IEEE 2007 Custom Intergrated Circuits Conference (CICC)

A Wide Range Spatial Frequency Analysis ofIntra-Die Variations with 4-mm 4000 x 1 Transistor
Arrays in 90nm CMOSDavid Levacq, Takuya Minakawa#, Makoto Takamiya*, and Takayasu Sakurai
Center for Collaborative Research, University of Tokyo, Japan*VLSI Design and Education Center, University of Tokyo, Japan#
now with Canon Inc.
E-mail: levacq@iis.u-tokyo.ac.jp

Abstract— In order to investigate the systematic intra-die
variations, the intra-die threshold voltage and on-current
variations are measured thanks to 4-mm 4000 x 1 transistor
arrays with 1 µm transistor-pitch in a 90nm CMOS technology,achieving the widest spatial distribution range. The spatial
frequency analysis of the variations indicates that bothvariations are random across 4 mm. The dependence of bothvariations on body bias is also measured and the relationships
between threshold voltage variations and on-current variations
are analyzed by using the alpha-power law model.

I. INTRODUCTION
The increased variation of the MOSFET transistor device
properties that come with the technology downscaling is amajor issue for VLSI designers. The variations usually havesystematic and random components. The systematic variations
can be compensated with the chip-level or the block-level
adaptive techniques, while managing the random variations isa difficult task because the correction of each transistor is notpractical. It is therefore very important to distinguish thesystematic variations and the random variations. Analyzing
the frequency distribution of the spatial variations is aneffective method to extract the systematic components. For
example, spatial frequency characterization provides anoptimum domain size when multi-domain adaptive body bias
within a chip [1] is used to compensate the intra-die variations.
Although the measurements of the intra-die variations withtransistor array circuits have been previously reported [2-6],
none of them shows the spatial frequency. Wide spectrum of
the spatial frequency is also required, because the minimum
pitch of the transistors and the chip size determine the highest
and the lowest frequency respectively. The conventional
circuits [2-6], provide narrow spectrum, because they have 2dimensional arrays and are not designed for the spatial
frequency measurement. In this paper, the wide spatial

1-4244-1623-X/07/$25.00 ©2007 IEEE

spectrums of intra-die variations are measured and analyzed
with the newly developed 4000 x 1 (= 1-dimensional)transistor arrays.
II. TRANSISTOR ARRAY CIRCUITS
Fig. 1 shows the structure of the transistor array circuits.
A unit consists of 1000 x 1 transistors array and four units areused for the measurement of the 4000 transistors. The pitch
between the transistors is 1 µm. The key challenge for thearray circuits is to reduce the subthreshold leakage current due
to the 999 unselected transistors well below the drain currentof a selected transistor. In order to obtain more than one
Vbs
GND
Vsel
VunselVD(1.0V)…
Data D Q

DQ

DQ

DQ

…

DQ

Vs(0V)

clk

MP-19-1

VDP1

Vsel

Measuredtransistor
Vbs

VunselVSFig. 1. Transistor array circuits for variability measurement.

257

1000 Transistor Array

4mm

Forward body bias (FBB) (Vbs=0.3V)# of Tr's

600
4001mm200
0

Zero body bias (ZBB) (Vbs=0V)# of Tr's

600
400
2004mm0Reverse body bias (RBB) (Vbs= -1.0V)

Fig. 2. Chip micrograph of transistor array circuits.

Fig. 2 shows the microphotograph of the chip fabricatedwith 1V 90nm CMOS process. Two sets of 4-mm 4000 x 1transistor arrays are horizontally and vertically placed on thechip respectively. This chip has four different transistor
arrays: both nMOSFETs and pMOSFETs with the minimum
gate length and 0.2 µm and 1.0 µm gate width. A deep n-wellis used to externally control the body bias (Vbs) of themeasured transistors to investigate the effect of Vbs onthreshold voltage (VTH) variations.
III. EXPERIMENTAL MEASUREMENT
In this chapter, nMOSFETs in the vertical array arediscussed, because the measured results show no significant
differences between the vertical array and the horizontal array,and the measured trends of pMOSFETs are similar to those of# of Tr's

hundred times drain current for the selected transistor of the999 unselected transistors, the drain voltages of unselectedtransistors are lowered by P1 to reduce the subthresholdleakage due to DIBL and the gate voltages of the unselectedtransistors are negatively biased with Vunsel. The voltage drops
across P1 and VD/VS lines are less than 1 mV by using wide
transistors and wide metal lines respectively.

600
400
20000.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Normalized Threshold Voltage

Fig. 4. Measured VTH histogram for different body bias (W = 0.2 µm).

nMOSFETs. VTH and the on-current (ION) of the 4000
nMOSFETs are measured and analyzed with 1 V drain-tosource bias. Fig. 3 shows ID – VGS characteristics and therelationship between ION and VTH for different body bias
conditions. The line at zero body bias (ZBB) is a starting line.
When the positive body bias is applied to the body, theforward body bias (FBB) decreases VTH and increases ION. Incontrast, when the negative body bias is applied to the body,the reverse body bias (RBB) increases VTH and decreases ION.
The relationships between VTH variations and ION variations
are measured and analyzed.

ID
ForwardIONbody bias

Zero
Reverse0VTH

VDD(1V)

VGS

Normalized σVTH (a.u.)

120
100

W = 0.2 µm

8060

W = 1 µm

40200FBBZBB

RBBFig. 5. Body bias dependence of the standard deviation (σVTH).

Fig. 3. ID – VGS characteristics for different body bias conditions.

MP-19-2258

Normalized Threshold Voltage# of Tr's

600 Forward body bias (FBB) (Vbs=0.3V)
400
200
0

Zero body bias (ZBB) (Vbs=0V)# of Tr's

600
400

1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0

Zero body bias
01000

3000

4000(a)200

Normalized On Current

1.4 Zero body bias

# of Tr's

0600 Reverse body bias (RBB) (Vbs= -1.0V)
400
200
00.0

0.2

0.4

0.6

0.8

1.0

1.2

Normalized On Current

1.4Fig. 4 shows the measured VTH histogram for different
body bias conditions when the gate width is 0.2 µm. The
histograms show Gaussian distribution. The body bias
dependence of the standard deviation (σVTH) of the measured
VTH is plotted in Fig. 5 for 0.2 µm and 1 µm gate widths.
FBB decreases both the average VTH and σVTH [7,8] because
FBB reduces the channel depletion width. In contrast, RBB
increases both the average VTH and σVTH. The increase ofσVTH does not directly result in an increase of leakage [9].
σVTH of 0.2-µm gate width is larger than that of 1-µm gate
width [10].
Fig. 6 shows the measured ION histogram for different

00.40.201000

2000

3000

4000

body bias conditions when the gate width is 0.2 µm. Fig. 7
show the body bias dependence of the relative standard
deviation (σION) of the measured ION when the gate width is0.2 µm and 1 µm. FBB increases the average ION but
decreases the relative σION (= σION / average ION). In contrast,RBB decreases the average ION but increases the relative σION.
In order to clarify the relationship between the relative σION
and σVTH, Fig. 7 also shows the calculated body bias
dependence of the relative ION by using the alpha-power law
model [11] in (1) and (2),
1.3

(1)

I ON + σI ON ∝ (V DD − VTH − σVTH ) .
1.3

Measured

W = 1 µm
FBB

CalculatedZBB(2)

VDD is a power supply voltage. The calculated body bias
dependence partially traces the measured body bias
dependence, which indicates that the relative σION is highlycorrelated with the σVTH. The error between the measured
and the calculated values at the gate width of 1 µm is less than14%, while the error at the gate width of 0.2 µm is less than27%. The large error is considered to be due to the mobility
modulation in the narrow channel MOSFETs.

W = 0.2 µm

310.6I ON ∝ (VDD − VTH ) .

620.8(b)Fig. 8. Position dependence of (a) VTH and (b) ION in the 4-mm 4000
transistor arrays with 1 µm pitch at ZBB (W = 0.2 µm).

741.0Transistor location(µm)

851.20.0

Fig. 6. Measured ION histogram for different body bias (W = 0.2 µm).

σION / (average ION) (%)

2000

Transistor location(µm)

RBB

Fig. 7. Body bias dependence of the relative σION.

Fig. 8 gives the position dependence of VTH and ION in the4-mm 4000 transistor arrays with 1 µm pitch at ZBB and thegate width of 0.2 µm. Fourier transform of the Fig. 8 gives theMP-19-3259

1Spatial frequency (µm)

Power(a.u.)

100

101Zero body bias
01E-30.010.1

0.01

1E-3

1E-4

0.1

Ref.[3]

Spatial frequency(1/µm)
(a)

[4]

[5]

[6] This Work

Fig. 10. Comparison of the measured spatial frequency range with the pastworks [3-6].

10Power(a.u.)

ACKNOWLEDGMENT
The authors gratefully acknowledge Dr. Kawahara and Dr.
Yamaoka from Hitachi, Ltd. and Prof. Hiramoto and Prof.
Fujishima from University of Tokyo for valuable suggestions
and discussions. This work is partially supported by MEXT
and Hitachi, Ltd. The test chips have been fabricated throughthe chip fabrication program of VLSI Design and Education
Center, the University of Tokyo, with the collaboration ofSTARC.

10
Zero body bias
01E-3

REFERENCES

0.01[1]0.1

Spatial frequency(1/µm)(b)Fig. 9. Spatial spectrum of (a) VTH and (b) ION in Fig. 8.

spatial spectrum shown in Fig. 9. The horizontal axes are thespatial frequency. For example, a 10-µm cycle VTH variations
gives a spatial frequency of 0.1 (1/µm). Fig. 9 shows noparticular peak, which indicates that the intra-die VTH and ION
variations are random across 4 mm. The intra-die randomvariations correspond with [5,6].
Fig. 10 shows the comparison of the measured spatial
frequency range with the past works [3-6]. This work covers
the widest spatial frequency range ever reported.
IV. CONCLUSIONS
The intra-die VTH and ION variations have beencharacterized with 4-mm 4000 x 1 transistor arrays with
reduced subthreshold leakage of unselected transistors, giving
the widest spatial frequency range ever reported.
Measurements on a 90nm CMOS technology demonstrate thatthose variations are not systematic but random across 4 mm.
The relationships between the measured σVTH and σION forvarious body biases are also analyzed by using the alphapower law model, and the close correlation between σION andσVTH are confirmed.

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