Exercise set 2

Exercise set 2
1. The measurement setup depicted below is used to measure the resistance of a two-dimensional
electron gas in GaAs (m∗ = 0.0067m) at the temperature T = 4.2K. The Hall bar sample
and a 10M Ω resistor are connected to a voltage source U0 = 1V . the resistance of the sample
is small compared to 10M Ω. A magnetic field B can be applied normal to the plane. Between
contacts 1 and 2 (separation L = 100µm) a voltage U = 10µV is measured at zero magnetic
field. Between contacts 2 and 3 the hall voltage UH = 200µV is measured at B = 1T . the
width of the sample is W = 30µm.
(a) What is the longitudinal (specific) resistivity ρxx and the transverse resistivity ρxy of the
electron gas?
(b) calculate the mobility µ, the scattering time τ , and the mean free path l of the electron
gas at the Fermi energy from the measured resistivities
2. Within the Drude model, the conductance GD of a diffusive wire made from a two-dimensional
electron gas in the quantum limit is given by
GD =
W ns e 2 τ
L m∗
(1)
where W is the width of the wire, L is it’s length, ns is the sheet electron density, and τ
is the Drude scattering time. Within the framework of the Landauer-Buttiker theory, the
corresponding conductance can be written as
GLB =
e2
MT
h
(2)
where M is the number of occupied modes in the wire, and T is the average transmission per
mode.
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(a) Estimate the number of modes in the wire, given the width W and the Fermi wavelength
λF
(b) Estimate the average transmission T per mode by comparing GD and GLB . Hint: rewrite
GD in terms of the mean free path l and the Fermi wavelength λF
3. Consider a perfectly ballistic very long quantum wire connected to two big electron reservoirs.
The central piece of the wire is narrower, hosting only M modes, while the main part of the
wire has N > M modes.
(a) For the purpose of estimating the 2 probe resistance of the wire, We need the resistance
of the central piece of the wire as a 4 probe resistance; explain why.
2 T
(b) Estimate this resistance by extending the single-channel 4 probe result G = 2eh R
to the
M
case of many channels assuming each channel has T = N . Motivate this assumption.
(c) Find the 2 probe resistance of the whole wire. Try to interpret the result.
(For a more elaborate derivation you may see Landauer 1989)
4. We investigate a three-terminal device which is based on a two-dimensional electron gas in
the quantum limit. A schematic of the sample with external circuit is depicted below. We
neglect the resistances of the ohmic contacts. The number of modes in the three ideal leads
is Ni , (i = 1, 2, 3), the reflection in contact i is Ri , and the transmission from contact i to j is
Tji .
(a) Calculate the relation between V /Vbias and the Tij within the Landauer-Buttiker theory.
(b) Discuss what it means for the transmissions T21 and T23 if R2 becomes very large. How
large can R2 be at most?
(c) Discuss which voltage V is measured if R2 tends towards it’s maximum value. Experimentally this could, for example, be achieved by placing a gate across lead 2 and depleting
the electron gas with a negative voltage.
(d) Show that, in this limit, the conductance of the structure approaches that of a twoterminal device.
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