Exerci
1.2
I V −
Fig.E1.2
Problem:Is there another way to derive the sheet resistance expression?
Solution:Consider a sample of thickness t and resistivity ρ.The four probes are arranged as in Fig.E1.2.Current I is injected at probe I +and spreads out cylindrically symmet-ric.By symmetry and current conrvation,the current density at distance r from the probe is
J =
I 2πrt The electric field is
E =Jρ=
Iρ
2πrt =−dV dr
Integrating this expression gives the voltage drop between probes V +and V −,located at
distances s 1and s 2from I +as
V s 2
V s 1
dV =−
Iρ
2πt
s 2
s 1
dr r ⇒V s 1−V s 2=V 12=Iρ2πt ln s 2
s 1
By the principle of superposition,the voltage drop due to current injected at I −is
V 34=−Iρ2πt ln s 3
s 4 leading to
V =V 12−V 34
=Iρ2πt ln s 2s 3
s 1s 4
For a collinear arrangement with s 1=s 4=s and s 2=s 3=2s
ρ=
πt ln (2)V I ;R sh =
πln (2)V I
Exerci 1.3
Problem:What does sheet resistance mean and why does it have such strange units?
Fig.E1.3
Solution:To understand the concept of sheet resistance,consider the sample in Fig.E1.3. The resistance between the two ends is given by
R=ρL
A
=ρL
W t
=ρ
t
L
W
ohms
Since L/W has no units,ρ/t should have units of ohms.Butρ/t is not the sample resis-tance.To distinguish between R andρ/t,the ratioρ/t is given the units of ohms/square and is named sheet resistance,R sh.Hence the sample resistance can be written as
R=R sh L
W
ohms
The sample is sometimes divided into squares,as in Fig.E1.4.The resistance is then given as
R=R sh(ohms/square)×Number of squares=5R sh ohms
Looking at it this way,the“square”cancels.
The sheet resistance of a miconductor sample is commonly ud to characterize ion implanted and diffud layers,metalfilms,etc.The depth variation of the dopant atoms need not be known,as is evident from Eq.(1.19).The sheet resistance can be thought of as the depth integral of the dopant atom density in the sample,regardless of its vertical spatial doping density variation.A few sheet resistances are plotted in Fig.E1.5versus sample thickness as a function of sample resistivity.Also shown are typical values for Al,Cu and heavily-doped Si.
Exerci1.4
Problem:For the carrier density profiles in Fig.E1.6,do the sheet resistances of the
three
layers differ?
Fig.E1.4
10−3
1010103
10−210−1100101102S h e e t R e s i s t a n c e (Ω/s q u a r e )
Thickness (cm)
Fig.E1.5
Solution:Eq.(1.19)shows the sheet resistance to be inverly proportional to the
conductivity-thickness product.For constant mobility,R sh is inverly proportional to the area under the curves in Fig.E1.6.Since the three areas are equal,this implies that R sh is the same for all three cas.In other words,it does not matter what the carrier distribution is,only the integrated distribution matters for R sh .
Four-point probe measurements are subject to further sample size correction factors.For circular wafers of diameter D ,the correction factor F 2in Eq.(1.12)is given by 16
F 2=
ln (2)
ln (2)+ln {[(D/s)2+3]/[(D/s)2−3]}
(1.20)
F 2is plotted in Fig.1.6for circular wafers.The sample must have a diameter D ≥40s for F 2to be unity.For a probe spacing of 0.1588cm,this implies that the wafer must be at least 6.5cm in diameter.Also shown in Fig.1.6is the correction factor for rectangular samples.6
The correction factor 4.532in Eq.(1.17)is for collinear probes with the current flowing into probe 1,out of probe 4,and with the voltage nd across probes 2and 3.For the current applied to and the voltage nd across other probes,different correction factors obtain.17For probes perpendicular to and a distance d from a non-conducting boundary ,the correction factors,for infinitely thick samples,are shown in Fig.1.7.2It is obvious from the figures that as long as the probe distance from the wafer boundary is at
least
n (x )
Fig.E1.6
00
10203040
0.20.4
0.6
0.81F 2
D /s
Fig.1.6Wafer diameter correction factors versus normalized wafer diameter.For circular wafers:D =wafer diameter;for rectangular samples:D =sample width,s =probe spacing.
0.5
1
1.5
2
F 31, F 32, F 33, F 34
d /s
Fig.1.7Boundary proximity correction factors versus normalized distance d (s =probe spacing)from th
e boundary.F 31and F 32are for non-conducting boundaries,F 33and F 34are for conducting boundaries.
three to four probe spacings,the correction factors F 31to F 34reduce to unity.For most four-point probe measurements this condition is easily satisfied.Correction factors F 31to F 34only become important for small samples in which the probe is,of necessity,clo to the sample boundary.
Other corrections must be applied when the probe is not centered even in a wafer of substantial diameter.16For rectangular samples it has been found that the nsitivity of the geometrical correction factor to positional error is minimized by orienting the probe with its electrodes within about 10%of the center.11For square arrays the error is minimized by orienting the probe array with its electrodes equidistant from the midpoints of the sides.There is also an angular dependence of the placement of a square array on the rectangular
