Comprehensive noise performance of ultrathin oxide ...

Solid-State Electronics 48 (2004) 61–71 www.elsevier.com/locate/sse

Comprehensive noise performance of ultrathin oxide MOSFETs at low frequencies q Jonghwan Lee, Gijs Bosman

*

Department of Electrical and Computer Engineering, University of Florida, 585 NEB, Gainesville, FL 32611, USA Received 24 April 2003; accepted 12 May 2003

Abstract An analytical model and analysis of the low frequency noise performance of ultrathin oxide MOSFETs are presented. The model includes the effects of gate tunneling leakage current, low frequency noise sources, and their cross correlation coefficient between the input and the output. The correlation coefficient is not negligible in the existence of direct tunneling current through the gate oxide, and has an impact on the overall noise performance at low frequencies. Based on the partition noise theory and BSIM4 gate leakage current model with the source–drain partition, the correlation coefficient is obtained from a physical and rigorous modeling of intrinsic noise sources. The calculated results are compared with correlation noise measurements and good agreement is observed. An analytical formulation of low frequency noise performance is newly derived using the frequency-dependent noise parameters P LF , RLF , and C LF . The noise performance parameters, such as minimum noise figure, optimum source resistance and reactance, are graphically presented to accurately predict the influence of the gate leakage current on the low frequency noise performance. From quantitative results simulated with typical device parameters, it is shown that the proposed model can be used for optimum design of ultrathin oxide MOS devices and circuits at low frequencies. 2003 Elsevier Ltd. All rights reserved. Keywords: Ultrathin oxide MOSFETs; Gate tunneling leakage current; Low frequency noise; Cross correlation coefficient; Partition noise theory

1. Introduction The gate tunneling leakage current resulting from downsizing CMOS technologies is increasingly important in terms of device operation and reliability [1–4]. As the gate oxide thickness is scaled down to the direct tunneling regime, the tunneling leakage current increases drastically and deteriorates the device performance. Moreover, the current source causes a low frequency noise component which not only determines the lower

q This work was supported by the Semiconductor Research Corporation. * Corresponding author. Tel.: +1-352-392-0910; fax: +1-352392-8381. E-mail addresses: jhlee77@ufl.edu (J. Lee), bosman@ ece.ufl.edu (G. Bosman).

end of the dynamic range of device operation in the base band, but also has potentially a major impact on the phase noise of nonlinear circuits in the GHz region through up-conversion [5]. Since the impact of gate leakage current can play a significant role in the overall device characteristics, an exhaustive evaluation of device performance is required over the entire frequency range. From the noise modeling point of view, the gate leakage current introduces a shot noise current source at high frequencies [6], in addition to the existing channel thermal noise source at the output and induced thermal noise source at the input. The high frequency noise sources are partially correlated, because the MOS transistor can be treated as an RC distributed network meaning that the high frequency gate admittance contains a conductive component [7,8]. At low frequencies, since the gate conductance in ultrathin gate oxide MOSFETs can be substantial due to the oxide traps, it is

0038-1101/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0038-1101(03)00237-5

62

J. Lee, G. Bosman / Solid-State Electronics 48 (2004) 61–71

the output, SITd ðf Þ, and the induced thermal noise source at the input, SITg ðf Þ, as shown in Fig. 1. These two sources are correlated due to capacitive coupling, and the corresponding correlation coefficient C T ðf Þ is complex [6–8]. In the low frequency range, the two dominant noise sources at the output are the 1=f c noise and the generation–recombination (G–R) noise modeled as follows " # X q1=f qGR LF 2 d i þ gmg SId ðf Þ ¼ 4kT 2 f 2 s2 f cd 1 þ 4p i i " # q1=f 2 d 4kT ð1Þ gmg f cd

also observed that a partial correlation between the drain and the gate current noise sources exists. Therefore, in order to accurately predict the noise performance at low frequencies, a physical and rigorous modeling of the correlation effects is required. The correlation noise measurements at low frequencies have been previously presented for hetero-junction bipolar transistors [9–13], and the influence of the gate leakage current on high frequency noise performance was investigated for MESFETs and MODFETs [14,15]. To date, no detailed study of low frequency correlation and noise performance with an effect of gate leakage current has been given for ultrathin oxide MOSFETs. In this paper, a comprehensive noise performance at low frequencies is presented, including a physical model for correlation coefficient and its measurements, as well as the minimum noise figure and the optimum source impedance in the presence of a gate leakage current. The correlation coefficient is derived on the basis of partition noise theory [16–18] and the BSIM4 gate leakage current model with the source–drain partition [19], and is compared with correlation noise measurements in terms of the current ratio of the gate to the drain. The analytical expression of low frequency noise performance is newly formulated using the frequency-dependent noise parameters P LF , RLF , and C LF , and the simulation results are graphically presented to investigate how the overall noise performance depends on the gate leakage current and the correlated noise sources at low frequencies. It is confirmed that accurate modeling and understanding of physical noise sources at low frequencies are indispensable for optimum designs of ultrathin oxide MOS devices and circuits.

2.1. Noise sources

where k is the Boltzmann constant, T , the absolute temperature, gmg ¼ oId =oVgs , the transconductance, ðq1=f d = f cd Þ, the 1=f c equivalent input noise resistance, and qGR i and si characterize a G–R noise resistance and a time constant for the ith component, respectively. The physical origins of 1=f c noise can be explained by the correlated number and mobility fluctuation model. The correlated model is based on the fact that fluctuation in the occupancy of the oxide traps induces correlated fluctuations in the carrier number and surface mobility [20]. G–R noise in MOSFETs is caused by random trapping–detrapping of carriers at the defect centers in the depletion region. The G–R noise is usually overshadowed by the higher 1=f c surface noise component. In the case of MOSFETs with ultrathin gate oxide, however, the low frequency noise performance should be modified due to the presence of the gate leakage current Ig , as schematically represented in Fig. 1. The low frequency noise source at the input, SILF ðf Þ, physically g follows the same mechanism as observed in the output, and is represented as " # q1=f g LF SIg ðf Þ ¼ 4kT ð2Þ gg2 f cg

The high frequency noise behavior of MOSFETs basically arises from the channel thermal noise source at

cg c where ðq1=f g =f Þ is the 1=f gate noise resistance and gg ¼ oIg =oVgs is the gate conductance. Note that the low

2. Noise theory

C T

CL

F

D

G

S ISg

S ITg

Noiseless

S ILF g S

MOSFET

S ILF d

S ITd

S

Fig. 1. Admittance representation of the physical noise sources of ultrathin oxide MOSFETs with a gate leakage current noise source.


frequency gate noise is proportional to 1=f c , while the high frequency gate noise shows the quadratic dependence on frequency [7,8]. The low frequency noise sources between the input and the output are partly correlated, and the cross correlation coefficient C LF may not be negligible. The Lorentzian-modulated shot noise source SISg ðf Þ is generated by a trap-assisted tunneling (TAT) through the gate oxide, and can be analyzed by the Fano factor F Fano defined as F Fano ¼

SISg ðf Þ

SILF ðf Þ ¼ d

1 1 þ kp

2

SILF ðf Þ þ s

1 1 þ kp

SILF ðf Þ g ð4aÞ

SILF ðf Þ ¼ g

kp 1 þ kp

2

SILF ðf Þ þ s

1 1 þ kp

SILF ðf Þ g ð4bÞ

ð3Þ

2qIg

63

SILF ðf Þ d Ig

¼

2.2. Noise models and correlation function

with

In a direct tunneling gate oxide MOSFET, there is a gate leakage current component Ig besides the drain current component Id . The ratio of gate leakage current to drain current, Ig =Id , increases approximately in proportion to the square of channel length, L2 and typically lies between 1 and 106 , depending on channel length [1]. As shown in Fig. 2, the source current Is is partitioned into Id flowing to the drain at the partition factor k, and Ig flowing to the gate at the partition factor (1 k). The average value of the partition factor k represents the probability that electrons arrive at the drain. Assuming that electrons arriving at the drain or at the gate depend only on the starting point of the electrons in the channel, the drain current noise SILF ðf Þ, the gate leakage current d noise SILF ðf Þ and the cross noise spectrum SILF ðf Þ at low g d Ig frequencies are given by [16–18]

ðf Þ ¼ SILF s

kp ð1 þ kp Þ2

SILF ðf Þ s

þ

1 1 þ kp

SILF ðf Þ g

ð4cÞ

1 þ kp LF Ig 1 k ½S ðf Þ SILF ðf Þ and kp ¼ ¼ g k 1 kp Id Id ð5Þ

ðf Þ is the source current noise before partiwhere SILF s tioning into the drain and the gate. The cross correlation coefficient at low frequencies between the drain and the gate current noise is defined as SILF ðf Þ d Ig C LF ðf Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LF ðf Þ SId ðf Þ SILF g

ð6Þ

Using (4a)–(6), and rearranging in terms of the ratio of the gate leakage current noise to the drain current noise, SILF ðf Þ=SILF ðf Þ, we obtain an expression for the correg d lation function as follows

G

-Ig= -(1-λ) Is Trap

Si-SiO2

S

D

-Id=-λIs

- Is ∆SI ( y,y’, f )

y 0

Substrate

y1

L

Fig. 2. Noise current components in ultrathin oxide MOSFETs. Is is partitioned into Id at the partition factor, and Ig at the partition factor (1 k).

64


" # " LF # SILF ðf Þ SIg ðf Þ kp 1 g 1 LF þ 1 þ kp SILF SId ðf Þ ð1 þ kp Þð1 kp Þ ðf Þ d C LF ðf Þ ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # " #2 9ffi u8 2 2 LF LF u> S ðf Þ S ðf Þ > k 1 Ig Ig > p u> > > 1 LF þ > > u< = LF 2 2 2 S S ðf Þ ðf Þ ð1 þ kp Þ u ð1 þ kp Þ ð1 kp Þ Id Id " # " # u u> > SILF ðf Þ SILF ðf Þ ð1 þ k2p Þ > g g t> >þ > > > 1 : ; LF LF 2 S S ðf Þ ðf Þ ð1 þ kp Þ ð1 kp Þ Id Id

For the evaluation of the correlation function at low frequencies, it is necessary to obtain a physical model of SILF ðf Þ=SILF ðf Þ, taking the noise generation process into g d account. Under the assumption that at low frequencies both the drain and gate noise are caused by a random trapping–detrapping of carriers in localized trap states near the Si–SiO2 interface, we can use the KlaassenÕs formula for the 1=f c noise current generator at y and y 0 along the channel [21] ðy; y 0 ; f Þ ¼ H d;g ðyÞ DSILF d;g

f

cd;g

I2 dðy y 0 Þ gðVi ðyÞÞ

for i ¼ 0; 1

ð8Þ

where dðyÞ is a delta function, H ðyÞ the local noise source related to a trap efficiency, gðVi Þ the channel conductance for unit length at y, V0 ðyÞ the channel voltage at y, and V1 ðyÞ the perturbation voltage induced due to the gate leakage current (V0 ðyÞ V1 ðyÞ). If the noise source is distributed uniformly over the range 0 < y < y1 the noise spectrum ratio may be represented by Z

y1

Z

Ig2

y1

1 dðy y 0 Þ dy dy 0 H g ðyÞ cg L2 0 f gðV1 ðyÞÞ 0 ¼ Z y1 Z y1 SILF ðf Þ 1 I2 d dðy y 0 Þ dy dy 0 H d ðyÞ c d 2 L 0 f d gðV0 ðyÞÞ 0 ð9Þ

V0 ðyÞ ¼

Id ¼ gðV0 ðyÞÞ

dV0 ðyÞ dy

and Ig ¼ gðV1 ðyÞÞ

dV1 ðyÞ dy

ð10Þ

where m is the body-effect coefficient ranging from 1.1 to 1.4, VT the threshold voltage, Vgs the gate-to-source voltage, and Vds the drain-to-source voltage. According to BSIM4 gate leakage current model including the source–drain partition [19], the perturbation voltage V1 ðyÞ is rewritten by Jg0 ½expðB KyÞ 1 þ Ly ½1 expðB KyÞ V1 ðyÞ ¼ lCox B2 K 2 ðVgs VT KyÞ ð13Þ with Pi gcd tox B ; B ¼ Vgs2

K¼

SILF ðf Þ g SILF ðf Þ d

¼ kp

V1 ðy1 Þ ðcd cg Þ f V0 ðy1 Þ

ð11Þ

pffiffiffiffiffiffiffiffiffiffiffiffi 8p 2qmox /b3=2 B¼ 3h

Vgs VT V2ds Vds ðVgs VT ÞL

ð14aÞ

ð14bÞ

where q is the electron charge, /b the tunneling barrier height, h the Planck constant, tox the gate oxide thickness, mox the electron effective mass of the conduction band in the oxide, Pi gcd a fitting parameter with a default value of 1, the mobility, and Cox the gate oxide capacitance per unit area. Jg0 is the direct tunneling current density with Vds ¼ 0 and is given by [19] Jg0 ¼

yields a simple expression as follows

Vgs VT m ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Vgs VT y Vgs VT y 2 Vds þ Vds 2 L L m m ð12Þ

SILF ðf Þ g

where L is the effective channel length. Assuming H d ðyÞ ¼ H g ðyÞ and carrying out the integration, bearing in mind that the drain and gate current are respectively [22],

ð7Þ

q3 Cð/b ; tox ; Fox ; Vg Þ 8ph/b eox h i3 2 pffiffiffiffiffiffiffiffiffiffi 8p 2mox /3=2 1 ð1 jVox j=/b Þ3=2 b 5 exp 4 3hqjFox j ð15Þ

In this formula, the channel voltage V0 ðyÞ as a function of y is given by [23]

with a correction function


"

k1 tox jFox j /b þ1 /bo # tox jFox j Vg N 1 /b tox

Cð/b ; tox ; Fox ; Vg Þ ¼ exp

20 /b

ð16Þ

where Fox is the electric field in the oxide layer, k1 a fitting parameter, /bo the Si–SiO2 barrier height (3.1 eV for electrons and 4.5 eV for holes), and N an indicator of the density of the tunneling carriers.

LF LF Fmin ¼ 1 þ 2gnLF ðRLF c þ Rs;opt Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! rnLF 2 ðRLF ¼ 1 þ 2gnLF RLF c Þ þ LF c þ gn

R¼

SILF ðf Þ g 4kTgg

þ

jY21 j SITg ðf Þ 4kT jY11 j2

! LF gmg SIg ðf Þ ½1 ðC LF Þ2 gg 4kT ! !1=2 LF LF gmg SIg ðf Þ gmg SIg ðf Þ LF 2C gg SILF ðf Þ gg SILF ðf Þ d d

ð17aÞ

1þ

ð21aÞ

gnLF

¼

f fc

2

2 SILF ðf Þ d 41 þ 4kT

2C LF ¼R

LF

T

þR

ð17bÞ

where jY21 j gmg and jY11 j 2pfCgs . At low frequencies, R is not due to the capacitive coupling, but due to the TAT through the gate oxide layer. The gate-to-source capacitance Cgs in the linear region and in the saturation region is calculated according to the following formula, respectively " # 2 ðVgd VT Þ2 Cgs ¼ WLCox 1 and 3 ðVgs þ Vgd 2VT Þ2 2 Cgs ¼ WLCox 3

LF gmg SIg ðf Þ gg SILF ðf Þ d

LF gmg SIg ðf Þ gg SILF ðf Þ d !1=2 3

!

5

ð21bÞ

LF ZcLF ¼ RLF c þ jXc

¼ ðRg þ Rf Þ þ

1 j2pfCgs 1C

LF


1þ

!


2C

LF

!1=2


!1=2

ð21cÞ

ð18Þ

where W is the effective channel width and Vgd is the gate-to-drain voltage. The cross correlation coefficient Cðf Þ between the drain and the gate can be written as SId Ig ðf Þ Cðf Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SId ðf Þ SIg ðf Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi P LF RLF P T RT þ C T ðf Þ ¼ C LF ðf Þ PR PR

1 gmg

rnLF ¼ ðRg þ Rf Þ þ

For the calculation of noise performance at low frequencies, the dimensionless noise coefficients P and R are defined as [15]. SId ðf Þ 1 ¼ fS LF ðf Þ þ SITd ðf Þg 4kT jY21 j 4kT jY21 j Id ¼ P LF þ P T

ð20Þ

with

2.3. Noise performance at low frequencies

P¼

65

where Rg and Rss are the parasitic gate and source resistance, respectively, and fc is the intrinsic cut-off frequency ð¼ gmg =2pCgs Þ. For Rg ¼ Rss ¼ 0, the minimum noise figure is simplified to the form

LF Fmin ¼1þ

ð19Þ

With knowledge of the basic noise sources, it is possible to calculate the low frequency noise performance using P LF , RLF , and C LF . We shall consider fundamental noise parameters as a function of the noise spectrum ratio at LF low frequencies. The minimum noise figure Fmin can be obtained in terms of the optimum source resistance LF LF RLF s;opt , the noise resistance rn , the noise conductance gn , and the correlation impedance ZcLF as follows [15]

f fc

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # u u SILF ðf Þ SILF ðf Þ g mg 2 g d t ½1 ðC LF Þ gg SILF ðf Þ 2kTgmg d

ð22Þ

The low frequency noise figure F LF can be written in the form

LF F LF ¼ Fmin þ

gnLF LF 2 jZs Zs;opt j Rs

ð23Þ

66


with 1

LF Zs;opt ¼

BLF s;opt

LF ¼ RLF s;opt þ jXs;opt

LF Ys;opt sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rnLF 2 LF ¼ ðRLF c Þ þ LF jXc gn

x 10 -5

Id

8

6

7

4

6

2

-8

2

Vgs =1.0V

0

Vds =0.3V

1

2

Id Ig

4

-2

0

3 0

0.2

0.4

0.6

(a)

0.8

1

1.2

1.4

1.6

1.8

0

2

Drain Voltage, Vds (V) x 10-4

x 10-7 2

0

0

-2

gd

x 10 -4

-2

-4 0.2

0.4

0.6

0.8

1

1.2

Drain Voltage, Vds (V)

1.4

1.6

1.8

x 10 -6 1.4

Vds =0.3V

1.2

1

1

0.8

0.8

gmg

0.6

0.6

0.4

0.4

0.2

Vgs =1.0V

gg

0

2

0

(d)

0 1.5

1 Gate Voltage, Vgs (V)

1.4

Conductance, g mg (A/V), g g (A/V)

Conductance, g md (A/V), g d (A/V)

gmd 2

0

0.5

(b)

1.2

(c)

x 10 -7 4

x 10 -4

5

Ig

4

ð26aÞ

9

8

ð25bÞ

GLF s;opt jleakage vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # u 2 u ðf Þ 2 2qF Fano Ig gmg gmg SILF 2 g t LF ¼ 2pfCgs ½1 ðC Þ þ LF LF gg SId ðf Þ SId ðf Þ

Current, Id (A), Ig(A)

10

Current, Id (A), Ig(A)

x 10 10

LF t

In addition to the influence of the correlation on the low frequency noise performance, the shot noise due to the gate leakage current, SIsg ðf Þ ¼ 2qIg F Fano , has an impact on the noise behavior by modifying the noise parameLF ters, GLF s;opt and Fmin , as follows [6]

ð24Þ

LF where Zs ¼ Rs þ jXs is the source impedance, and Zs;opt LF LF LF and Ys;opt ¼ Gs;opt þ jBs;opt are the optimum external source impedance and the optimum source admittance at low frequencies, respectively. From (17), (21), and LF (24), expressing Ys;opt for Rg ¼ Rf ¼ 0 in terms of the noise ratio, we can get vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # u u ðf Þ gmg SILF 2 g LF t LF ð25aÞ Gs;opt ¼ 2pfCgs ½1 ðC Þ gg SILF ðf Þ d

12

v" ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #ffi9 u u gmg SILF ðf Þ = g ¼ 2pfCgs 1 C : gg SILF ðf Þ ; d 8