Model the vacuum gap between a scanning tunnelling microscope tip and a conducting…

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction

Solid State Physics, Ch Seeing atoms: electron and scanning-probe microscopy

Problem

Model the vacuum gap between a scanning tunnelling microscope tip and a conducting sample as a rectangular potential barrier of height

\phi

(the work function) above the common Fermi energy

E_F

of the two electrodes, and of width

d

. The applied bias

V

is small compared with

\phi

(

eV\ll\phi

), so

\phi

may be treated as constant over the range of energies that contribute to the current. At zero temperature only electrons within the bias window of width

eV

near

E_F

can tunnel elastically into empty states on the other side. Take the WKB transmission coefficient for a thick rectangular barrier as

T(E)\approx e^{-2\kappa d}

, with

\kappa=\sqrt{2m_e(E_F+\phi-E)}/\hbar

for an electron of energy

E

. ((a)) Show that, under these conditions, the tunnel current obeys

I\propto Ve^{-2\kappa d}

, with

\kappa

evaluated at

E=E_F

. ((b)) Show that

\kappa=0.512\sqrt{\phi[\mathrm{eV}]}\ \mathrm{\AA^{-1}}

, with

\phi

in electronvolts, using

m_e=9.109\times10^{-31}\,\mathrm{kg}

\hbar=1.055\times10^{-34}\,\mathrm{J\,s}

and

e=1.602\times10^{-19}\,\mathrm{C}

. ((c)) Evaluate

\kappa

for a tip-sample pair with an average work function of

4.5\,\mathrm{eV}

Related theory

Quantum Mechanics

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