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The previous discussion is repeated slightly more formally here. The two conductivity tensors are written explicitly for an isotropic 2D gas. The boundary condition (bc)
J
x = 0 leads to
S
= \alpha/\sigma (neglecting the small off-diagonal terms). The current along the
y
-axis is comprised of four terms, proportional to
E
x,
E
y, \partial_x
T
and \partial_y
T
. By good fortune, the Hall thermal gradient \partial_y
T
is very small in the cuprates (because the thermal conductivity
k
xx is dominated by the phonons). Recent measurements of
k
xy in the cuprates allow \partial_y
T
to be estimated in this experiment [Y. Zhang et al. PRL 1999]. This is not true in conventional metals. One is either restricted to using thin-film samples only, or (for crystals) resorts to heroic measures to enforce the condition \partial_y
T
= 0 (for e.g. by heat-sinking the two side edges with highly conducting strips). Fortunately, in the cuprates, the term in \partial_y
T
is ~30 times smaller than the other 3. Hence,
J
y is given as shown below. Setting
J
y to zero and solving for
E
y, we arrive at the expression for
vN
given in the last line. We note that the first term -
S
*tan(\theta)/
B
can be measured independently.
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