# US 9,519,741 B2 · Method of characterizing and modeling leakage statistics and threshold voltage

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- Number: US 9,519,741 B2
- Document ID: US9519741B2
- Status: Granted
- Kind: B2
- Application number: 14/553,228
- Filed: 2014-11-25
- Published: 2016-05-26
- Granted: 2016-12-13
- Earliest claimed: 2014-11-25
- Assignees: International Business Machines Corporation
- Applicants: International Business Machines Corporation
- Inventors: Ning Lu
- Web page: https://patentgrep.com/patent/US9519741B2

## CPC

- G06F 30/367: Design verification, e.g. using simulation, simulation program with integrated circuit emphasis [SPICE], direct methods or relaxation methods
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)

## Abstract

An approach includes deriving an uplift factor as a function of a width of the device for each leakage current component based on an amount of uncorrelated random variations in the leakage current component for one specific width and using the uplift factor as a multiplier for the leakage current component. The approach includes using the uplift factor for sub-threshold drain current as a multiplier of the sub-threshold drain current so that a lowering of nominal threshold voltage of the device occurs in a single simulation run. The approach further includes deriving a threshold voltage mismatch expression related to an amount of an uncorrelated random variation in sub-threshold drain current which is not directly inversely proportional to a square root of the width. The uplift factors and the threshold voltage mismatch expression within a model are used to predict statistical characteristics of the leakage current.

## Claims

### Claim 1

A method implemented in a computer infrastructure having computer executable code tangibly embodied on a computer readable storage medium having programming instructions operable to:
derive an uplift factor as a function of a width of a single-finger semiconductor device for each leakage current component based on an amount of uncorrelated random variations in the leakage current component for one specific width of the semiconductor device and use the uplift factor as a multiplier for the leakage current component;
when the semiconductor device is a single-finger transistor, use the uplift factor for sub-threshold drain current as a multiplier of the sub-threshold drain current so that a lowering of nominal threshold voltage of the single-finger device occurs in a single simulation run;
derive a threshold voltage mismatch expression which is related to an amount of an uncorrelated random variation in sub-threshold drain current and which is not directly inversely proportional to a square root of the width of the single-finger device;
use the uplift factors and the threshold voltage mismatch expression within a semiconductor device model to predict at least statistical characteristics of the leakage currents and
use the deriving steps and the using steps in manufacturing an integrated circuit chip.

### Claim 2 (depends on 1)

The method of claim 1, wherein the leakage current component comprises sub-threshold drain current, gate tunneling currents, and diode currents.

### Claim 3 (depends on 1)

The method of claim 1, wherein the amount of uncorrelated random variations comprises a random dopant fluctuation.

### Claim 4 (depends on 1)

The method of claim 1, wherein an amount of the lowering of the nominal threshold voltage is a function of a total width of the single-finger device.

### Claim 5 (depends on 1)

The method of claim 1, wherein using the uplift factors and the threshold voltage mismatch expression predict average threshold voltage and threshold voltage mismatch of the single-finger device based on a plurality of width segments of the single-finger device.

### Claim 6 (depends on 1)

The method of claim 1, wherein the semiconductor device model is a compact model of the single-finger device.

### Claim 7 (depends on 1)

The method of claim 1, wherein the leakage current component comprises a plurality of width segments of the single-finger device.

### Claim 8 (depends on 1)

The method of claim 1, wherein the uplift factor is a function of two ratios, Wnom/w0 and σ0/nvt, where:
Wnom is a nominal electrical width of the single-finger device,
w0 is a nominal electrical width of a reference single-finger device,
σ0 is an uncorrelated variation of threshold voltage in a reference device,
vt=kBT/q is thermal voltage,
kB is Boltzmann's constant,
T is absolute temperature,
q is electronic charge, and
nvt is related to sub-threshold slope S through S=(ln 10)nvt.

### Claim 9 (depends on 8)

The method of claim 8, wherein the usage of uplift factor and the threshold voltage mismatch expression in the semiconductor device model takes the following form:
I
leak
=
U
plift
⁡
(
W
nom
)
⁢
i
leak
⁡
(
L
,
W
,
V
g
,
V
d
,
V
s
,
V
x
,
T
)
⁢
exp
⁡
(
-
g
⁢
⁢
σ
uncor
⁡
(
W
nom
)
nv
t
)
,
wherein:
Wnom(=Wdes+dw) is a nominal electrical width of the single-finger device that has a design width Wdes,
W is an electrical width of the single-finger device, including width tolerance,
Ileak is a leakage current of the single-finger device after including the uplift factor and threshold voltage mismatch expression,
ileak is a leakage current of the single-finger device before including the uplift factor and with setting threshold voltage mismatch to zero,
L is a channel length of the single-finger device;
Vg is a gate voltage;
Vd is a drain voltage;
Vs is a source voltage;
Vx is a substrate voltage;
Vgs=Vg−Vs is a gate bias;
T is absolute temperature; and
Vth is a threshold voltage.

### Claim 10 (depends on 9)

The method of claim 9, further comprising treating the single-finger device as multiple (m) narrower devices of a same type and same channel length connected in parallel and applying the threshold voltage mismatch expression in the model to each of m individual narrower devices.

### Claim 11 (depends on 10)

The method of claim 10, wherein the electrical features are at least one of leakage current and threshold voltage, and the method further comprises determining an uncorrelated component and a correlated component for the single-finger device.

### Claim 12 (depends on 10)

The method of claim 10, wherein the single-finger device is one of low Vt NFET, high Vt PFET, analog Vt NFET, thick oxide PFET, MOS capacitor, MOS varactor, and diode.

### Claim 13

A computer program product for determining leakage current, the computer program product comprising a computer usable storage medium having program code embodied in the storage medium, the program code readable/executable by a computing device to:
identify a nominal or average or median electrical effective width Wnom (e.g., Wnom=Wdes+dw) of a single finger device that has a design width Wdes;
insert an uplift factor Uplift(Wnom) as a multiplier into a statistical leakage current model for the single-finger device;
use expression σuncor(Wnom) in the statistical leakage current model as a standard deviation of uncorrelated threshold voltage variations;
adjust Uplift(Wnom) and σuncor(Wnom) such that two averages of leakage current are identical and two standard deviations of leakage current are the same when the statistical leakage current model is used in different approaches; and
use the identifying, inserting, using and adjusting steps in manufacturing an integrated circuit chip.

### Claim 14 (depends on 13)

The computer program product of claim 13, wherein one average of leakage current and one standard deviation of leakage current are the result of applying leakage current model
I
leak
=
U
plift
⁡
(
W
nom
)
⁢
i
leak
⁡
(
L
,
W
,
V
g
,
V
d
,
V
s
,
V
x
,
T
)
⁢
exp
⁡
(
-
g
⁢
⁢
σ
uncor
⁡
(
W
nom
)
nv
t
)
,
once to a whole width of the single-finger device, and the other average of leakage current and standard deviation of leakage current are the result of applying the leakage current model multiple times, each time to one segment of the single-finger device.

### Claim 15 (depends on 13)

The computer program product of claim 13, further comprising constructing the uplift factor Uplift(Wnom) for leakage current of the single-finger device, wherein the uplift factor is a function of two ratios Wnom/w0 and of σ0/nvt, where
Wnom(=Wdes+dw) is a nominal electrical width of the single-finger device that has a design width Wdes,
w0(=wdes,ref+dw) is a nominal electrical width of a reference single-finger device that has a design width wdes,ref,
σ0 is an uncorrelated variation of threshold voltage in the reference device,
vt=kBT/q is thermal voltage,
kB is Boltzmann's constant,
T is absolute temperature,
q is electronic charge, and
nvt is related to sub-threshold slope S through S=(ln 10)nvt.

### Claim 16 (depends on 15)

The computer program product of claim 15, wherein the uplift factor Uplift(Wnom) has the following properties:
(i) there is no uplift at reference design width Wdes,ref, Uplift(Wnom)=1;
(ii) there is no uplift when there is no uncorrelated random variations, Uplift(Wnom)=1 when σ0=0;
(iii) for a given ratio σ0/nvt, the uplift factor increases monotonically with an increase of the ratio Wnom/w0; and
(iv) for a given ratio Wnom/w0, the uplift factor increases monotonically with an increase of the ratio σ0/nvt.

### Claim 17 (depends on 16)

The computer program product of claim 16, wherein an amount of the uncorrelated threshold voltage variation, σuncor(Wnom), is a function of σ0, thermal voltage nvt, and the ratio Wnom/w0.

### Claim 18 (depends on 17)

The computer program product of claim 17, wherein the expression σuncor(Wnom), has the following properties:
(i) it reduces to a given amount of uncorrelated threshold voltage variation at a reference design width wdes,ref, σuncor(w0)=σ0;
(ii) there is no uncorrelated threshold voltage variation when σ0 is zero, σuncor(Wnom)=0 when σ0=0;
(iii) it is not inversely proportional to the square root of Wnom;
(iv) the ratio σuncor(Wnom)/nvt is a function of two ratios Wnom/w0 and σ0/nvt;
(v) for a given ratio σ0/nvt, the ratio σuncor(Wnom)/nvt decreases monotonically with the increase of the ratio Wnom/w0; and
(vi) for a given ratio Wnom/w0, the ratio σuncor(Wnom)/nvt increases monotonically with the increase of the ratio σ0/nvt.

### Claim 19 (depends on 13)

The computer program product of claim 13, wherein the different approaches comprises applying the model once to a whole single-finger device and treating the whole single-finger device as multiple (m) narrower devices of a same type connected in parallel and apply the model to each of m individual narrower devices.

### Claim 20

A system comprising:
a CPU and a computer readable memory;
first program instructions to derive an uplift factor as a function of a width of a single-finger semiconductor device for each leakage current component based on an amount of uncorrelated random variations in the leakage current component for one specific width of the semiconductor device and use the uplift factor as a multiplier for the leakage current component;
second program instructions to use the uplift factor for sub-threshold drain current as a multiplier of the sub-threshold drain current so that a lowering of nominal threshold voltage of the single-finger device occurs in a single simulation run when the semiconductor device is a single-finger transistor;
third program instructions to derive a threshold voltage mismatch expression which is related to an amount of an uncorrelated random variation in sub-threshold drain current and which is not directly inversely proportional to a square root of the width of the single-finger device; and
fourth program instructions to use the uplift factors and the threshold voltage mismatch expression within a model to predict at least statistical characteristics of the leakage currents,
wherein the first through fourth program instructions are used in manufacturing an integrated circuit chip, and
wherein the program instructions are executed by the CPU via the computer readable memory.