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A compact and accurate MOSFET model with simple expressions for linear, saturation and sub-threshold regions
Solid-State Electronics 50 (2006) 301–308 www.elsevier.com/locate/sse
A compact and accurate MOSFET model with simple expressions for linear, saturation and sub-threshold regions Hisao Katto
*
Department of Electronic Systems Engineering, Tokyo University of Science, Suwa, 5000-1 Toyohira, Chino, Nagano 391-0292, Japan Received 19 May 2004; accepted 19 January 2006 Available online 20 March 2006
The review of this paper was arranged by Prof. Y. Arakawa
Abstract A compact channel-current model is proposed for the linear, saturation and sub-threshold regions of MOSFETs with eight parameters at the maximum. To derive new formulas both physically reasonable and analytically simple, the core part of the known theories and formulas including BSIM is carefully examined, and the comparison with the exact gradual model is made. A simple formula for the linear region is obtained considering the velocity saturation effect, the bias dependent mobility and the series resistance in the source and drain junctions. It is theoretically predicted and experimentally confirmed that the two new parameters in the denominator strongly depend on the channel length. Simple expressions are additionally advised for the saturation and the sub-threshold regions. By applying the model to a set of devices covering a wide range of channel length, the parameters are extracted, and good agreement between theory and measurement is demonstrated. 2006 Elsevier Ltd. All rights reserved. Keywords: MOSFETs; Parameter estimation; Modeling
1. Introduction The current–voltage (I–V) characteristics of MOSFETs have remained an important subject because of wide applications including circuit design, device design and reliability studies. BSIM of UC Berkeley, now available in the WEB site, accumulates and combines theories on various aspects of the device structure and maintains integrity as an excellent framework to describe the I–V characteristics of a MOSFET. A potential problem of the model will be that it has so many parameters with the intention of achieving ultimate accuracy, and it is sometimes hard for us to extract all the parameters from a device because the parameters are often redundant. For some applications, it is helpful to minimize the number of fitting parameters and have a very compact model like the physical alpha-power *
Tel.: +81 266 73 1201; fax: +81 266 73 1230. E-mail address: [email protected]
0038-1101/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.sse.2006.01.014
law model [1] as long as the accuracy is not excessively degraded. Compact models are important not only for circuit applications but for device and reliability study where one wants to quickly look into the basic device parameters. The model in [1] is analytically simple, and can combine all linear, saturation and sub-threshold regions. Recently, the sub-threshold region is important because the off current became an important topic of circuit design [2]. A drawback of this model will be that the theory is over-simplified by neglecting the series resistance that is practically important as discussed later. The purpose of the present paper is to advise simple formulas that are physically more reasonable than the alphapower law model. One of the basic rules in carrying out the study is that the comparison with, or verification by, the measurement is simple because it is considered an important ingredient of compactness. The linear region is the most important part of a MOSFET I–V model. One of the approaches taken was to
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H. Katto / Solid-State Electronics 50 (2006) 301–308
carefully re-examine the existing theories to extract only the core part without excessively sacrificing accuracy. A combined analysis of the bias dependent mobility and the vertical and horizontal electric field effects is made, and a simple formula is derived for the linear region. The effect of parasitic resistance, R, is naturally included in the simple formula. The surface depletion charge is carefully examined by using the exact gradual channel model [3] and it is decided that we can temporarily do without it except the contribution to the threshold voltage. A formula is successfully derived for the boundary between the linear and saturation regions, VDsat. Similarly compact formulas are adopted for saturation and sub-threshold regions. Only the core part of BSIM3 is covered, and the topics like the following are not included: (1) the substrate bias (VB) dependence, (2) such device structure effects as the poly gate depletion effect and the narrow channel effect, and (3) any new topics for ultra-scaled devices in BSIM4. The substrate bias is not discussed because increasing a measurement parameter makes the study strenuous, and omitting it does not seem to degrade the usefulness of the model. The poly-gate depletion and narrow channel effects are not important in our samples with the gate oxide of 5.5 nm and the gate width of 15 lm. It is expected that the model will make a good basis for discussing the additional effects. 2. Theory 2.1. Linear region To discuss the mobility dependent on the gate bias, VG, and the drain bias, VD, the basic equation for the channel current, I, of a MOSFET is given by I ¼ W G Qv
ð1Þ
where WG is the gate width, Q ¼ C OX ðV Gt V Þ
ð2Þ
is the carrier density, VGt VG VT is the gate voltage as measured from the threshold voltage, VT, V is the surface potential along the channel, COX (=eOX/tOX) is the oxide capacitance where eOX is the permittivity of oxide, tOX is the gate oxide thickness, and v is the velocity of the induced carriers. The velocity v is given by [4] v ¼ leff E=ð1 þ E=EC Þ
ð3Þ
where E = dV/dx is the electric field, EC = vsat/leff is the critical field, leff is the effective mobility, and vsat is the saturation velocity that is a constant. The effective mobility is assumed to be leff ¼ l0 =½1 þ h0 ðV Gt V þ DV Þ
ð4Þ
where l0, h0 and DV are constant. Though such a formula as leff = l0/(1 + h0VGt) is widely assumed for simplicity [1], the vertical field should exactly be a function of (VGt V) instead of VGt. The parameter DV (2VT) is added be-
cause, as discussed in BSIM or more exactly in [5], the vertical field becomes zero not at VG = VT but at another gate bias VG = VT DV (VT 2VT = VT). Finally, note that BSIM proposes a quadratic formula of the form leff ¼ l0 =ð1 þ a1 V Gt þ a2 V 2Gt Þ where a1 and a2 are constant, but for simplicity, we do not include higher order terms of VGt assuming they are not critically important. Combining Eqs. (1)–(4), the differential equation I¼
W G C OX l0 ðV Gt V Þ ðl0 =vsat ÞI E 1 þ h0 ðV Gt V þ DV Þ
ð5Þ
is derived. The integration is performed over the channel length, LG, and the equation I=Ai ¼
1 V D ð1 þ h0 DV Þ=h0 ln f h0 1 þ ðb0 =h0 Þ ln f
ð6Þ
with ln f ln([1 + h0(VGt + DV)]/[1 + h0(VGt VD + DV)]) is derived. Ai = COXl0(WG/LG) and b0 = l0/(LGvsat) are constant. Using the Tailor expansion ln f (h0VD)/A + (h0VD/A)2/2 with A 1 + h0(VGt + DV), and neglecting minor terms, a useful formula: I ffi ðAi =f0 Þ
V Gt V D V 2D =2 1 þ ðh0 =f0 ÞV Gt þ ðb0 =f0 ÞV D þ e
ð7Þ
is derived where f0 1 + h0DV. Note that the simple expression in the denominator, a linear combination of VGt and VD, is derived. It is confirmed that a very similar expression can be derived within BSIM if one carefully manipulates the complex formulas there. In Eq. (7), a constant factor f0, being slightly greater than unity, is added in three places. Since it applies to all three parameters, we can temporarily remove it by rewriting Ai/f0 as Ai, h0/f0 as h0 and b0/f0 as b0, with the understanding that the factor f0 can be restored any time when required. The parameter e (>0) is here temporarily incorporated to mathematically assess the difference between Eqs. (6) and (7), and it is confirmed that e is an increasing function of h0 and is generally small. In the following, Eq. (7) with e = 0 will be used instead of Eq. (6), and the factor f0 will be omitted. The effect of the parasitic resistance, R, at the source and drain terminals, is incorporated by replacing VGt by (VGt RI) and VD by (VD 2RI), and by neglecting the second order terms of I. Then, we have I=Ai ffi
V Gt V D V 2D =2 1 þ hV Gt þ bV D
ð8Þ
where the parameters in the denominator are given by h ¼ h0 þ 2RAi ¼ h0 þ ð2RC OX l0 W G Þ=LG b ¼ b0 RAi ¼ l0 ð1=vsat RC OX W G Þ=LG
ð9Þ
The equations suggest that both h and b depend on LG, and are larger in shorter channel devices. It is easy to confirm that R of as small as 10 X (or 150 X-lm if normalized to the channel width) gives a large h value. On the other hand, the effect of R on b is less great assuming vsat 105 m/s, and the first term l0/(vsatLG) will be the major term of b.
H. Katto / Solid-State Electronics 50 (2006) 301–308
pffiffiffiffiffiffiffiffiffiffi GU S ;
303
2.2. Boundary between linear and saturation regions
U Tfb ¼ U S þ
VDsat to give the boundary between the linear and saturation regions is derived from Eq. (8) by solving oI/oVD = 0, and is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ V Dsat ¼ ð2V Gt Þ=½1 þ 1 þ ð2bV Gt Þ=ð1 þ hV Gt Þ
is newly adopted based on the exact gradual channel model [3]. UTfb = bVTfb and UF = bVF = ln(NA/ni) are nondimensional because they are normalized to kT/q (=1/b), and G ¼ b ð2qeSi N A =C 2OX Þ is the non-dimensional body factor. NA is the substrate doping density, ni is the intrinsic carrier density, and eSi is the permittivity of silicon, k is Boltzmann’s constant, q is the electron charge and T is the temperature. The expression of Eq. (13) is already proposed in [3] as the boundary between sub-threshold and normal regions (U1 in Eq. (32)). The addition of the logarithmic term may not be critically important in discussing the linear region, but is important in deriving the coefficient asub for the sub-threshold region as discussed in the following section. Experimentally, VT is extracted in the linear region with a small VD. Thereby, note that not only the term V 2D =2 in the numerator but also h and b in the denominator in Eq. (8) should be taken into account. Fig. 1 shows the theoretical I–VGt curves assuming typical values of h and b. The offset of VD/2 is visible, but note that the offset is not exactly equal to VD/2. The effect of b is found in the broken lines. For example, for VD = 100 mV, we have an offset of 47 mV instead of 50 mV. The influence of h is detected by comparing the solid and broken lines. In the present paper, we always use VD = 20 mV, and thereby neglect the small offset for simplicity.
If one assumes h = 0 (and R = 0), this reduces to the traditional expression as in [1] or [6] not to speak of the body factor. The denominator increases with VGt, but the growth is not great because of the new term (1 + hVGt), and so VDsat is greater compared with the case of h = 0. With oI/oVD = 0, the I–VD curve may become unnaturally flat at the boundary, but the mathematical smoothing as in BSIM is not used here to simplify the discussion. 2.3. Saturation region In the saturation region as defined by VD > VDsat, the current IDsat is theoretically given by replacing VD by VDsat in Eq. (8). This holds in long channel devices, and IDsat depends only on VG. In shorter channel devices, however, the measured current increases gradually with VD, and we need a simple analytical expression with a minimized number of parameters. In our devices, it is empirically found that the dependence I=I Dsat ¼ 1 þ d ½ðV D V Dsat Þ=V Gt
ð11Þ
roughly holds as far as VGt is not too small. Here, d (1) is a fitting parameter, and is zero for long channel devices. We can briefly discuss the plausibility of Eq. (11) based on BSIM. In BSIM, the saturation region is described through channel-length-modulation (CLM) and drain-induced-barrier-lowering (DIBL) modes, and the analytical expression is given by I=I Dsat ¼ 1 þ ðV D V Dsat Þ=V A
ð13Þ
2.5. Sub-threshold region In the sub-threshold region, the formula I=Ai ¼ ðasub =b2 Þðb0 þ b1 V D Þð1 ebV D Þeðb=gÞV Gt
ð14Þ
0.01
ð12Þ
solid
θ =0.8V–1
broken θ =0 0.008 VD=100mV VD=50mV 0.006
I/Ai
where VA is a rational function of VD and VG. Though the general expression is complex, the formula for VA becomes simple if we consider the DIBL mode only. In the DIBL mode, V A / V 2G =ðV Dsat þ V G Þ roughly holds neglecting minor terms, and then VA / VG holds by assuming VDsat / VG. Eq. (11) covers the entire saturation region, and further adding the CLM mode near VDsat does not seem critically important. We can possibly test such an extended expression as I/IDsat = 1 + d Æ [(VD VDsat)/VGt]Asat with an extra parameter Asat to partly include the substrate current induced body effect (SCBE) region.
U S ¼ 2U F þ lnð2U F Þ
0.004 VD=20mV 0.002
2.4. VT The theoretical definition of threshold voltage, VT, is important mainly for the sub-threshold region where the current is sensitive to ffi VT. Against the popular definition pffiffiffiffiffiffiffiffiffi of U Tfb ¼ U S þ GU S (as measured from the flat band voltage) and US = 2UF, the definition of
0 0
0.1
0.2
0.3
VGt (V) Fig. 1. Theoretical I/Ai–VGt curves with small VD values. b = 1 is assumed. Marks show calculations with the interval of DVGt = 50 mV, and linear curves are fitted against the first six marks.
H. Katto / Solid-State Electronics 50 (2006) 301–308
pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi with g ¼ 1 þ G=U F =2 and asub ¼ ð2U F Þ1=g G=U F =4 is newly adopted. The parameter g is identical to the one adopted in [1], whereas the parameter asub is newly derived using the exact gradual channel model [3] assuming Eq. (13). The coefficient asub (2.8 in our devices) is greater than the corresponding factor adopted in [1] (=g 1.5) mainly because the newly defined VT is larger and should give a larger current. Further, Eq. (14) does not assume the offset voltage adopted in [1]. The analytical derivation of asub will be briefly described in Appendix. The validity of Eq. (14) is confirmed by graphically comparing with the exact model with tOX and NA as parameters. Errors may increase with G, but are tolerably small in the area of interest G 6 20. We have the term (b0 + b1VD) just to empirically express the VD dependence. Note that b0 = 1 and b1 = 0 should ideally hold. In our samples, it is found that the linear factor (b0 + b1VD) explains the VD dependence better than another possible formula of eqV D where q is constant. Eq. (11) is applied e.g. for the bVGt > 2 region, and Eq. (14) applies e.g. for the bVGt < 2 region. In the area between saturation and sub-threshold regions where jVGtj is small (2 < bVGt < 2), logarithmic averaging as proposed in [1] will be effective. 2.6. Charge in surface depletion region If we consider the charge in the surface depletion region [7], Eq. (2) becomes as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ¼ C OX ðV Gt V Þ ðG=bÞ ð2V F þ V Þ ð15Þ We go through integration, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and two new terms are obtained one with 2V F þ V D and the other logarithmic. Adopting up to the third order term of the Taylor expansion for the logarithmic term, we obtain pffiffiffiffiffiffiffiffiffi V D V Gt V 2D =2 þ ð2=3Þ G=b½ð2V F Þ1:5 ð2V F þ V D Þ1:5 ð16Þ as the numerator of Eq. (8). The surface depletion charge (SDC) term in this equation is not small in magnitude, but will not be adopted in our model by the reasons as follows. The most important effect of the SDC term given by Eq. (16) is to cause a ‘‘VT shift’’ of about 0.8 V as shown in Fig. 2. To make the matter clearer, assume that VD ispsmall. ffiffiffiffiffiffiffiffiffi Then, we can simplify the SDC term as ð2=3Þ G=b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ð2V F Þ1:5 ð2V F þ V D Þ1:5 ðG=bÞ ð2V F ÞV D . This is equivalent to the threshold voltagepshift ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffi of DV T ¼ ðG=bÞ ð2V F Þ as given by the term GU S in Eq. (13) because US b Æ (2VF). In other words, Eq. (13) already contains the SDC term, and further adding another DVT through Eq. (16) simply double-counts the term DVT. The SDC term has another effect. The higher order terms of Taylor expansion of the SDC term may not be negligible when VD is large, and we must consider the unabridged SDC term if errors are not tolerated. However, it is reminded that the error can largely be compensated
3 VD=1V
without SDC 2
I/Ai
304
1
with SDC
0
0
1
2
3
4
VGt [V] Fig. 2. The ‘‘VT shift’’ of about 0.8 V due to the SDC effect.
through adjusting the parameters in the denominator, h and b. Further, we have complicacy in including the SDC term: (1) we need a prior knowledge of NA to theoretically predict the SDC term, or need an additional parameter to fit. (2) Including the SDC term in VDsat may complicate the analysis. (3) Another series resistance term should be added in the denominator. We would rather stick to the simpler formula as far as the errors in b, h and VDsat are tolerable and the overall agreement with the experimental I–V characteristics remains satisfactory. 3. Experiment The devices evaluated are NMOSFETs with the dimensions of tOX = 5.5 nm, WG = 15 lm and LG = 0.375 15 lm. The nominal LG value for the product is 0.35 lm, and the nominal power supply voltage is 3.3 V. We have devices with LG 0.375 lm, but the data are withheld because short channel effects may complicate the discussion. The device characteristics are measured at the room temperature using HP4142B. The measurement bias interval is 50 mV for VG whereas VD starts from 20 mV and ends in 4 V with irregular intervals as shown in the figures appearing later. Though the data will not be presented, the devices with tOX = 8 nm and LG = 0.34 lm are also evaluated and the conclusions drawn on the tOX = 5.5 nm devices, especially the LG dependence of h0 and b are confirmed also in these devices. Using a spreadsheet program, Excel, four-step fitting is made between theory and measurement. First, VT is decided at VD = 20 mV by extrapolating the linear portion of the I–VG curve near VT. Then in the linear region, the three parameters h0, b and l0 are decided. Then in the saturation region, the parameter d is decided. The last step is
H. Katto / Solid-State Electronics 50 (2006) 301–308
305
Table 1 Device parameters extracted by comparing the compact model with the tOX = 5.5 nm devices covering a wide LG range of 0.375–15 lm LG (lm) VT (V) l0 (m2/V s) h (V1) b (V1) d g NA (cm3) b0 b1/b0
15 0.600 0.0269 0.0733 0.087 0 1.495 5.40 · 1017 2.921 0.081
4 0.606 0.0284 0.131 0.098 0 1.537 6.18 · 1017 1.955 0.094
2 0.607 0.0349 0.283 0.165 0 1.536 6.36 · 1017 1.482 0.104
0.7 0.624 0.0298 0.239 0.349 0.0636 1.487 5.21 · 1017 2.236 0.141
0.5 0.633 0.0288 0.330 0.383 0.0827 1.483 5.11 · 1017 2.080 0.172
0.375 0.646 0.0317 0.394 0.605 0.105 1.506 5.65 · 1017 1.677 0.207
VT is first extracted, and the next three parameters are extracted in the linear region. d is then extracted in the saturation region, and the remaining parameters are extracted in the sub-threshold region.
4. Experimental results and discussions The parameters extracted for the eight devices with LG = 0.37515 lm are summarized in Table 1. Note that NA is theoretically derived from the exponential slope factor g in the sub-threshold region neglecting the contribution from surface states, and is used to derive the coefficient asub using the equation shown just below Eq. (14). The value of asub is needed simply because b0 is derived only if we know asub. 4.1. VT versus NA In Table 1, VT is only weakly dependent on LG in the LG range shown. It is confirmed that the VT–NA relation is roughly explained by assuming the flat-band voltage VFB = 1. When VT is plotted againstNA, we observe the scattering of ±15% in NA. If we have an error of 15% in NA, we will theoretically have an error of 15% in the parameter G, and then have an error in g of 2.4% and in asub of 1.8%. The relatively small errors in g and asub indicate that the theoretical prediction of the sub-threshold characteristics is not critically influenced by the scattering of NA.
The table shows that the mobility is not strongly dependent on LG, and the average is close to 0.03 m2/V s. The value seems reasonably smaller than the reported bulk value for electrons in p-Si, 0.035 m2/V s (NA = 6 · 1017/ cm3) [8]. The parameters h and b are plotted in Fig. 3 against 1/LG. As expected, they are linearly dependent on 1/LG. Comparing with Eq. (9), h0 0.08 V1, R 21 X and vsat 1.2 · 105 m/s are obtained assuming l0 = 0.03 m2/ V s. h0 is small but may not exactly be zero considering the data scattering. The value of R is similar to the reported value of 27 X for the tOX = 5.9 nm and WG/LG = 20/
0.4
1
θ : 0.119/LG+ 0.081 b : 0.190/LG + 0.062
0.8
0.3
0.6 0.2
b(V–1)
4.2. Linear region
0.2 lm device extracted from the I–VG characteristics at VD = 0.1 V [9]. Note that h for LG = 0.2 lm (gray circle) is not included in drawing the line because it deflects from the line. A large h may possibly come from the parasitic resistance of the needle contact. Since the concept of vertical field dependent mobility is popularly accepted, the idea of small or zero h0 will be discussed in some more detail. Assume h0 = 0 in Eq. (8) and temporarily remove the term containing b that is not needed for the discussion. Then, we have I=Ai ¼ ðV Gt V D V 2D =2Þ=½1 þ f2RC OX l0 ðW G =LG Þg V Gt . This expression is equivalent to the known mobility reduction model with h0 = l0/(2tOXvnorm) [1] if l0/(2tOXvnorm) = 2RCOXl0(WG/LG) or vnorm = (LG/WG)/(4ReOX) practically holds. R = 21 X and LG/WG = 1/15 is equivalent to vnorm 2.3 · 107 m/s. This value is close to 2.2 · 107 m/s assumed in [1]. If R and LG similarly scale, the ratio LG/R is kept constant, and the ‘‘equivalent’’ vnorm value does not change with scaling. The scaling law on R is not accurately known, but R is expected to scale with
θ (V–1)
the sub-threshold region where the three parameters g, b0 and b1 are decided. As for the device dimensions, WG, LG, tOX, design values are adopted. eOX/e0 = 3.9 and eSi/e0 = 11.7 are assumed.
0.4
0.1 0.2
0 0
0.5
1
1.5
2
2.5
3
0
1/LG (µm–1) Fig. 3. Extracted h and b values in Table 1 plotted against 1/LG. Linear relations are observed as suggested by Eq. (9). The h curve gives h0 = 0.08 V1 and R = 21 X, and the b curve gives vsat 1.2 · 105 m/s.
306
H. Katto / Solid-State Electronics 50 (2006) 301–308 0.2
technology (Figs. 6 and 7 [10]). It is very natural to assume that R scales because otherwise h becomes unnaturally great and b can become zero or negative as the device is scaled and COX increases in Eq. (9).
1 δ b1 /b0
0.15
4.3. Saturation region
0.1
0.1
b1/b0
δ
0.0811+0.0426•LG–1.09
–0.152•LOG(LG/1.81) 0.05
As shown in Fig. 4, the parameter d of the LG 6 1 lm devices logarithmically depends on LG [lm] while d is zero in longer channel devices (LG P 2 lm). We can regard 1.81 lm shown in Fig. 4 as the short/long channel boundary on d in this device set. The simple physical model for the logarithmic dependence is left for future study. 4.4. Sub-threshold region
0 0.1
1
0.01
10
L G (µm)
Fig. 4. Parameters d and b1/b0 for the VD dependence in the saturation and sub-threshold regions plotted against LG. For short channel devices, LG < 1 lm, logarithmic dependence of d on LG is observed. Power law dependence of b1/b0 on LG is observed.
As discussed earlier, b0 = 1 should ideally hold. In the table, b0 is closer to 2, but considering the exponential nature of the sub-threshold current, the error is not considered great. The VD dependence factor b1/b0 as shown in Fig. 4 is found a strong function of LG. The simple physical model for this dependence is left for future study. 4.5. Over-all characteristics Figs. 5 and 6 compare the theory with the measurement for the LG = 0.375 lm device. Fig. 5 shows the normal cur-
VGt=3V 1.2
VGt= -0.1V
2.5V 10 -3
1
-0.2V 2V
10 -4
I/Ai
0.8
-0.3V 0.6
10 -5
I/Ai
1.5V
0.4
10 -6
-0.4V
1V VDsat
0.2
10 -7
0.5V
0V 0
0
1
2
-0.5V
10 -8 -0.6V
3
VD (V) Fig. 5. Over-all I–V characteristics for the device with LG = 0.375 lm in the normal current region. Measured data are shown by cross marks with the gate bias interval of 0.1 V and by black circles with the interval of 0.5 V. Theoretical curves are shown with the gate bias interval of 0.5 V. White circles indicate the theoretical VDsat–IDsat relation. Note that the gate bias is measured from VT, and the current is normalized against Ai.
10 -9 0
1
2
3
VD (V) Fig. 6. Over-all I–V characteristics in the sub-threshold region for the same device as shown in Fig. 5.
H. Katto / Solid-State Electronics 50 (2006) 301–308
rent region. In the theoretical curves, the data are calculated only for the biases corresponding to the experimental points and are smoothly connected by the function of the spreadsheet program. Agreement between theory and measurement is good especially if one considers the minimized number of parameters, or compared with the physical alpha-power law model (Fig. 4 in [1]). Fig. 6 shows the I–V characteristics of the sub-threshold region. Good agreement between theory and measurement is obtained also in this region. 5. Concluding summary The new compact MOSFET model has eight parameters to fit, VT, h, b, l0, d, g, b0 and b1. They are extracted without difficulty through the comparison with measurement for the tOX = 5.5 nm devices with LG between 0.375 lm and 15 lm through the four steps of assessing (1) VT and (2) linear, (3) saturation, and (4) sub-threshold regions. Good agreement between theory and measurement is obtained. In the linear region, the simple formula using the two parameters h and b naturally combines the vertical and horizontal electric field effects and the series resistance, R. The parameters h and b are theoretically predicted and experimentally confirmed to depend linearly on 1/LG. It is found that the intrinsic part of h, h0 (0.08 V1), is relatively small and the parasitic resistance R (21 X) mainly accounts for the magnitude of h. This shows that such an approach as the alpha-power law model that neglects the series resistance has a drawback in explaining the LG dependence. Further, a reasonable value of saturation velocity vsat 1.2 · 105 m/s is derived from the b LG dependence data. Only a minimized number of parameters are adopted in other regions so that the model becomes compact. We have at present such empirical factors as R, d, b0 and b1/b0 that can only be obtained through fitting. Theoretical prediction of these factors will be an important subject of future study. Some minor but conceptually meaningful factors are theoretically discussed. Firstly, a constant factor f0 that reflects the vertical electric field is discussed. Secondly, issues on introducing the SDC term is discussed. It is discussed that the error caused by neglecting the higher order terms of SDC are largely compensated by a minor adjustment of other fitting parameters, and so the terms are temporarily neglected to make the theory simple and compact. In the sub-threshold region, the formula for the coefficient asub is newly proposed. Acknowledgement The author is indebted to the members of the Device Development Center, Hitachi Ltd., who supported the work especially by supplying the devices and discussing the results.
307
Appendix In the exact gradual channel model [3], the channel diffusion current is approximately given by ) pffiffiffiffi ( U ð0Þ2U F G e S eU S ðU D ÞU D 2U F ð17Þ I diff =Ai 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2b U S ð0Þ 1 U S ðU D Þ 1 where US(0) and US(UD) are surface potentials at the source and drain junctions and UD is the drain bias all normalized to kT/q (=1/b). By taking into account the relation between the gate bias as measured from the flat-band condition, UGfb (=bVGfb), US(0) and US(UD) and through mathematical manipulations using the Taylor expansion for exponential functions assuming that US(0) and US(UD) UD are smaller than 2UF, the equation can be modified as ! pffiffiffiffi G eU S ð0Þ2U F ð1 eU D Þ G I diff =Ai 2 ð1 þ e1 Þ 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4b 2 U S ð0Þ 1 U Gfb U S ð0Þ þ G=2 ð18Þ
For our applications, the term e1 1/{2 Æ [US(0) 1]} is small and can be neglected. Also, we neglect ‘‘1’’ in such an expression as [US(0) 1] for simplicity. Eq. (18) must be equivalent to the compact expression I diff =Ai ¼ ðasub =b2 Þ 1 eU D eðU Gfb U Tfb Þ=g ð19Þ with asub and g as parameters. Eq. (18) is so complicated to be compared with Eq. (19), and for the ease of analysis, we take US(0) = UF as a representative point where pffiffiffiffiffiffiffiffiffiffithe two equations exactly agree. Then, U Gfb U F þ GU F holds. By using Eq. (13), the difference of the indexes of the exponential functions is transformed as x ½U S ð0Þ 2U F ½ðU Gfb U Tfb Þ=g pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 pffiffiffiffiffiffiffiffiffiffi g ¼ GU F þ lnð2U F Þ þ G½2U F þ lnð2U F Þ 2 ½lnð2U F Þ=g pffiffiffiffiffiffiffiffiffiffiffiffiffi where g 1 þ ð1=2Þ G=U F . Therefore, ex = (2UF)1/g hold.pffiffiffiffi In pthe rightffi hand side of Eq. (18), ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1p þffiffiffiffiffiffiffiffiffiffi G=½2 U S ð0Þ 1 g and U Gfb U S ð0Þ þ G=2 g GU F hold, and asub as defined in the main text is derived. References [1] Bowman KA, Austin BL, Eble JC, Tang X, Meindl JD. A physical alpha-power law MOSFET model. IEEE J Solid-State Circ 1999; 34(10):1410–4. [2] Josephson D, Storey M, Dixon D. Microprocessor IDDQ testing: a case study. IEEE Des Test Comput 1995;12:42–52. [3] Katto H, Itoh Y. Analytical expressions for the static MOS transistor characteristics based on the gradual channel model. Solid-State Electron 1974;17:1283–92. [4] Sodini CG, Ko P-K, Moll JL. The effect of high fields on MOS device and circuit performance. IEEE Trans Electron Dev 1984;ED-31(10): 1386–93.
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[5] Katto H. Device parameters extracted in the linear region of MOSFET by comparing with the exact gradual channel model. Solid-State Electron 2000;44:969–76. [6] Arora ND, Sharma MS. MOSFET substrate current model for circuit simulation. IEEE Trans Electron Dev 1991;38(6):1392–8. [7] Sze SM. Semiconductor devices: physics and technology. 2nd ed. John Wiley & Sons; 2001 [Chapter 6].
[8] Wolf HF. Silicon semiconductor data. Pergamon Press; 1969. p. 77. [9] Lim KY, Zhou X. A physically-based semi-empirical series resistance model for deep-submicron MOSFET I–V modeling. IEEE Trans Electron Dev 2000;47(6):1300–2. [10] Kim S-D, Park C-M, Woo JCS. Advanced model and analysis of series resistance for CMOS scaling into nanometer regime—Part II: quantitative analysis. IEEE Trans Electron Dev 2002;49(3):467–72.
A compact and accurate MOSFET model with simple expressions for linear, saturation and sub-threshold regions Hisao Katto
*
Department of Electronic Systems Engineering, Tokyo University of Science, Suwa, 5000-1 Toyohira, Chino, Nagano 391-0292, Japan Received 19 May 2004; accepted 19 January 2006 Available online 20 March 2006
The review of this paper was arranged by Prof. Y. Arakawa
Abstract A compact channel-current model is proposed for the linear, saturation and sub-threshold regions of MOSFETs with eight parameters at the maximum. To derive new formulas both physically reasonable and analytically simple, the core part of the known theories and formulas including BSIM is carefully examined, and the comparison with the exact gradual model is made. A simple formula for the linear region is obtained considering the velocity saturation effect, the bias dependent mobility and the series resistance in the source and drain junctions. It is theoretically predicted and experimentally confirmed that the two new parameters in the denominator strongly depend on the channel length. Simple expressions are additionally advised for the saturation and the sub-threshold regions. By applying the model to a set of devices covering a wide range of channel length, the parameters are extracted, and good agreement between theory and measurement is demonstrated. 2006 Elsevier Ltd. All rights reserved. Keywords: MOSFETs; Parameter estimation; Modeling
1. Introduction The current–voltage (I–V) characteristics of MOSFETs have remained an important subject because of wide applications including circuit design, device design and reliability studies. BSIM of UC Berkeley, now available in the WEB site, accumulates and combines theories on various aspects of the device structure and maintains integrity as an excellent framework to describe the I–V characteristics of a MOSFET. A potential problem of the model will be that it has so many parameters with the intention of achieving ultimate accuracy, and it is sometimes hard for us to extract all the parameters from a device because the parameters are often redundant. For some applications, it is helpful to minimize the number of fitting parameters and have a very compact model like the physical alpha-power *
Tel.: +81 266 73 1201; fax: +81 266 73 1230. E-mail address: [email protected]
0038-1101/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.sse.2006.01.014
law model [1] as long as the accuracy is not excessively degraded. Compact models are important not only for circuit applications but for device and reliability study where one wants to quickly look into the basic device parameters. The model in [1] is analytically simple, and can combine all linear, saturation and sub-threshold regions. Recently, the sub-threshold region is important because the off current became an important topic of circuit design [2]. A drawback of this model will be that the theory is over-simplified by neglecting the series resistance that is practically important as discussed later. The purpose of the present paper is to advise simple formulas that are physically more reasonable than the alphapower law model. One of the basic rules in carrying out the study is that the comparison with, or verification by, the measurement is simple because it is considered an important ingredient of compactness. The linear region is the most important part of a MOSFET I–V model. One of the approaches taken was to
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carefully re-examine the existing theories to extract only the core part without excessively sacrificing accuracy. A combined analysis of the bias dependent mobility and the vertical and horizontal electric field effects is made, and a simple formula is derived for the linear region. The effect of parasitic resistance, R, is naturally included in the simple formula. The surface depletion charge is carefully examined by using the exact gradual channel model [3] and it is decided that we can temporarily do without it except the contribution to the threshold voltage. A formula is successfully derived for the boundary between the linear and saturation regions, VDsat. Similarly compact formulas are adopted for saturation and sub-threshold regions. Only the core part of BSIM3 is covered, and the topics like the following are not included: (1) the substrate bias (VB) dependence, (2) such device structure effects as the poly gate depletion effect and the narrow channel effect, and (3) any new topics for ultra-scaled devices in BSIM4. The substrate bias is not discussed because increasing a measurement parameter makes the study strenuous, and omitting it does not seem to degrade the usefulness of the model. The poly-gate depletion and narrow channel effects are not important in our samples with the gate oxide of 5.5 nm and the gate width of 15 lm. It is expected that the model will make a good basis for discussing the additional effects. 2. Theory 2.1. Linear region To discuss the mobility dependent on the gate bias, VG, and the drain bias, VD, the basic equation for the channel current, I, of a MOSFET is given by I ¼ W G Qv
ð1Þ
where WG is the gate width, Q ¼ C OX ðV Gt V Þ
ð2Þ
is the carrier density, VGt VG VT is the gate voltage as measured from the threshold voltage, VT, V is the surface potential along the channel, COX (=eOX/tOX) is the oxide capacitance where eOX is the permittivity of oxide, tOX is the gate oxide thickness, and v is the velocity of the induced carriers. The velocity v is given by [4] v ¼ leff E=ð1 þ E=EC Þ
ð3Þ
where E = dV/dx is the electric field, EC = vsat/leff is the critical field, leff is the effective mobility, and vsat is the saturation velocity that is a constant. The effective mobility is assumed to be leff ¼ l0 =½1 þ h0 ðV Gt V þ DV Þ
ð4Þ
where l0, h0 and DV are constant. Though such a formula as leff = l0/(1 + h0VGt) is widely assumed for simplicity [1], the vertical field should exactly be a function of (VGt V) instead of VGt. The parameter DV (2VT) is added be-
cause, as discussed in BSIM or more exactly in [5], the vertical field becomes zero not at VG = VT but at another gate bias VG = VT DV (VT 2VT = VT). Finally, note that BSIM proposes a quadratic formula of the form leff ¼ l0 =ð1 þ a1 V Gt þ a2 V 2Gt Þ where a1 and a2 are constant, but for simplicity, we do not include higher order terms of VGt assuming they are not critically important. Combining Eqs. (1)–(4), the differential equation I¼
W G C OX l0 ðV Gt V Þ ðl0 =vsat ÞI E 1 þ h0 ðV Gt V þ DV Þ
ð5Þ
is derived. The integration is performed over the channel length, LG, and the equation I=Ai ¼
1 V D ð1 þ h0 DV Þ=h0 ln f h0 1 þ ðb0 =h0 Þ ln f
ð6Þ
with ln f ln([1 + h0(VGt + DV)]/[1 + h0(VGt VD + DV)]) is derived. Ai = COXl0(WG/LG) and b0 = l0/(LGvsat) are constant. Using the Tailor expansion ln f (h0VD)/A + (h0VD/A)2/2 with A 1 + h0(VGt + DV), and neglecting minor terms, a useful formula: I ffi ðAi =f0 Þ
V Gt V D V 2D =2 1 þ ðh0 =f0 ÞV Gt þ ðb0 =f0 ÞV D þ e
ð7Þ
is derived where f0 1 + h0DV. Note that the simple expression in the denominator, a linear combination of VGt and VD, is derived. It is confirmed that a very similar expression can be derived within BSIM if one carefully manipulates the complex formulas there. In Eq. (7), a constant factor f0, being slightly greater than unity, is added in three places. Since it applies to all three parameters, we can temporarily remove it by rewriting Ai/f0 as Ai, h0/f0 as h0 and b0/f0 as b0, with the understanding that the factor f0 can be restored any time when required. The parameter e (>0) is here temporarily incorporated to mathematically assess the difference between Eqs. (6) and (7), and it is confirmed that e is an increasing function of h0 and is generally small. In the following, Eq. (7) with e = 0 will be used instead of Eq. (6), and the factor f0 will be omitted. The effect of the parasitic resistance, R, at the source and drain terminals, is incorporated by replacing VGt by (VGt RI) and VD by (VD 2RI), and by neglecting the second order terms of I. Then, we have I=Ai ffi
V Gt V D V 2D =2 1 þ hV Gt þ bV D
ð8Þ
where the parameters in the denominator are given by h ¼ h0 þ 2RAi ¼ h0 þ ð2RC OX l0 W G Þ=LG b ¼ b0 RAi ¼ l0 ð1=vsat RC OX W G Þ=LG
ð9Þ
The equations suggest that both h and b depend on LG, and are larger in shorter channel devices. It is easy to confirm that R of as small as 10 X (or 150 X-lm if normalized to the channel width) gives a large h value. On the other hand, the effect of R on b is less great assuming vsat 105 m/s, and the first term l0/(vsatLG) will be the major term of b.
H. Katto / Solid-State Electronics 50 (2006) 301–308
pffiffiffiffiffiffiffiffiffiffi GU S ;
303
2.2. Boundary between linear and saturation regions
U Tfb ¼ U S þ
VDsat to give the boundary between the linear and saturation regions is derived from Eq. (8) by solving oI/oVD = 0, and is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ V Dsat ¼ ð2V Gt Þ=½1 þ 1 þ ð2bV Gt Þ=ð1 þ hV Gt Þ
is newly adopted based on the exact gradual channel model [3]. UTfb = bVTfb and UF = bVF = ln(NA/ni) are nondimensional because they are normalized to kT/q (=1/b), and G ¼ b ð2qeSi N A =C 2OX Þ is the non-dimensional body factor. NA is the substrate doping density, ni is the intrinsic carrier density, and eSi is the permittivity of silicon, k is Boltzmann’s constant, q is the electron charge and T is the temperature. The expression of Eq. (13) is already proposed in [3] as the boundary between sub-threshold and normal regions (U1 in Eq. (32)). The addition of the logarithmic term may not be critically important in discussing the linear region, but is important in deriving the coefficient asub for the sub-threshold region as discussed in the following section. Experimentally, VT is extracted in the linear region with a small VD. Thereby, note that not only the term V 2D =2 in the numerator but also h and b in the denominator in Eq. (8) should be taken into account. Fig. 1 shows the theoretical I–VGt curves assuming typical values of h and b. The offset of VD/2 is visible, but note that the offset is not exactly equal to VD/2. The effect of b is found in the broken lines. For example, for VD = 100 mV, we have an offset of 47 mV instead of 50 mV. The influence of h is detected by comparing the solid and broken lines. In the present paper, we always use VD = 20 mV, and thereby neglect the small offset for simplicity.
If one assumes h = 0 (and R = 0), this reduces to the traditional expression as in [1] or [6] not to speak of the body factor. The denominator increases with VGt, but the growth is not great because of the new term (1 + hVGt), and so VDsat is greater compared with the case of h = 0. With oI/oVD = 0, the I–VD curve may become unnaturally flat at the boundary, but the mathematical smoothing as in BSIM is not used here to simplify the discussion. 2.3. Saturation region In the saturation region as defined by VD > VDsat, the current IDsat is theoretically given by replacing VD by VDsat in Eq. (8). This holds in long channel devices, and IDsat depends only on VG. In shorter channel devices, however, the measured current increases gradually with VD, and we need a simple analytical expression with a minimized number of parameters. In our devices, it is empirically found that the dependence I=I Dsat ¼ 1 þ d ½ðV D V Dsat Þ=V Gt
ð11Þ
roughly holds as far as VGt is not too small. Here, d (1) is a fitting parameter, and is zero for long channel devices. We can briefly discuss the plausibility of Eq. (11) based on BSIM. In BSIM, the saturation region is described through channel-length-modulation (CLM) and drain-induced-barrier-lowering (DIBL) modes, and the analytical expression is given by I=I Dsat ¼ 1 þ ðV D V Dsat Þ=V A
ð13Þ
2.5. Sub-threshold region In the sub-threshold region, the formula I=Ai ¼ ðasub =b2 Þðb0 þ b1 V D Þð1 ebV D Þeðb=gÞV Gt
ð14Þ
0.01
ð12Þ
solid
θ =0.8V–1
broken θ =0 0.008 VD=100mV VD=50mV 0.006
I/Ai
where VA is a rational function of VD and VG. Though the general expression is complex, the formula for VA becomes simple if we consider the DIBL mode only. In the DIBL mode, V A / V 2G =ðV Dsat þ V G Þ roughly holds neglecting minor terms, and then VA / VG holds by assuming VDsat / VG. Eq. (11) covers the entire saturation region, and further adding the CLM mode near VDsat does not seem critically important. We can possibly test such an extended expression as I/IDsat = 1 + d Æ [(VD VDsat)/VGt]Asat with an extra parameter Asat to partly include the substrate current induced body effect (SCBE) region.
U S ¼ 2U F þ lnð2U F Þ
0.004 VD=20mV 0.002
2.4. VT The theoretical definition of threshold voltage, VT, is important mainly for the sub-threshold region where the current is sensitive to ffi VT. Against the popular definition pffiffiffiffiffiffiffiffiffi of U Tfb ¼ U S þ GU S (as measured from the flat band voltage) and US = 2UF, the definition of
0 0
0.1
0.2
0.3
VGt (V) Fig. 1. Theoretical I/Ai–VGt curves with small VD values. b = 1 is assumed. Marks show calculations with the interval of DVGt = 50 mV, and linear curves are fitted against the first six marks.
H. Katto / Solid-State Electronics 50 (2006) 301–308
pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi with g ¼ 1 þ G=U F =2 and asub ¼ ð2U F Þ1=g G=U F =4 is newly adopted. The parameter g is identical to the one adopted in [1], whereas the parameter asub is newly derived using the exact gradual channel model [3] assuming Eq. (13). The coefficient asub (2.8 in our devices) is greater than the corresponding factor adopted in [1] (=g 1.5) mainly because the newly defined VT is larger and should give a larger current. Further, Eq. (14) does not assume the offset voltage adopted in [1]. The analytical derivation of asub will be briefly described in Appendix. The validity of Eq. (14) is confirmed by graphically comparing with the exact model with tOX and NA as parameters. Errors may increase with G, but are tolerably small in the area of interest G 6 20. We have the term (b0 + b1VD) just to empirically express the VD dependence. Note that b0 = 1 and b1 = 0 should ideally hold. In our samples, it is found that the linear factor (b0 + b1VD) explains the VD dependence better than another possible formula of eqV D where q is constant. Eq. (11) is applied e.g. for the bVGt > 2 region, and Eq. (14) applies e.g. for the bVGt < 2 region. In the area between saturation and sub-threshold regions where jVGtj is small (2 < bVGt < 2), logarithmic averaging as proposed in [1] will be effective. 2.6. Charge in surface depletion region If we consider the charge in the surface depletion region [7], Eq. (2) becomes as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ¼ C OX ðV Gt V Þ ðG=bÞ ð2V F þ V Þ ð15Þ We go through integration, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and two new terms are obtained one with 2V F þ V D and the other logarithmic. Adopting up to the third order term of the Taylor expansion for the logarithmic term, we obtain pffiffiffiffiffiffiffiffiffi V D V Gt V 2D =2 þ ð2=3Þ G=b½ð2V F Þ1:5 ð2V F þ V D Þ1:5 ð16Þ as the numerator of Eq. (8). The surface depletion charge (SDC) term in this equation is not small in magnitude, but will not be adopted in our model by the reasons as follows. The most important effect of the SDC term given by Eq. (16) is to cause a ‘‘VT shift’’ of about 0.8 V as shown in Fig. 2. To make the matter clearer, assume that VD ispsmall. ffiffiffiffiffiffiffiffiffi Then, we can simplify the SDC term as ð2=3Þ G=b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ð2V F Þ1:5 ð2V F þ V D Þ1:5 ðG=bÞ ð2V F ÞV D . This is equivalent to the threshold voltagepshift ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffi of DV T ¼ ðG=bÞ ð2V F Þ as given by the term GU S in Eq. (13) because US b Æ (2VF). In other words, Eq. (13) already contains the SDC term, and further adding another DVT through Eq. (16) simply double-counts the term DVT. The SDC term has another effect. The higher order terms of Taylor expansion of the SDC term may not be negligible when VD is large, and we must consider the unabridged SDC term if errors are not tolerated. However, it is reminded that the error can largely be compensated
3 VD=1V
without SDC 2
I/Ai
304
1
with SDC
0
0
1
2
3
4
VGt [V] Fig. 2. The ‘‘VT shift’’ of about 0.8 V due to the SDC effect.
through adjusting the parameters in the denominator, h and b. Further, we have complicacy in including the SDC term: (1) we need a prior knowledge of NA to theoretically predict the SDC term, or need an additional parameter to fit. (2) Including the SDC term in VDsat may complicate the analysis. (3) Another series resistance term should be added in the denominator. We would rather stick to the simpler formula as far as the errors in b, h and VDsat are tolerable and the overall agreement with the experimental I–V characteristics remains satisfactory. 3. Experiment The devices evaluated are NMOSFETs with the dimensions of tOX = 5.5 nm, WG = 15 lm and LG = 0.375 15 lm. The nominal LG value for the product is 0.35 lm, and the nominal power supply voltage is 3.3 V. We have devices with LG 0.375 lm, but the data are withheld because short channel effects may complicate the discussion. The device characteristics are measured at the room temperature using HP4142B. The measurement bias interval is 50 mV for VG whereas VD starts from 20 mV and ends in 4 V with irregular intervals as shown in the figures appearing later. Though the data will not be presented, the devices with tOX = 8 nm and LG = 0.34 lm are also evaluated and the conclusions drawn on the tOX = 5.5 nm devices, especially the LG dependence of h0 and b are confirmed also in these devices. Using a spreadsheet program, Excel, four-step fitting is made between theory and measurement. First, VT is decided at VD = 20 mV by extrapolating the linear portion of the I–VG curve near VT. Then in the linear region, the three parameters h0, b and l0 are decided. Then in the saturation region, the parameter d is decided. The last step is
H. Katto / Solid-State Electronics 50 (2006) 301–308
305
Table 1 Device parameters extracted by comparing the compact model with the tOX = 5.5 nm devices covering a wide LG range of 0.375–15 lm LG (lm) VT (V) l0 (m2/V s) h (V1) b (V1) d g NA (cm3) b0 b1/b0
15 0.600 0.0269 0.0733 0.087 0 1.495 5.40 · 1017 2.921 0.081
4 0.606 0.0284 0.131 0.098 0 1.537 6.18 · 1017 1.955 0.094
2 0.607 0.0349 0.283 0.165 0 1.536 6.36 · 1017 1.482 0.104
0.7 0.624 0.0298 0.239 0.349 0.0636 1.487 5.21 · 1017 2.236 0.141
0.5 0.633 0.0288 0.330 0.383 0.0827 1.483 5.11 · 1017 2.080 0.172
0.375 0.646 0.0317 0.394 0.605 0.105 1.506 5.65 · 1017 1.677 0.207
VT is first extracted, and the next three parameters are extracted in the linear region. d is then extracted in the saturation region, and the remaining parameters are extracted in the sub-threshold region.
4. Experimental results and discussions The parameters extracted for the eight devices with LG = 0.37515 lm are summarized in Table 1. Note that NA is theoretically derived from the exponential slope factor g in the sub-threshold region neglecting the contribution from surface states, and is used to derive the coefficient asub using the equation shown just below Eq. (14). The value of asub is needed simply because b0 is derived only if we know asub. 4.1. VT versus NA In Table 1, VT is only weakly dependent on LG in the LG range shown. It is confirmed that the VT–NA relation is roughly explained by assuming the flat-band voltage VFB = 1. When VT is plotted againstNA, we observe the scattering of ±15% in NA. If we have an error of 15% in NA, we will theoretically have an error of 15% in the parameter G, and then have an error in g of 2.4% and in asub of 1.8%. The relatively small errors in g and asub indicate that the theoretical prediction of the sub-threshold characteristics is not critically influenced by the scattering of NA.
The table shows that the mobility is not strongly dependent on LG, and the average is close to 0.03 m2/V s. The value seems reasonably smaller than the reported bulk value for electrons in p-Si, 0.035 m2/V s (NA = 6 · 1017/ cm3) [8]. The parameters h and b are plotted in Fig. 3 against 1/LG. As expected, they are linearly dependent on 1/LG. Comparing with Eq. (9), h0 0.08 V1, R 21 X and vsat 1.2 · 105 m/s are obtained assuming l0 = 0.03 m2/ V s. h0 is small but may not exactly be zero considering the data scattering. The value of R is similar to the reported value of 27 X for the tOX = 5.9 nm and WG/LG = 20/
0.4
1
θ : 0.119/LG+ 0.081 b : 0.190/LG + 0.062
0.8
0.3
0.6 0.2
b(V–1)
4.2. Linear region
0.2 lm device extracted from the I–VG characteristics at VD = 0.1 V [9]. Note that h for LG = 0.2 lm (gray circle) is not included in drawing the line because it deflects from the line. A large h may possibly come from the parasitic resistance of the needle contact. Since the concept of vertical field dependent mobility is popularly accepted, the idea of small or zero h0 will be discussed in some more detail. Assume h0 = 0 in Eq. (8) and temporarily remove the term containing b that is not needed for the discussion. Then, we have I=Ai ¼ ðV Gt V D V 2D =2Þ=½1 þ f2RC OX l0 ðW G =LG Þg V Gt . This expression is equivalent to the known mobility reduction model with h0 = l0/(2tOXvnorm) [1] if l0/(2tOXvnorm) = 2RCOXl0(WG/LG) or vnorm = (LG/WG)/(4ReOX) practically holds. R = 21 X and LG/WG = 1/15 is equivalent to vnorm 2.3 · 107 m/s. This value is close to 2.2 · 107 m/s assumed in [1]. If R and LG similarly scale, the ratio LG/R is kept constant, and the ‘‘equivalent’’ vnorm value does not change with scaling. The scaling law on R is not accurately known, but R is expected to scale with
θ (V–1)
the sub-threshold region where the three parameters g, b0 and b1 are decided. As for the device dimensions, WG, LG, tOX, design values are adopted. eOX/e0 = 3.9 and eSi/e0 = 11.7 are assumed.
0.4
0.1 0.2
0 0
0.5
1
1.5
2
2.5
3
0
1/LG (µm–1) Fig. 3. Extracted h and b values in Table 1 plotted against 1/LG. Linear relations are observed as suggested by Eq. (9). The h curve gives h0 = 0.08 V1 and R = 21 X, and the b curve gives vsat 1.2 · 105 m/s.
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H. Katto / Solid-State Electronics 50 (2006) 301–308 0.2
technology (Figs. 6 and 7 [10]). It is very natural to assume that R scales because otherwise h becomes unnaturally great and b can become zero or negative as the device is scaled and COX increases in Eq. (9).
1 δ b1 /b0
0.15
4.3. Saturation region
0.1
0.1
b1/b0
δ
0.0811+0.0426•LG–1.09
–0.152•LOG(LG/1.81) 0.05
As shown in Fig. 4, the parameter d of the LG 6 1 lm devices logarithmically depends on LG [lm] while d is zero in longer channel devices (LG P 2 lm). We can regard 1.81 lm shown in Fig. 4 as the short/long channel boundary on d in this device set. The simple physical model for the logarithmic dependence is left for future study. 4.4. Sub-threshold region
0 0.1
1
0.01
10
L G (µm)
Fig. 4. Parameters d and b1/b0 for the VD dependence in the saturation and sub-threshold regions plotted against LG. For short channel devices, LG < 1 lm, logarithmic dependence of d on LG is observed. Power law dependence of b1/b0 on LG is observed.
As discussed earlier, b0 = 1 should ideally hold. In the table, b0 is closer to 2, but considering the exponential nature of the sub-threshold current, the error is not considered great. The VD dependence factor b1/b0 as shown in Fig. 4 is found a strong function of LG. The simple physical model for this dependence is left for future study. 4.5. Over-all characteristics Figs. 5 and 6 compare the theory with the measurement for the LG = 0.375 lm device. Fig. 5 shows the normal cur-
VGt=3V 1.2
VGt= -0.1V
2.5V 10 -3
1
-0.2V 2V
10 -4
I/Ai
0.8
-0.3V 0.6
10 -5
I/Ai
1.5V
0.4
10 -6
-0.4V
1V VDsat
0.2
10 -7
0.5V
0V 0
0
1
2
-0.5V
10 -8 -0.6V
3
VD (V) Fig. 5. Over-all I–V characteristics for the device with LG = 0.375 lm in the normal current region. Measured data are shown by cross marks with the gate bias interval of 0.1 V and by black circles with the interval of 0.5 V. Theoretical curves are shown with the gate bias interval of 0.5 V. White circles indicate the theoretical VDsat–IDsat relation. Note that the gate bias is measured from VT, and the current is normalized against Ai.
10 -9 0
1
2
3
VD (V) Fig. 6. Over-all I–V characteristics in the sub-threshold region for the same device as shown in Fig. 5.
H. Katto / Solid-State Electronics 50 (2006) 301–308
rent region. In the theoretical curves, the data are calculated only for the biases corresponding to the experimental points and are smoothly connected by the function of the spreadsheet program. Agreement between theory and measurement is good especially if one considers the minimized number of parameters, or compared with the physical alpha-power law model (Fig. 4 in [1]). Fig. 6 shows the I–V characteristics of the sub-threshold region. Good agreement between theory and measurement is obtained also in this region. 5. Concluding summary The new compact MOSFET model has eight parameters to fit, VT, h, b, l0, d, g, b0 and b1. They are extracted without difficulty through the comparison with measurement for the tOX = 5.5 nm devices with LG between 0.375 lm and 15 lm through the four steps of assessing (1) VT and (2) linear, (3) saturation, and (4) sub-threshold regions. Good agreement between theory and measurement is obtained. In the linear region, the simple formula using the two parameters h and b naturally combines the vertical and horizontal electric field effects and the series resistance, R. The parameters h and b are theoretically predicted and experimentally confirmed to depend linearly on 1/LG. It is found that the intrinsic part of h, h0 (0.08 V1), is relatively small and the parasitic resistance R (21 X) mainly accounts for the magnitude of h. This shows that such an approach as the alpha-power law model that neglects the series resistance has a drawback in explaining the LG dependence. Further, a reasonable value of saturation velocity vsat 1.2 · 105 m/s is derived from the b LG dependence data. Only a minimized number of parameters are adopted in other regions so that the model becomes compact. We have at present such empirical factors as R, d, b0 and b1/b0 that can only be obtained through fitting. Theoretical prediction of these factors will be an important subject of future study. Some minor but conceptually meaningful factors are theoretically discussed. Firstly, a constant factor f0 that reflects the vertical electric field is discussed. Secondly, issues on introducing the SDC term is discussed. It is discussed that the error caused by neglecting the higher order terms of SDC are largely compensated by a minor adjustment of other fitting parameters, and so the terms are temporarily neglected to make the theory simple and compact. In the sub-threshold region, the formula for the coefficient asub is newly proposed. Acknowledgement The author is indebted to the members of the Device Development Center, Hitachi Ltd., who supported the work especially by supplying the devices and discussing the results.
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Appendix In the exact gradual channel model [3], the channel diffusion current is approximately given by ) pffiffiffiffi ( U ð0Þ2U F G e S eU S ðU D ÞU D 2U F ð17Þ I diff =Ai 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2b U S ð0Þ 1 U S ðU D Þ 1 where US(0) and US(UD) are surface potentials at the source and drain junctions and UD is the drain bias all normalized to kT/q (=1/b). By taking into account the relation between the gate bias as measured from the flat-band condition, UGfb (=bVGfb), US(0) and US(UD) and through mathematical manipulations using the Taylor expansion for exponential functions assuming that US(0) and US(UD) UD are smaller than 2UF, the equation can be modified as ! pffiffiffiffi G eU S ð0Þ2U F ð1 eU D Þ G I diff =Ai 2 ð1 þ e1 Þ 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4b 2 U S ð0Þ 1 U Gfb U S ð0Þ þ G=2 ð18Þ
For our applications, the term e1 1/{2 Æ [US(0) 1]} is small and can be neglected. Also, we neglect ‘‘1’’ in such an expression as [US(0) 1] for simplicity. Eq. (18) must be equivalent to the compact expression I diff =Ai ¼ ðasub =b2 Þ 1 eU D eðU Gfb U Tfb Þ=g ð19Þ with asub and g as parameters. Eq. (18) is so complicated to be compared with Eq. (19), and for the ease of analysis, we take US(0) = UF as a representative point where pffiffiffiffiffiffiffiffiffiffithe two equations exactly agree. Then, U Gfb U F þ GU F holds. By using Eq. (13), the difference of the indexes of the exponential functions is transformed as x ½U S ð0Þ 2U F ½ðU Gfb U Tfb Þ=g pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 pffiffiffiffiffiffiffiffiffiffi g ¼ GU F þ lnð2U F Þ þ G½2U F þ lnð2U F Þ 2 ½lnð2U F Þ=g pffiffiffiffiffiffiffiffiffiffiffiffiffi where g 1 þ ð1=2Þ G=U F . Therefore, ex = (2UF)1/g hold.pffiffiffiffi In pthe rightffi hand side of Eq. (18), ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1p þffiffiffiffiffiffiffiffiffiffi G=½2 U S ð0Þ 1 g and U Gfb U S ð0Þ þ G=2 g GU F hold, and asub as defined in the main text is derived. References [1] Bowman KA, Austin BL, Eble JC, Tang X, Meindl JD. A physical alpha-power law MOSFET model. IEEE J Solid-State Circ 1999; 34(10):1410–4. [2] Josephson D, Storey M, Dixon D. Microprocessor IDDQ testing: a case study. IEEE Des Test Comput 1995;12:42–52. [3] Katto H, Itoh Y. Analytical expressions for the static MOS transistor characteristics based on the gradual channel model. Solid-State Electron 1974;17:1283–92. [4] Sodini CG, Ko P-K, Moll JL. The effect of high fields on MOS device and circuit performance. IEEE Trans Electron Dev 1984;ED-31(10): 1386–93.
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