Editing Electron mobility (section)

==Measurement of semiconductor mobility==

===Hall mobility===
{{main|Hall effect}}

[[File:Hall Effect Measurement Setup for Holes.png|right|frame|Hall effect measurement setup for holes]]
[[File:Hall Effect Measurement Setup for Electrons.png|right|frame|Hall effect measurement setup for electrons]]

Carrier mobility is most commonly measured using the [[Hall effect]]. The result of the measurement is called the "Hall mobility" (meaning "mobility inferred from a Hall-effect measurement").

Consider a semiconductor sample with a rectangular cross section as shown in the figures, a current is flowing in the ''x''-direction and a [[magnetic field]] is applied in the ''z''-direction. The resulting Lorentz force will accelerate the electrons (''n''-type materials) or holes (''p''-type materials) in the (−''y'') direction, according to the [[right hand rule]] and set up an electric field ''ξ<sub>y</sub>''. As a result there is a voltage across the sample, which can be measured with a [[High impedance|high-impedance]] voltmeter. This voltage, ''V<sub>H</sub>'', is called the [[Hall effect|Hall voltage]]. ''V<sub>H</sub>'' is negative for ''n''-type material and positive for ''p''-type material.

Mathematically, the [[Lorentz force]] acting on a charge ''q'' is given by

For electrons:
<math display="block">\mathbf F_{Hn} = -q(\mathbf v_n \times \mathbf B_z)</math>

For holes:
<math display="block">\mathbf F_{Hp} = +q(\mathbf v_p \times \mathbf B_z)</math>

In steady state this force is balanced by the force set up by the Hall voltage, so that there is no [[net force]] on the carriers in the ''y'' direction. For electrons,

<math display="block">\mathbf F_y = (-q)\xi_y + (-q)[\mathbf v_n \times\mathbf B_z] = 0</math>

<math display="block">\Rightarrow -q\xi_y + qv_xB_z = 0</math>

<math display="block"> \xi_y = v_xB_z</math>

For electrons, the field points in the −''y'' direction, and for holes, it points in the +''y'' direction.

The [[Electric current|electron current]] ''I'' is given by <math>I = -qnv_xtW</math>. Sub ''v''<sub>''x''</sub> into the expression for ''ξ''<sub>''y''</sub>,

<math display="block">\xi_y = -\frac{IB}{nqtW} = +\frac{R_{Hn}IB}{tW}</math>

where ''R<sub>Hn</sub>'' is the Hall coefficient for electron, and is defined as
<math display="block">R_{Hn} = -\frac{1}{nq}</math>

Since <math>\xi_y = \frac{V_H}{W}</math>
<math display="block">R_{Hn} = -\frac{1}{nq} = \frac{V_{Hn}t}{IB}</math>

Similarly, for holes
<math display="block">R_{Hp} = \frac{1}{pq} = \frac{V_{Hp}t}{IB}</math>

From the Hall coefficient, we can obtain the carrier mobility as follows:
<math display="block">\begin{align}
\mu_n &= \left(-nq\right) \mu_n \left(-\frac{1}{nq}\right) \\
&= -\sigma_n R_{Hn} \\
&= -\frac{\sigma_n V_{Hn} t}{IB}
\end{align}</math>

Similarly,
<math display="block">\mu_p = \frac{\sigma_p V_{Hp}t}{IB}</math>

Here the value of ''V<sub>Hp</sub>'' (Hall voltage), ''t'' (sample thickness), ''I'' (current) and ''B'' (magnetic field) can be measured directly, and the conductivities ''σ''<sub>n</sub> or ''σ''<sub>p</sub> are either known or can be obtained from measuring the resistivity.

===Field-effect mobility===
{{See also|MOSFET}}
{{Distinguish|Wien effect}}
The mobility can also be measured using a [[field-effect transistor]] (FET). The result of the measurement is called the "field-effect mobility" (meaning "mobility inferred from a field-effect measurement").

The measurement can work in two ways: From saturation-mode measurements, or linear-region measurements.<ref name=Rost>{{cite book |author=Constance Rost-Bietsch|title=Ambipolar and Light-Emitting Organic Field-Effect Transistors|url=https://books.google.com/books?id=Xxvt0CkVKaIC&pg=PA17|access-date=1 March 2011|date=August 2005|publisher=Cuvillier Verlag |isbn=978-3-86537-535-3 |pages=17–}}. This reference mistakenly leaves out a factor of 1/V<sub>DS</sub> in eqn (2.11). The correct version of that equation can be found, e.g., in {{cite journal|last1=Stassen|first1=A. F.|last2=De Boer|first2=R. W. I.|last3=Iosad|first3=N. N.|last4=Morpurgo|first4=A. F.|title=Influence of the gate dielectric on the mobility of rubrene single-crystal field-effect transistors|journal=Applied Physics Letters|volume=85|issue=17|pages=3899–3901|year=2004|doi=10.1063/1.1812368|arxiv = cond-mat/0407293 |bibcode = 2004ApPhL..85.3899S |s2cid=119532427|url=http://resolver.tudelft.nl/uuid:868f9c8e-b994-47e8-b2fd-69cca21b1415}}</ref> (See [[MOSFET]] for a description of the different modes or regions of operation.)

====Using saturation mode====
In this technique,<ref name=Rost/> for each fixed gate voltage V<sub>GS</sub>, the drain-source voltage V<sub>DS</sub> is increased until the current I<sub>D</sub> saturates. Next, the square root of this saturated current is plotted against the gate voltage, and the slope ''m''<sub>sat</sub> is measured. Then the mobility is:
<math display="block">\mu = m_\text{sat}^2 \frac{2L}{W} \frac{1}{C_i}</math>
where ''L'' and ''W'' are the length and width of the channel and ''C''<sub>''i''</sub> is the gate insulator capacitance per unit area. This equation comes from the approximate equation for a MOSFET in saturation mode:
<math display="block">I_D = \frac{\mu C_i}{2}\frac{W}{L}(V_{GS}-V_{th})^2.</math>
where ''V''<sub>th</sub> is the threshold voltage. This approximation ignores the [[Early effect]] (channel length modulation), among other things. In practice, this technique may underestimate the true mobility.<ref name=Rost2>{{cite book|author=Constance Rost-Bietsch|title=Ambipolar and Light-Emitting Organic Field-Effect Transistors|url=https://books.google.com/books?id=Xxvt0CkVKaIC&pg=PA19|access-date=20 April 2011|date=August 2005|publisher=Cuvillier Verlag|isbn=978-3-86537-535-3 |pages=19–}} "Extracting the field-effect mobility directly from the linear region of the output characteristic may yield larger values for the field-effect mobility than the actual one, since the drain current is linear only for very small VDS and large VG. In contrast, extracting the field-effect mobility from the saturated region might yield rather conservative values for the field-effect mobility, since the drain-current dependence from the gate-voltage becomes sub-quadratic for large VG as well as for small VDS."</ref>

====Using the linear region====
In this technique,<ref name=Rost/> the transistor is operated in the linear region (or "ohmic mode"), where V<sub>DS</sub> is small and <math>I_D \propto V_{GS}</math> with slope ''m''<sub>lin</sub>. Then the mobility is:
<math display="block">\mu = m_\text{lin} \frac{L}{W} \frac{1}{V_{DS}} \frac{1}{C_i}.</math>
This equation comes from the approximate equation for a MOSFET in the linear region:
<math display="block">I_D= \mu C_i \frac{W}{L} \left( (V_{GS}-V_{th})V_{DS}-\frac{V_{DS}^2}{2} \right)</math>
In practice, this technique may overestimate the true mobility, because if V<sub>DS</sub> is not small enough and V<sub>G</sub> is not large enough, the MOSFET may not stay in the linear region.<ref name=Rost2/>

===Optical mobility===
Electron mobility may be determined from non-contact laser [[photo-reflectance technique]] measurements. A series of photo-reflectance measurements are made as the sample is stepped through focus.  The electron diffusion length and recombination time are determined by a regressive fit to the data. Then the Einstein relation is used to calculate the mobility.<ref>W. Chism, "Precise Optical Measurement of Carrier Mobilities Using Z-scanning Laser Photoreflectance," [https://arxiv.org/abs/1711.01138 arXiv:1711.01138] [physics:ins-det], Oct. 2017.</ref><ref>W. Chism, "Z-scanning Laser Photoreflectance as a Tool for Characterization of Electronic Transport Properties," [https://arxiv.org/abs/1808.01897 arXiv:1808.01897] [cond-mat:mes-hall], Aug. 2018.</ref>

===Terahertz mobility===
Electron mobility can be calculated from time-resolved [[terahertz time-domain spectroscopy|terahertz probe]] measurement.<ref name="UlbrichtHendry2011">{{cite journal|last1=Ulbricht|first1=Ronald|last2=Hendry|first2=Euan| last3=Shan|first3=Jie| last4=Heinz|first4=Tony F.|last5=Bonn|first5=Mischa|title=Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy|journal=Reviews of Modern Physics|volume=83|issue=2|year=2011|pages=543–586|issn=0034-6861 |doi=10.1103/RevModPhys.83.543| bibcode=2011RvMP...83..543U|hdl=10871/15671|url=https://ore.exeter.ac.uk/repository/bitstream/10871/15671/2/RevModPhys.83.543.pdf|hdl-access=free}}</ref><ref name="Lloyd-HughesJeon2012">{{cite journal|last1=Lloyd-Hughes|first1=James| last2=Jeon|first2=Tae-In|title=A Review of the Terahertz Conductivity of Bulk and Nano-Materials| journal=Journal of Infrared, Millimeter, and Terahertz Waves|volume=33|issue=9|year=2012|pages=871–925 |issn=1866-6892|doi=10.1007/s10762-012-9905-y|bibcode=2012JIMTW..33..871L|s2cid=13849900}}</ref> [[Femtosecond laser]] pulses excite the semiconductor and the resulting [[photoconductivity]] is measured using a terahertz probe, which detects changes in the terahertz electric field.<ref name="EversSchins2015">{{cite journal|last1=Evers|first1=Wiel H.|last2=Schins|first2=Juleon M.| last3=Aerts| first3=Michiel|last4=Kulkarni|first4=Aditya|last5=Capiod|first5=Pierre|last6=Berthe|first6=Maxime|last7=Grandidier|first7=Bruno|last8=Delerue|first8=Christophe|last9=van der Zant|first9=Herre S. J.|last10=van Overbeek|first10=Carlo| last11=Peters| first11=Joep L.|last12=Vanmaekelbergh|first12=Daniel|last13=Siebbeles|first13=Laurens D. A.|title=High charge mobility in two-dimensional percolative networks of PbSe quantum dots connected by atomic bonds|journal=Nature Communications|volume=6| year=2015|pages=8195|issn=2041-1723|doi=10.1038/ncomms9195|pmid=26400049|pmc=4598357|bibcode=2015NatCo...6.8195E}}</ref>

=== Time resolved microwave conductivity (TRMC) ===
{{main|Time resolved microwave conductivity}}

A proxy for charge carrier mobility can be evaluated using time-resolved microwave conductivity (TRMC).<ref>{{Cite journal| last1=Savenije|first1=Tom J.|last2=Ferguson|first2=Andrew J.|last3=Kopidakis|first3=Nikos|last4=Rumbles|first4=Garry| date=2013-11-21| title=Revealing the Dynamics of Charge Carriers in Polymer:Fullerene Blends Using Photoinduced Time-Resolved Microwave Conductivity|url=https://doi.org/10.1021/jp406706u|journal=The Journal of Physical Chemistry C|volume=117|issue=46 |pages=24085–24103|doi=10.1021/jp406706u|issn=1932-7447|url-access=subscription}}</ref> A pulsed optical laser is used to create electrons and holes in a semiconductor, which are then detected as an increase in photoconductance. With knowledge of the sample absorbance, dimensions, and incident laser fluence, the parameter <math>\phi\Sigma\mu=\phi(\mu_{e}+\mu_{h})</math> can be evaluated, where <math>\phi</math> is the carrier generation yield (between 0 and 1), <math>\mu_{e}</math> is the electron mobility and <math>\mu_{h}</math> is the hole mobility. <math>\phi\Sigma\mu</math> has the same dimensions as mobility, but carrier type (electron or hole) is obscured.