### The Inversion of the Tunneling Conductance Formula

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relation
The Inversion of the Tunneling Conductance
Formula
Institut für Theoretische Physik, Universität Frankfurt
Frankfurt (Main)
(Z. Naturforsch. 26 a, 1763 [1971] ; received 2 June 1971)
OR
Inversion formulas are given for determining the density
of states by tunneling conductance measurements.
The finite-temperature tunneling current between
two metals with densities of states Nt (x) and N2 (x)
is proportional to
+ oo
— oo
N2(x + v) (fix)-f(x
+ v))
(1)
where / (x) denotes the Fermi function.
When one of the two metals has a slowly-varying
density of states at the Fermi level the normalized
tunneling conductance can be written as
« w - Ä - i V l ^ w «
1
' ^2 T-
* sign (x — fi) f d3r Im G(r,r, x).
(7)
stands for the retarded and advanced case respectively.
We have here omitted that part of the integrand which
has no poles in the analytical region of GA(GR) and
gives no contribution to OR (OA) . Applying the functional equation
ip'(x)-rp'(l+x)
=x~2
(8)
we find
o(x + ijiT) —o(x—inT)
= 2 n i T(dN/dx).
(9)
Another inversion formula is obtained by expanding
(9) in powers of temperature and integration over v
N(v) =o( v)
<2>
The density of states is connected with the single-particle Green's function via 1
N(x) = -
(6)
where xp (x) is the first derivative of the digamma
function, which has poles on the negative real axis,
we get
H A N S ENGLER
I(v) <x f dxNt(x)
TI2 sech2 n x = ip'(h+ i x) +yj'(h — ix)
(3)
In terms of the retarded and advanced Green's functions, which are analytic in the upper and lower half
planes respectively, the density of states is given as
+
(xT)'
3!
d2a
{ti T)4
d4a
-
Av2
5!
dv*
+
(10)
In the general case of the formula (1) the density of
states of the one metal can be obtained by means of
Fourier transform when that of the other is known.
Performing (1) we find
+oo
I (v) cosech (v/2 T)<x.
f dx Nt (x) sech (x/2 T)
-oo
•N2(X
+ V)
(11)
sech((x + v)/2 T) .
This equation can be solved by Fourier transformaThe tunneling conductance can be represented by the
difference of the two functions
ÖR>A(V)
=
J^iT
+jdxGR,A(p,x)
sech2^
tion2-
Wi, 2 (y) = f dx
A . A . ABRIKOSOV, L . P . GORKOV, and I . Y E . DZYALOSHIN-
SKII, Quantum Field Theoretical Methods
Physics, Pergamon Press, New York 1965.
in Statistical
eixy
Nh 2 (x) sech (x/2 T),
(12)
— oo
+ oo
Hy) = —foo dx eixy I (x) cosech (x/2 T).
(5)
1
.N.Cy),
+ oo
.
The hyperbolic secant is a meromorphic function with
simple poles on the imaginary axis. Making use of the
%)
I would like to thank Prof. Dr. P. FULDE for his interest
in this problem.
2
E. C.. TITCHMARSH, Introduction to the Theory of Fourier
Integrals, Oxford 1959.
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