We have fitted our data to analytical expressions obtained from Borodin
et al. [10] who discuss the potential profile on the face of the crystal with lateral current contacts. They considered a metal electrode of widthl with its middle atx= 0. The other contact is at x=∞. The potential U(x) on the bottom face of the crystal (y= 0) is
U(x) =−Et √ A π arc cosh cosh πl 2t√A −exp πx t√A sinh πl 2t√A for |x|> l 2, (2.1) where E is the electric field far from the current contact (x → ∞). A negativex denotes positions beyond current contacts.
We have performed numerical calculations of the Laplace equation, which agree with the analytical potential profile on the bottom face of the crystal given by Eq. 2.1. Fig. 2.1a shows the current distribution near a current contact as determined from numerically solving the Laplace equation in two dimensions
d2U(x, y)
dx2 +
d2U(x, y)
dy2 = 0, (2.2)
whereU(x, y) is the potential distribution. The direction of x is indicated in Fig. 2.1;yis directed in the thickness direction. The boundary conditions are:
• U(x,0) = 0 for|x| ≤l/2
• dUdy(x,0) = 0 for|x|> l/2 • dUdy(x,t) = 0
• dUdx(x,y) =E forx→ ∞ and 0≤y≤t
The current is injected perpendicular to the length of the crystal. The elec- tric field just above the contact is very high and the current even flows beyond the contact. As a consequence, the voltage probes beyond the cur- rent contacts detect a voltage of opposite sign and therefore the spreading resistance is negative beyond the contacts. However, the net current through the entire cross section is zero beyond current contacts.
We use Eq. 2.1 to derive an estimate for the resistance ratio RS
R0, which
is the quantity obtained from our measurements. Suppose the potential difference between two voltage probes equals ∆U. Then, for small probe spacingL, the potential difference ∆U ≈ dUdxL, where dUdx is the local electric field taken in the middle of two adjacent voltage probes and the resistance
References 33
ratio RS
R0 equals
dU/dx·L
E·L . We introduce a dimensionless contact widthY = πl
4t√A. For narrow contacts (Y 1), RS R0 can be expressed as RS R0 = exp (4Y x/l) 2Y r 1−exp(4Y x/l) 2Y 2 −1 for |x|> l 2. (2.3)
The spreading resistance RS approximatesR0 when measuring far from
the current contact, so that RS
R0 goes to unity forx→ ∞. Beyond the current
contact the spreading resistance is negative and goes to zero for x → −∞. The assumption that L is small is valid for all our samples. This has been checked by fitting our data to Eq. 2.1 using the exact ∆U and comparing them to the fits to Eq. 2.3.
We have fitted our data with l = 100 nm and using Y as the single fit parameter within and beyond contacts. Good fits were obtained on all samples in the temperature range studied, see the solid lines in Fig. 2.1b. The values of Y obtained at T = 120 K are listed in Table 2.1 and all are consistent with the assumptionY 1.
From the definition of Y the longitudinal length scale t√A and the anisotropy A are deduced. The anisotropy at T = 120 K of o-TaS3 and
NbSe3is on the order of 103 and 102, respectively. The measured anisotropy
of the thin samples are smaller than those of the thicker samples. Most likely, this is because a more reliable fit can be made to the data of thicker samples and samples that have a higher anisotropy.
We have also determined Y at other temperatures and Y is approxi- mately temperature independent. To get a better estimate of A(T), thicker crystals and a more accurate determination of the crystal’s thickness are needed. In Chapter 3, another technique is used to measure the anisotropy of NbSe3 in both the crystallographic a*-axis as the c-axis.
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