A simple compact model for long-channel junctionless Double Gate MOSFETs

Solid-State Electronics 80 (2013) 28–32

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A simple compact model for long-channel junctionless Double Gate MOSFETs François Lime ⇑, Ernesto Santana, Benjamin Iñiguez Universitat Rovira i Virgili (URV), ETSE DEEEA, Avda. Països Catalans 26, 43007 Tarragona, Spain

a r t i c l e

i n f o

Article history: Received 19 March 2012 Received in revised form 13 September 2012 Accepted 14 October 2012 Available online 19 December 2012 The review of this paper was arranged by S. Cristoloveanu Keywords: Compact model DG MOSFET Drain current Junctionless Long channel

a b s t r a c t This paper presents a simple explicit compact model for the drain current of long channel symmetrical junctionless Double Gate MOSFETs. Our approach leads to very simple equations compared to other models, while retaining high accuracy and physical consistency. Explicit and analytical solutions are also given. Compared to TCAD simulations, the model gives excellent results in accumulation regime. Although the accuracy decreases in depletion regime for very high doping and semiconductor thicknesses, it still remains very good and it is shown that this issue can be neglected because it can only be seen on devices with both high doping and semiconductor thicknesses, that are unlikely to be used as a real device, because of their negative threshold voltage. Finally, it is shown that the model reproduces the two observed different conduction modes, related to accumulation and depletion regimes and that the effective gate capacitance and threshold voltage are different in those regimes, which explains the change of slope observed in the Id(Vg) characteristics. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Since CMOS scaling is approaching its limits, a lot of interest has been put lately into post-CMOS devices. The junctionless transistor is one of those. It consists in a multi-gate depletion and accumulation mode field effect transistor, ideally a nanowire, without source and drain p–n junctions. This device presents some theoretical advantages compared to classic CMOS, like a simpler fabrication process, a volume conduction for higher mobilities and drive currents, and a simpler scaling rule [1,2]. Some models for this device have already been presented, but they feature complicated formulations and in some cases accuracy problems [3–6]. For instance, an extensive description of the device was given in [3], but at the expense of a somewhat complicated formulation. On the other hand, a much simpler model was presented in [4] but only the bulk current is described. This paper presents a compact model for the long channel junctionless symmetrical Double Gate MOSFET, valid for depletion and accumulation, and consisting of simpler physically based equations, for better understanding of this device, and also easier implementation and better computation speed as a compact model. 2. Model The goal is to develop a core model using the same charge based formulation presented in [7] for the junctionless device presented ⇑ Corresponding author. Tel.: +34 977 25 6190; fax: +34 977 55 9605. E-mail address: [email protected] (F. Lime). 0038-1101/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.sse.2012.10.017

Fig. 1. Thus we are considering a long channel device which means that Poisson equation reduces to its 1D form. Poisson equation for a N-type device, with a doping impurities concentration Nd is written:

    @ 2 w qN d wV  1 ; ¼ exp @x2 esc mt

ð1Þ

with mt = kBT/q. Integrating from the middle of the silicon film x0 = tsc/2, where the electric field is 0, to the surface x = 0, we obtain:

  2       @w 2mt qNd ws  V w0  V w  w0  exp  s ; ¼ exp  @x x¼0 esc mt mt mt ð2Þ where ws and w0 are the potentials at the surface and the middle of the semiconductor film. From the boundary conditions, we have, for the symmetrical case:

ws ¼ V g 

Qm  C ox

Q dop 2

¼ V g 

QT ; C ox

ð3Þ

where QT is the algebraic sum of the fixed and mobile charges densities Qdop/2 and Qm, with Qdop = qNdtsc, V g ¼ V gs  V FB , with VFB = /ms + mt ln (Nd/ni). Qm as defined here is half of the total mobile charges in the channel. It is interesting to see that compared to the intrinsic case, Qdop/2 can be considered as an offset to the mobile charge values, due to the doping fixed charges. For junctionless devices, negative values of QT corresponds to the depletion regime,

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Q m ¼ Q cp Q dop exp

  ws  V 1  B exp QT mt



Q T 2Q cp

 ð9Þ

;

qffiffiffiffiffiffiffiffiffiffiffiffi

 Q Qm . However, this equation gives a with B ¼ 1 þ 12 p 4Qdopcp exp  2Q cp singularity at flat-band. To avoid it, (9) is further approximated as:

 Q T   ws  V 1  B exp 2Q cp Q m ¼ Q cp Q dop exp ; mt Q T  2Q cp lnðBÞ

Fig. 1. Schematic representation of the DGMOS junctionless device.

while positive values corresponds to accumulation. For traditional MOSFETs, QT as expressed in (3) is always positive. Assuming a parabolic potential profile in the semiconductor, we have:

 2 w w t sc wðxÞ ¼ s 2 0 x  þ w0 : t sc 2

ð4Þ

2

Applying Gauss theorem around the semiconductor, at the interface with the gate oxide (see Fig. 1):

 @w @x 

¼

Q T

x¼0

esc

ð5Þ

:

Injecting (4) in (5), it is found:

ws  w0

mt

Q

¼

Q m  2dop QT ¼ ; 2Q cp 2Q cp

ð6Þ

with Qcp = 2Cscmt. Replacing (6) in (2) seems to be quite reasonable at this point, and is what was done for inversion-mode DG MOSFETs [8]. However, for junctionless devices, this is not true anymore. Indeed, we noticed that it gives accurate results in accumulation but overestimates the threshold voltage. This occurs when the doping level is high for a given value of tsc. In other words, when Qdop  Qcp, which is generally the case for this device. To avoid this, Qm in (6) must be expressed in terms of the potential at the middle of the channel w0, but only for sub-threshold regime. This can be achieved in the following manner: for the depletion case, w0 > ws and if a parabolic potential profile is considered, then the mobile charge exhibits a Gaussian distribution centered in the middle of the channel. The mobile charge density in this particular regime can then be obtained integrating Boltzmann’s distribution as:

  wðxÞ  V dx mt 0   1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w V  ; pQ dop Q cp exp 0 2 mt

Qm ¼

Z

tsc 2

qN dop exp

ð7Þ

so that the fraction is always positive, the term 2Qcpln(B) being negligible for high doping. With all of the previous approximations, (10) now appropriately reduces to (7) below threshold. Applying to (10) the boundary condition (3) with (6), it yields:

  1 0 Q Q  dop   2Q cp 1  B exp  m2Q cp2 C B 2Q m Q C; þ m  ln B m ¼ ln A @ Q dop Q dop C ox mt  2Q cp lnðBÞ Qm 

ws  w0

mt



1 2

 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q Qm pQ dop Q cp exp w0mV  2Q þ Q m  dop 2 t cp 2Q cp

;

ð8Þ

where the term Qm/2Qcp has been added to cancel the Qm given by (7) above threshold, extending its range of validity to the partially depleted regime, so that (8) reduces to (6). The Qm in (8) was kept to ensure the good behavior above threshold. Replacing in (2) gives:

ð11Þ

2

Q dop

V þ

V

where m ¼ g 2Cmtox contains all the voltages dependences, so that (11) can be solved by plotting m(Qm), with Qm varying from a very low value below threshold to a sufficiently high value above threshold (1015 C/cm2 to 3Coxmt, for example). This curve is independent of the applied biases, so it only has to be computed once for each device. Remember that Qm represents half of the mobile charge. Considering drift–diffusion, the drain current for each value of Vgs is easily obtained from the corresponding Qm(m) curve, by interpolation and integration between the two values of m that correspond to the drain and source bias, md and ms respectively (the drain value of m is lower):

Ids ¼ 2mt l

W L

Z

ms

md

Q m ðmÞ dm:

ð12Þ

This solution of the drain current is not fully analytic because it requires a numerical integration, but as it does not require any iterations, it is still quite fast to compute, and the computation speed is still compatible with the requirements for a compact model. However, to address the cases where speed is the most important criterion, an approximated analytical solution is given in the next section. 3. Analytical and explicit solution It is interesting to see that Eq. (11) reduces to two simpler similar relations depending on if we are in accumulation or depletion regime, the last logarithmic term doing the transition. As such, in depletion regime (11) reduces to:

mþ using (4) and (6) in depletion and the following property of GaussR1 pffiffiffi ian distributions: 0 expðax2 Þdx ¼ 12 pa . Here an assumption has been made that the tail of the Gaussian distribution do not overlap with the gate insulator, which is reasonable for the full depletion case considered here. Eq. (7) is then the limiting trend for the mobile charge below threshold, when w0 > ws. To incorporate (7) in (2) and have a better accuracy in sub-threshold regime, we modify (6) as follows:

ð10Þ

  Q dop Qm Q þ m;  lnðBÞ ¼ ln 4Q cp 2Q cp Q eq

ð13Þ

where Qeq = ((Coxmt)1 + (2Qcp)1)1, while for accumulation we have:

m ¼ ln



  Q m 2Q m Q þ1 þ m ; 2Q cp Q dop C ox mt

ð14Þ

where the term 2Qcpln(B) of (11), only relevant at flat-band, has been neglected. If we neglect the quadratic term in (14), which can be justified by the low vertical field in junctionless compared to inversion mode devices, where it cannot be neglected, we can see that these equations are similar but with a threshold voltage shift of

Q dop 8C sc

 mt lnðBÞ between depletion and accumulation regimes

and an effective gate capacitance of

1 ½ðC ox Þ1 þð4C sc Þ1 

for depletion and

Cox for accumulation. Eqs. (13) and (14) can be solved using Lambert function LW. A useful analytical approximation for Lambert function with an error

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F. Lime et al. / Solid-State Electronics 80 (2013) 28–32

This higher precision is needed to compute the mobile charge in all regimes of operations. This one can be analytically obtained solving (11) in the following way:

0

1 Q dop Qa m 2Q cp 4Q cp Q Q e  B e cp dop m A; Q m ¼ Q eq LW@ Be a Q Q eq Q m  2dop  2Q cp lnðBÞ

Fig. 2. Mobile charges Qm(m) with its respective approximations in depletion and acc accumulation regimes Q dep m ðmÞ and Q m ðmÞ, from (15) and (17).

ð19Þ

where Q am is the approximation for the mobile charge in accumulation, as given by (18). This way of solving (11) is justified by the fact that the terms with Q am are only dominant in accumulation. It might be necessary to do one iteration of (19), especially for low channel doping, to obtain the required precision. That is done applying one more time (19) with the previous result of (19) in place of Q am . The drain currents for depletion regime Idep ds and accumulation acc Ids are obtained differentiating (11) to obtain dm as a function of dQ and then analytically carrying the integration in (12) as a function of dQ. The exact mathematical solution involves polylogarithms and discontinuities that make it difficult to evaluate numerically. The problem was solved replacing the discontinuous functions by continuous ones, found by trial and error, that give the same results with only a small error around the transition point between accumulation and depletion. The final result is given below:

Ids ¼ 2

W lmt ðf ðQ s Þ  f ðQ d ÞÞ; L

ð20Þ

with

f ðQ Þ ¼

  Q Q dop =2 Q dop Q2 þ þ 2Q  Q ln 1 þ e 4Q cp 2Q eq 2 1 0 Q dop C B Q 2 C  ln B QQ dop =2 A: @ 4Q cp 1  e 4Q cp

Fig. 3. Validation of the analytical solution on Id(Vd) characteristics. Lines = analytical solution (20). Dashed lines = numerical integration solution from (12).

of 2% can be found in [9]. For depletion, the solution gives the mobile charges in this regime:

Q m  Q eq LW

   2Q cp Q dop ; B exp m þ Q eq 4Q cp

ð15Þ

where the attenuation of B on Qm has been modified to render it explicit in v:

B¼1þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q p dop

4Q cp 2 1 þ

Q eq 2Q cp

1



  : Q ln exp v þ 4Qdopcp þ 1

ð16Þ

For accumulation, neglecting the quadratic term, which is a good assumption for highly doped devices, it is found:

Q m  C ox mt LW



 2Q cp m e : C ox mt

ð17Þ

These approximated solutions are compared with the general one in Fig. 2. This quadratic term, that is only dominant in strong accumulation, can be reintroduced considering that the mobile charge in this regime can be roughly approximated as Coxmtln(em + 1). Thus a better approximation than (17) in the accumulation region can be written as:

Q am

0 1 Q dop C m  ox t lnðem þ1Þ m 4Q cp 2Qcp Q Q e 1  e dop cp A: ¼ C ox mt LW@ C ox mt C ox mt lnðem þ 1Þ  Q dop 2

ð18Þ

It must be stated that this simple analytical drain current model should be compatible with any junctionless DGMOS charge model because it only depends on the mobile charge values at the source and at the drain sides Qs and Qd. These charges are obtained iterating one time (19) as explained below this equation. The validation of this explicit analytical model for the drain current is illustrated Fig. 3. 4. Results The model was compared with TCAD simulations made with exactly the same parameter values, for the long channel case, considering W = L = 1 lm. These simulations were made using a constant mobility of 100 cm2/V s whose same value was used in the model. We compared the model with TCAD simulations because we wanted to check the electrostatic behavior of the model and because process dependent parameters like the mobility would otherwise be considered as complex fitting parameters. Another important issue is the series resistance. Indeed, a standard junctionless DG MOSFET should have the same doping concentration in the source and drain areas and in the channel. However, we noticed that this configuration exhibits very high series resistance for low channel doping. Our model is a long channel one, and, as such, it does not take into account the source and drain configuration. To have a reliable comparison with the model, we then decided to decrease the series resistance as much as possible by increasing the doping in the source and drain regions, knowing that the series resistance can be incorporated as an gate voltage dependent attenuation factor in the mobility [10]. In the simulations, we also tried to have the highest threshold voltage possible. The threshold voltage of these devices depends on the semiconductor layer thickness,

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F. Lime et al. / Solid-State Electronics 80 (2013) 28–32

Fig. 6. Comparison of the model with TCAD simulations for various semiconductor thicknesses, for Id(Vg) characteristics in linear and logarithmic scales. Lines = model. Dashed lines = TCAD.

solutions, one can achieve a very high computation speed if the following approximation for Lambert function is used [9]:

  lnð1 þ lnð1 þ zÞÞ : LWðzÞ  lnð1 þ zÞ 1  2 þ lnð1 þ zÞ

Fig. 4. Comparison of the model with TCAD simulations. Lines = model. Dashed lines = TCAD. (a) Id(Vg) characteristics in logarithmic and linear scales. (b) Id(Vd) characteristics in linear scale.

Fig. 5. Comparison of the model with TCAD simulations for various channel doping, for Id(Vg) characteristics in linear and logarithmic scales. Lines = model. Dashed lines = TCAD.

the doping and the work-function of the gate. A gate work-function of 5.27 eV was chosen in order to have the highest threshold voltage at a given channel doping level. This specific value should align the Fermi level of the gate with the valence band of the semiconductor, in the TCAD software Silvaco–Atlas we used. In the following, the comparisons with TCAD simulations were made relative to the numerical integration of (12). A standard trapezoidal integration worked fine, and led a very good speed. As for the analytical

ð21Þ

We can see from Fig. 4 that compared to TCAD simulations, the model gives very good results for the dimensions that are expected for this device. Only a small discrepancy is observed for the highest gate voltage value on the Id(Vd) characteristic. Fig. 5 shows the dependence with channel doping. An excellent accordance is observed for the lowest doping levels, while small discrepancies start to appear in depletion for very high doping, but the accuracy is still very good. On Fig. 6 we can see the excellent agreement of the model with TCAD simulations, for small semiconductor thicknesses, while small discrepancies start to appear for high thicknesses. To our knowledge, all the models for this device that have been reported so far exhibit a decreased accuracy in the same conditions. However, this is not an important issue because it is only seen on devices that cumulate high doping and thick semiconductor thickness, that are unlikely to be used as a real device, because of their very negative threshold voltage. Comparing with [3], that is among the most accurate models for this device, the accuracy is equivalent for the threshold voltage: our model tends to slightly underestimate the current, while [3] tends to overestimate it. In accumulation our model has an equivalent and maybe better accuracy, and for partial depletion the overestimation of the current is a little better for [3]. Our model has the main advantage of being much simpler, and also propose analytical approximated solutions. From these approximated solutions and from Fig. 2, it can be seen that the drain current in junctionless devices consists in the transition from a depletion mode that corresponds to volume conduction, to an accumulation mode that corresponds to surface conduction. Each mode has its own effective gate capacitance and threshold voltage, which is the reason why a change of slope in the Id(Vg) characteristics is observed, at the transition point between depletion and accumulation. In fact from Eqs. (15) and (17), comparing with the junction based case well above threshold, we can deduce that for accumulation the effective gate capacitance Q

and threshold voltage are respectively Cox and V FB  2Cdop  ox

 cp 1 mt ln 4Q and V , while for depletion they are FB  Q dop ððC ox Þ1 þð4C sc Þ1 Þ

 Q dop Q dop 4Q cp . As can be seen in Figs. 5 and 6, this means  8C sc  mt ln Q 2C ox dop

that the Id(Vg) characteristics in linear scale can exhibit two

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F. Lime et al. / Solid-State Electronics 80 (2013) 28–32

different slopes, whose transition point depends on the doping and the semiconductor thickness. Although this model has been developed for symmetrical Double Gate MOSFET, it could eventually be used for Gate-All-Around and other multi-gate FETs using the approach presented in [11]. 5. Conclusion We presented a simple compact model for the symmetric junctionless DG MOSFET transistor. The model is physically based with relatively simple equations, and show excellent agreements with TCAD simulations. In addition, an approximated explicit analytical solution is given. Furthermore, it was shown that the model reproduces the two observed different conduction regimes, accumulation and depletion and that the effective gate capacitance and threshold voltage are different in those regimes, which explains the change of slope observed in the Id(Vg) characteristics. Acknowledgements This work was supported by the following research projects: European Union (EU) COmpact MOdeling Network (COMON) Industry Academia Partnerships and Pathways under Project FP7IAPP-218255, and EU Silicon Quantum Wire Transistors (SQWIRE) under Project FP7-ICT-STREP 257111.

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