chapter 4 series resistance, channel length and width, and threshold voltage
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CHAPTER 4
SERIES RESISTANCE,CHANNEL LENGTH AND WIDTH,
AND THRESHOLD VOLTAGE
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4.1 INTRODUCTION
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• Semiconductor devices is degraded by series resistance.
• Series resistance depends on semiconductor resistivity, contact resistance, and the geometrical factors.
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4.2 PN JUNCTION DIODES
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Current-Voltage
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Equivalent circuit of a
diode.
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Log(Ι) versus V for a diode with series resistance. The upper dashed line is for rs=0
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Open-Circuit Voltage Decay
Open-circuit voltage decay of a pn junction showing the voltage discontinuity at t=0
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Capacitance-Voltage
1. f low, such that 2πfrsC<<1, to obtain C 2. f high, such that 2πfrsC>>1, calculate rs from the data of Cm vs. f.3. This method is used when DC method is unavailable, such as in MOS capacitor.
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4.3 SCHOTTKY BARRIER DIODES
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Series Resistance
The method of extracting rs for pn diode
can also be used for Schottky diode.
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Norde function F is defined as:
It can also be written as:
For low voltage Irs<<V, dF/dV 1/2-1=-1/2≒
For high voltage Irs ≒ V, dF/dV -1/2+1=1/2≒
Therefore, there exists a Fmin.
From Fmin. to determine Vmin. and then find
out Imin..
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Another method is define H as:
Plot H vs. I, the slope is rs and the intercept
is nψB.
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A modified Norde function is defined as:
Plot F1 vs. V for different Temp., each curve
has a F1min , which has a corresponding Vmin
and Imin.
Plot the left side of the above equation vs. q/kT,then n and A* can be extracted from the slope and the intercept, respectively.
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4.4 SOLAR CELLS
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SOLAR CELLS
Equivalent circuit of a solar cell.
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Multiple Light Intensities
Current-voltage characteristic of a solar cell.
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Series resistance determination of a solar cell.
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For Idk=Iph=Isc
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Constant Light Intensity
This method (area method) is suitable
for concentrator cells with high Iph, it
overestimates rs at 1-sun.
This method (area method) is suitable
for concentrator cells with high Iph, it
overestimates rs at 1-sun.
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Constant Light Intensity
For high intensity flash
illumination method,
neglecting rsh, I V≒ oc/(RL+rs).
By varying the load
resistance RL at constant
light intensity, we have
RL should be on the order of rs.
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Shunt Resistance
Rewrite the above equation as
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For rs<<rsh and Iscrs<<nkT/q,
rs 0.1Ω, I≒ sc 3mA it becomes≦
At very low light intensities, the second term is neglected, then
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4.5 BIPOLAR JUNCTION TRANSISTORS
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BIPOLAR JUNCTION TRANSISTORS
An npn bipolar junction transistor and its parasitic resistances.The base resistance is composed of intrinsic and extrinsic resistance.
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Gummel plots showing the effects of emitter-base space-charge region recombination (n≈2), quasineutral region recombination (n≈1), and series resistance.
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Emitter Resistance
Emitter resistance measurement setup and IB -VCE plot.
RE≒1Ω for discrete BJT and around 5~100Ω for IC transistors.
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Emitter ResistanceAnother method is to supply current from B1 only, and no current flow through B2. Then
VBE2=VBEeff+IERE.
RE=(VBE2-VBEeff)/IE
Where )1)nkT
qV(exp(
II BEeff0C
1B
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Collector Resistance
Common emitter output characteristics. The two lines show the limiting values of RC .
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Base Resistance
Plot ΔVBE/IB vs β has a slope of RE and an intercept on the ΔVBE/IB axis of RB+RE.But it is difficult to change β of a transistor without changing other parameters.
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Base Resistance
Measured device characteristics for a self-aligned, high-speed digital BJT. The βmust be varied in this measurements, and RBi/β must be a constant.
Rewrite the equation of IB in the following form
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Base Resistance
The intrinsic base resistance for rectangular emitter with 1 contact is: RBi=RshiW/3L.
The intrinsic base resistance for rectangular emitter with 2 contact is: RBi=RshiW/12L.
Square emitter with contact on all sides:RBi=Rshi/32. (W=L)
Circular emitter with contact all around:RBi=Rshi/8π.
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Equivalent emitter-base portion of the “two-base contact” BJT.
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Measured base resistance versus emitter window width as a function of base-emitter voltage. d is the deviation between emitter window and the effective base width. Rshi is a function of VBE due to base conductivity modulation.
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Frequency Measurements on RB
The input impedance circle method measures the Zin as a function of frequency. The real axis intersections give:
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4.6 MOSFETs
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Series Resistance and Channel Length— I-V
(a)A MOSFET with source and drain resistances, (b) device cross section showing the actual gate length L and Leff =L-∆L with ∆L=2δL. The substrate resistance is not shown.
ξ=0.37, 0.58, 0.75, 0.9;
xch: channel thickness
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DSTGS
DSTGSD RVVk
VVVkI
)(1
)(
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Method 1
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Rm versus L as a function of gate voltage.
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• Short channel devices have a channel length dependent threshold voltage.
• VG , L↗ eff , R↗ SD .↘
• Keep VG at a fixed value, change VT by changing VSB.
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Method 2
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Plot E vs. (VGS-VT) as a function of L
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(a) E versus (VGS –VT )as a function of gate length. These lines have intercept Ei and slope m; (b) Ei and m versus gate length. Ei=0 when L=ΔL, and m=RSD when L=ΔL. The slopes give μo and θ.
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Rmi=RSD+θm
Method 3
(a)Rm versus 1/(VGS -VT), the slope is m=(L-ΔL)/WeffμoCox;
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(b) slope m versus L, the slope is 1/WeffμoCox, (c) Rmi versus m.
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Method 4
)(1
)(
TGSo
DSTGSoD VVRk
VVVkI
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Plot vs. VGS the intercept gives V
T and the slope gives ko.
mD gI /
Plot 1/ ko vs. L, the intercept gives ΔL.
R’ is obtained from ID= ko(VGS-VT)(VDS-IDR’)
Plot R’ vs. 1/ ko, the intercept gives RSD and the slope gives θ.
OSD kRR
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Method 5
For Vds1>>ID1RSD and Vds2>>ID2RSD
Plot ID1/ID2 vs. (ID1-ID2), the slope is k1RSD/k2VDS,
the intercept is k1/k2.
If the assumption is not valid, then plot
(VDS2/ID2-VDS1/ID1) vs. (VDS1/ID1), the intercept is
RSD, the slope is (L2-L1)/(L1-ΔL) which gives
ΔL.
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Method 6
gm and gd are measured at linear region for VDS
= 25~50mV. Define r as
A long channel device with Lref and a short channel device with L are required.
Define Δλ as
Plot Δλ vs. (VGS-VT), the intercept is ΔL, and
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Method 7This shift and ratio method needs one large (Lo) and several constant width varying length (L) small devices. Redefine Rm as:
Neglect the effect of dRDS/dVGS
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Plot S vs. VGS for the large and one small devices to obtain L and VT. Shift the curve by δ and calculate r(VGS)=S(VGS)/S(VGS- δ).When δ=VT1-VT2 r is almost a constant, and the mobility will be identical, which yields :
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The conventional I-V methods reach their limit when Leff 0.1μm, because R≒ ch is no longer a linear function of Leff due to short channel effects.
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Drain Induced Barrier lowering (DIBL)
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Channel Length--C-V
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CGC –VGS curse; W=10μm, tox =10nm, NA =1.6×1017 cm-3 .
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In order to determine the channel width, devices with constant length and varying width are used.The drain conductance is given by:
Does a plot of gd vs. W gives ΔW?RS and RD varies with W!!
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The drain current can be written by:
Plot ID vs. W gives ΔW for ID=0. The measured drain resistance is:
From which ΔW can be obtained.
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Differentiating the above equation and define
Plotting m vs. W gives ΔW.
We may also use the oxide capacitance of constant length and varying width transistors to determine ΔW.
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4.7 MESFETs AND MODFETs
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MESFETs AND MODFETs
Cross section of a MODFET showing the various resistances. RG is the resistance of the wide band gap semiconductor.
Method 1
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Define
then the drain acts as a voltage probe.
If ID=0, α 0.5.≒
For IG<<ID,
Method 2
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Method 3
Devices with various channel lengths are used,
and operated in linear region. I-V measurements
are made with one of the electrodes floating.
When the gate is floating, the resistances are:
With source floating,
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Define
where LG is the channel length, R and G are the resistance and conductance per unit length then
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Plot of RGS (fg), RGD (fg), RSD (fg) versus 1/ RGS (fg). Reprinted after Azzam et al.(Ref.92) by permission of IEEE.
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4.8 THRESHOLD VOLTAGE
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THRESHOLD VOLTAGE
ID -VGS curve of a MOSFET near the threshold voltage. Modeled using Leff 1.5μm, tox =25nm, VT,start =0.7V, VD =0.1V.
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DSDSTGSD VVVVL
WkI )5.0(
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Threshold voltage determination by the linear extrapolation technique. VDS =0.1V, tox =5.6nm, W/L=20μm/20μm.
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Plot gd/(gm)1/2 vs. VGS
Plot (ID, sat)1/2 vs. VGS
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Threshold voltage determination by the saturation extrapolation technique; tox=5.6nm, W/L=20μm/20μm.
2TGS
OXeffsat,D )VV(
L
CmWI
m is a function of doping density
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satTGSOXD v)VV(WCI
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Threshold voltage determination by the threshold drain current technique. (a) Measurement circuit, (b) experimental data; tox=5.6nm, W/L=20μm/20μm, VBS=0. Data courtesy of Y. B. Park, Arizona State University; MOSFET courtesy of J. Sanchez, Intel Corporation.
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Threshold voltage determination by the subthreshold technique; tox=5.6nm, W/L=20μm/20μm.Data courtesy of Y. B. Park,
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Drain Current Ratio
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PSEUDO MOSFET