 1500 共1999兲. 6 E. Cassan, S. Galdin, P. Dollfus, and P. Hesto, J. Appl. Phys. 86, 3804 共1999兲. 7 R. Clerc, A. Spinelli, G. Ghibaudo, and G. Pananakakis, J. Appl. Phys. 91, 1400 共2002兲. 8 A. Gehring and S. Selberherr, IEEE Trans. Device Mater. Reliab. 4, 306 共2004兲. 9 E. P. Nakhmedov, K. Wieczorek, H. Burghardt, and C. Radehaus, J. Appl. Phys. 98, 024506 共2005兲. 4

J. Appl. Phys. 101, 073709 共2007兲

V. N. Do and P. Dollfus 10

C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James, J. Phys. C 4, 916 共1971兲. 11 Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512 共1992兲. 12 S. Datta, Superlattices Microstruct. 28, 4 共2000兲. 13 R. Venugopal, Z. Ren, S. Datta, M. S. Lundstrom, and D. Jovanovic, J. Appl. Phys. 92, 3730 共2002兲. 14 M. Bescond, K. Nehari, J. L. Autran, N. Cavassilas, D. Munteanu, and M. Lannoo, Tech. Dig. - Int. Electron Devices Meet. 2004, 617 共2004兲. 15 A. Svizhenko, M. P. Anantram, T. R. Govindan, and B. Biegel, J. Appl. Phys. 91, 2343 共2002兲. 16 V. N. Do, P. Dollfus, and V. Lien Nguyen, J. Appl. Phys. 100, 093705 共2006兲.

Downloaded 13 Apr 2007 to 129.175.97.14. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp