Local Density of States in Mesoscopic Samples from Scanning Gate Microscopy

Julian Threatt

EE235

Scanning Gate Microscopy (SGM)• Scanning probe microscopy (SPM) is used to measure the local

electronic properties of mesoscopic structures. • Scanning tunnel microscopy (STM) is a type of SPM that capable of

directly imaging the density of states of a system.• Scanning gate microscopy (SGM) was developed from SPM in

order to obtain information similar to information obtained from STM for structures buried beneath insulating material.

• The gate acts as a local electrostatic potential on the electronic system and allows us to obtain 2D conductance images.

• SGM is used to study nanostructures based on heterostructures such as quantum dots and quantum point contacts as well as nanotubes and nanowires.

LDOS-ΔG

ji

Rji

Rji

N

RL gGVGG2

,

2

,

)1(2

0

FRjiFi eGex ,Im

1;

FRjiFi eGUex

G

G,Re2;

The conductance G of a multistate system connected to single channel leads is given by:

The local density of states (LDOS) is proportional to the imaginary part of the diagonal elements of the retarded Green’s function:

Considering the effect of the charged scanning tip on the transmission we find that the conductance correction is linear in the charged tip U:

The variation in conductance due to the scanning potential and the LDOS are related in the same way as the real and imaginary part of greens function by a Kramers-Kronig relation:

Rji

Rji

Rji GdPGHG ,,, Im

11ImRe

Single Channel Transmission• In order for the potential to give

rise to a linear correction the potential must be less than the Fermi energy minus the on-site energy.

• Also the spatial range of the potential must be less than half the Fermi wavelength.

• In the single channel regime the Hilbert transmission adds π/2 phase shift.

• Finally the Hilbert transform is the convolution of the LDOS and the inverse of the Fermi energy and can be viewed as a filter for the states closest to the Fermi level.

Multi-Channel Transmission

• The LDOS is dominated by the states closest to the Fermi energy. Therefore in the multichannel regime the restrictions on spatial resolution and potential are dependent on the states closest to the Fermi energy.

Conclusion

• It has been demonstrated how LDOS can be determined through SGM.

• The relationship is direct in the case of single channel devices.• The relationship only holds up for systems where the states close to

the Fermi energy dominate the LDOS.