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The Physics of NanoelectronicsTransport and Fluctuation Phenomena at Low Temperatures$

Tero T. Heikkilä

Print publication date: 2013

Print ISBN-13: 9780199592449

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199592449.001.0001

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(p.253) E Reflection coefficient in electronic circuits

(p.253) E Reflection coefficient in electronic circuits

Source:
The Physics of Nanoelectronics
Publisher:
Oxford University Press

A popular means of measuring the impedance of nanoelectronic samples fast and accurately is to connect the sample as a termination impedance at the end of an electronic waveguide and measure the reflection1 coefficient of waves sent to the waveguide. This appendix presents the calculation of the reflection coefficient for a waveguide terminated with impedance Z(ω‎).

A waveguide is characterized by an inductance L and a capacitance C per unit length. We may derive the equation of motion for the voltages and currents in the waveguide by first discretizing it into units of length h as in Fig. E.1 The currents I(x i) = I i flowing through inductors and the stray currents I c(x i) = I ci flowing through the capacitors are related with the voltages V(x i) = V i of the nodes of the circuit by

I i = 1 h L t ( V i ( t ) V i 1 ( t ) ) d t
(E.1a)
I c i = C h V ˙ i ( t ) ,
(E.1b)

where V ˙ i ( t ) denotes the time derivative of the voltage. Current conservation into node i yields

I i I i + 1 + I c i = 1 h L t ( 2 V i ( t ) V i + 1 ( t ) V i 1 ( t ) ) d t C h V ˙ i ( t ) = 0.
(E.2)
E Reflection coefficient in electronic circuits

Fig. E.1 Discretization of a waveguide.

We can identify the first term as the discretization of the second derivative. Dividing both sides by h, taking the limit h→ 0 and differentiating once by time yields the wave equation

t 2 V ( x , t ) = 1 L C x 2 V ( x , t ) .
(E.3)

Here we identify the effective speed of light, c = 1 / L C . For a given voltage profile, the current can be obtained from eqn (E.1a) in the limit h → 0,

I ( x ) = 1 L t x V ( x , t ) d t .
(E.4)

Let us assume that the waveguide is terminated at x = 0 by an impedance Z(ω‎) as in Fig. E.2. There we obtain for the Fourier transformed voltages and currents V(0,ω‎) = Z(ω‎) I(0,ω‎). Combining this (p.254) with eqn (E.4) hence gives a boundary condition

Z ( ω ) x V ( x = 0 , ω ) = i ω L V ( x = 0 , ω ) ,
(E.5)

where the integral over time is replaced by 1/iω‎.

E Reflection coefficient in electronic circuits

Fig. E.2 Waveguide terminated with impedance Z(ω‎).

Now finding the reflection coefficient is rather straightforward. Assume we send a wave with amplitude V 0 and frequency ω‎ along the waveguide towards the sample, and get a reflected wave with amplitude rV 0. The total wave is thus

V ( x , t ) = V 0 e i ω t ( e i ( ω / c ) x + r e i ( ω / c ) x ) ,
(E.6)

where the wave number has been chosen so that the wave equation (E.3) is satisfied. The boundary condition (E.5) yields for the Fourier amplitude at frequency ω‎,

Z ( ω ) i ω V 0 ( 1 r ) = i ω L c V 0 ( 1 + r ) .
(E.7)

Solving this gives the reflection coefficient

r = Z ( ω ) L c Z ( ω ) + L c = Z ( ω ) Z 0 Z ( ω ) + Z 0 ,
(E.8)

where Z 0 = L c = L / C is the characteristic impedance of wave-guide the waveguide. Measuring the changes in the amplitude and/or the phase of r as a function of some control parameter, the sample impedance can be measured very accurately. For the coaxial cables typically used in microwave experiments, Z 0 is often of the order of 50 Ω‎.

Notes:

(1) Or the transmission coefficient, but here we only focus on the reflection coefficient.