View source for Shubnikov–de Haas effect

An [[oscillation]] in the [[Electrical conductivity|conductivity]] of a material that occurs at low temperatures in the presence of very intense [[magnetic fields]], the '''Shubnikov–de Haas effect''' ('''SdH''') is a [[macroscopic]] manifestation of the inherent [[quantum mechanical]] nature of matter. It is often used to determine the [[Effective mass (solid-state physics)|effective mass]] of [[charge carrier]]s ([[electron]]s and [[electron hole]]s), allowing investigators to distinguish among [[majority carrier|majority]] and [[minority carrier]] populations. 
The effect is named after [[Wander Johannes de Haas]] and [[Lev Shubnikov]].

== Physical process ==
At sufficiently low temperatures and high magnetic fields, the free electrons in the conduction band of a [[metal]], [[semimetal]], or narrow [[band gap]] [[semiconductor]] will behave like [[simple harmonic oscillator]]s.  When the magnetic field strength is changed, the oscillation period of the simple harmonic oscillators changes proportionally.  The resulting [[energy spectrum]] is made up of [[Landau quantization|Landau levels]] separated by the [[cyclotron]] energy.  These Landau levels are further split by the [[Zeeman effect|Zeeman energy]]. In each Landau level the cyclotron and Zeeman energies and the number of electron states (''eB''/''h'') all increase linearly with increasing magnetic field. Thus, as the magnetic field increases, the [[spin-split]] Landau levels move to higher energy. As each energy level passes through the [[Fermi energy]], it depopulates as the electrons become free to flow as current. This causes the material's [[Transport phenomena|transport]] and [[thermodynamic]] properties to oscillate periodically, producing a measurable oscillation in the material's conductivity.  Since the transition across the Fermi 'edge' spans a small range of energies, the waveform is square rather than [[sinusoidal]], with the shape becoming ever more square as the temperature is lowered.{{citation needed|reason=this claim needs a reliable source, as several examples in the literature do not appear to be square waves.|date=July 2016}}

== Theory ==
Consider a two-dimensional quantum gas of electrons confined in a sample with given width and with edges. In the presence of a magnetic flux density ''B'', the energy eigenvalues of this system are described by [[Landau quantization|Landau levels]]. As shown in Fig 1, these levels are equidistant along the vertical axis. Each energy level is substantially flat inside a sample (see Fig 1). At the edges of a sample, the [[work function]] bends levels upwards.
[[File:Sketch energy edge channels.svg|thumb|Fig 1: Edge channels of a sample with a two-dimensional electron gas]]
Fig 1 shows the [[Fermi level|Fermi energy]] ''E''<sub>F</sub> located in between<ref>Since defects in the sample will affect the position of the [[Fermi level|Fermi energy]] ''E''<sub>F</sub>, this is strictly speaking an approximation.  Any influence of defects and of temperatures above 0 K is neglected here for now.</ref> two [[Landau quantization|Landau levels]]. Electrons become mobile as their energy levels cross the [[Fermi level|Fermi energy]] ''E''<sub>F</sub>. With the [[Fermi level|Fermi energy]] ''E''<sub>F</sub> in between two [[Landau quantization|Landau levels]], scattering of electrons will occur only at the edges of a sample where the levels are bent. The corresponding electron states are commonly referred to as edge channels.

The Landauer–Büttiker approach is used to describe transport of electrons in this particular sample. The Landauer–Büttiker approach allows calculation of net currents ''I<sub>m</sub>'' flowing between a number of contacts {{nowrap|1 ≤ ''m'' ≤ ''n''}}. In its simplified form, the net current ''I<sub>m</sub>'' of contact ''m'' with [[chemical potential]] ''μ<sub>m</sub>'' reads
{{NumBlk|:|<math>I_m = 2 \frac{e\cdot i}{h} \left(\mu_m - \sum_{l\neq m} T_{ml} \mu_l \right),\,</math>|{{EquationRef|1}}}}
where ''e'' denotes the [[electron charge]], ''h'' denotes the [[Planck constant]], and ''i'' stands for the number of edge channels.<ref>The number of edge channels ''i'' is closely related to the filling factor {{math|''&nu;'' {{=}} ''i''}}. The factor 2 is due to [[Degenerate energy levels|spin degeneracy]].</ref> The matrix ''T<sub>ml</sub>'' denotes the probability of transmission of a negatively charged particle (i.e. of an electron) from a contact {{math|''l'' &ne; ''m''}} to another contact ''m''. The net current ''I<sub>m</sub>'' in relationship ({{EquationNote|1}}) is made up of the currents towards contact ''m'' and of the current transmitted from the contact ''m'' to all other contacts {{math|''l'' &ne; ''m'' }}. That current equals the voltage {{math|''&mu;''<sub>''m''</sub> / ''e''}} of contact ''m'' multiplied with the [[Quantum Hall effect|Hall conductivity]] of  {{math|2''e''<sup>2</sup> / ''h''}} per edge channel. 
[[File:Shubnikov hall contacts.png|thumb|Fig 2: Contact arrangement for measurement of SdH oscillations]]
Fig 2 shows a sample with four contacts. To drive a current through the sample, a voltage is applied between the contacts 1 and 4. A voltage is measured between the contacts 2 and 3. Suppose electrons leave the 1st contact, then are transmitted from contact 1 to contact 2, then from contact 2 to contact 3, then from contact 3 to contact 4, and finally from contact 4 back to contact 1. A negative charge (i.e. an electron) transmitted from contact 1 to contact 2 will result in a current from contact 2 to contact 1. An electron transmitted from contact 2 to contact 3 will result in a current from contact 3 to contact 2 etc. Suppose also that no electrons are transmitted along any further paths. The probabilities of transmission of ideal contacts then read
: <math>T_{21} = T_{32} = T_{43} = T_{14} = 1,</math>
and
: <math>T_{ml} = 0</math>
otherwise. With these probabilities, the currents ''I''<sub>1</sub> ... ''I''<sub>4</sub> through the four contacts, and with their [[chemical potential]]s ''μ''<sub>1</sub> ... ''μ''<sub>4</sub>, equation ({{EquationNote|1}}) can be re-written
: <math>\left(\begin{matrix} I_1 \\ I_2 \\ I_3 \\ I_4 \end{matrix}\right)=\frac{2e\cdot i}{h}\left(\begin{matrix} 1 & 0 & 0 & -1 \\ -1 & 1 & 0 & 0 \\ 0 & -1 & 1 & 0 \\ 0 & 0 & -1 & 1 \end{matrix}\right)\left(\begin{matrix} \mu_1 \\ \mu_2 \\ \mu_3 \\ \mu_4 \end{matrix}\right).</math>
A voltage is measured between contacts 2 and 3. The voltage measurement should ideally not involve a flow of current through the meter, so ''I''<sub>2</sub>&nbsp;=&nbsp;''I''<sub>3</sub>&nbsp;=&nbsp;0. It follows that
: <math>I_3 = 0 = \frac{2e\cdot i}{h} \left(-\mu_2 + \mu_3\right),</math>
: <math>\mu_2 = \mu_3.</math>
In other words, the [[chemical potential]]s ''μ''<sub>2</sub> and ''μ''<sub>3</sub> and their respective voltages {{math|''μ''<sub>2</sub> / ''e''}} and {{math|''μ''<sub>3</sub> / ''e''}} are the same. As a consequence of no drop of voltage between the contacts 2 and 3, the current ''I''<sub>1</sub> experiences zero resistivity ''R''<sub>SdH</sub> in between contacts 2 and 3
: <math>R_{\mathrm{SdH}} = \frac{\mu_2 - \mu_3}{e\cdot I_1}=0.</math>
The result of zero resistivity between the contacts 2 and 3 is a consequence of the electrons being mobile only in the edge channels of the sample. The situation would be different if a [[Landau quantization|Landau level]] came close to the [[Fermi level|Fermi energy]] ''E''<sub>F</sub>. Any electrons in that level would become mobile as their energy approaches the [[Fermi level|Fermi energy]] ''E''<sub>F</sub>. Consequently, scatter would lead to  ''R''<Sub>SdH</Sub>&nbsp;>&nbsp;0. In other words, the above approach yields zero resistivity whenever the [[Landau quantization|Landau levels]] are positioned such that the [[Fermi level|Fermi energy]]  ''E''<sub>F</sub> is in between two levels.

== Applications ==
Shubnikov–De Haas oscillations can be used to determine the two-dimensional electron density of a sample. For a given magnetic flux <math>\Phi</math> the maximum number ''D'' of electrons with spin ''S''&nbsp;=&nbsp;1/2 per [[Landau quantization|Landau level]]  is
{{NumBlk|:|<math>D = \left(2 S + 1\right) \frac{\Phi}{\Phi_0} = 2 \frac{\Phi}{\Phi_0}.\,</math>|{{EquationRef|2}}}}
Upon insertion of the expressions for the [[Landau quantization|flux quantum]] {{math|&Phi;<sub>0</sub> {{=}} ''h'' / ''e''}} and for the magnetic flux {{math|&Phi; {{=}} ''BA''}} relationship ({{EquationNote|2}}) reads
: <math> D = 2 \frac{e B A}{h}</math>
Let ''N'' denote the maximum number of states per unit area, so {{math|''D'' {{=}} ''NA''}} and
: <math> N = 2 \frac{e B}{h}.</math>
Now let each [[Landau quantization|Landau level]] correspond to an edge channel of the above sample. For a given number ''i'' of edge channels each filled with ''N'' electrons per unit area, the overall number ''n'' of electrons per unit area will read
: <math> n = i N = 2  i \frac{e B}{h}.</math>

The overall number ''n'' of electrons per unit area is commonly referred to as the electron density of a sample. No electrons disappear from the sample into the unknown, so the electron density ''n'' is constant. It follows that
: <math> B_i = \frac{n h}{2 e i},</math>
: <math> \frac{1}{B_i} = \frac{2 e i}{n h},</math>

{{NumBlk|:|<math>\Delta\left(\frac{1}{B}\right) = \frac{1}{B_{i+1}}-\frac{1}{B_i} = \frac{2 e}{n h}.\,</math>|{{EquationRef|3}}}}
[[File:InvBvsnplot.png|thumb|center|upright=2.0|Fig 3: Inverse magnetic flux densities 1/''B''<sub>''i''</sub> vs Shubnikov–De Haas minima as observed in highly doped [[Bismuth selenide|Bi<sub>2</sub>Se<sub>3</sub>]] ]]
For a given sample, all factors including the electron density ''n'' on the right hand side of relationship ({{EquationNote|3}}) are constant. When plotting the index ''i'' of an edge channel versus the reciprocal of its magnetic flux density 1/''B''<sub>''i''</sub>, one obtains a straight line with slope 2''e''/(''nh''). Since the [[electron charge]] ''e'' is known and also the [[Planck constant]] ''h'', one can derive the electron density ''n'' of a sample from this plot.<ref>Relationship ({{EquationNote|3}}) is expressed in [[SI units]]. In [[CGS units]], the same relationship reads <math>\Delta\left(\frac{1}{B}\right) = \frac{2 \cdot e}{n\cdot h\cdot c}.</math></ref>
Shubnikov–De Haas oscillations are observed in highly doped [[Bismuth selenide|Bi<sub>2</sub>Se<sub>3</sub>]].<ref name="cao2012">
{{cite journal
 |first1=Helin |last1=Cao
 |first2=Jifa |last2=Tian
 |first3=Ireneusz |last3=Miotkowski
 |first4=Tian |last4=Shen
 |first5=Jiuning |last5=Hu
 |first6=Shan |last6=Qiao
 |first7=Yong P. |last7=Chen
 |title=Quantized Hall Effect and Shubnikov–De Haas Oscillations in Highly Doped Bi2Se3: Evidence for Layered Transport of Bulk Carriers
 |journal=[[Physical Review Letters]]
 |volume=108 |pages=216803
 |year=2012
 |issue=21
 |doi=10.1103/PhysRevLett.108.216803
 |bibcode = 2012PhRvL.108u6803C |pmid=23003290|url=http://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=2193&context=nanopub|doi-access=free
}}</ref> Fig 3 shows the reciprocal magnetic flux density 1/''B''<sub>''i''</sub> of the 10th to 14th minima of a [[Bismuth selenide|Bi<sub>2</sub>Se<sub>3</sub>]] sample. The slope of 0.00618/T as obtained from a linear fit yields the electron density ''n''
: <math> n = \frac{2 e}{0.00618/\mathrm{T}\cdot h} \approx 7.82 \times 10^{14}/\mathrm{m}^2.</math>

Shubnikov–de Haas oscillations can be used to [[Fermi surface#Experimental determination|map the Fermi surface]] of electrons in a sample, by determining the periods of oscillation for various applied field directions.

== Related physical process ==
The effect is related to the [[De Haas–Van Alphen effect]], which is the name given to the corresponding oscillations in magnetization. The signature of each effect is a [[Wave period|periodic]] [[waveform]] when plotted as a function of inverse magnetic field. The "[[frequency]]" of the [[magnetoresistance]] [[oscillations]] indicate areas of extremal orbits around the [[Fermi surface]]. The area of the Fermi surface is expressed in [[Tesla (unit)|teslas]]. More accurately, the period in inverse Teslas is inversely proportional to the area of the extremal orbit of the Fermi surface in inverse m/cm.

== References ==
{{reflist}}
* {{cite journal
 |first1=L. |last1=Schubnikow
 |first2=W.J. |last2=De Haas
 |title=Magnetische Widerstandsvergrösserung in Einkristallen von Wismut bei tiefen Temperaturen
 |trans-title= Magnetic resistance increase in single crystals of bismuth at low temperatures
 |url=http://www.dwc.knaw.nl/DL/publications/PU00015868.pdf
 |language=German
 |journal=Proceedings of the Royal Netherlands Academy of Arts and Science
 |volume=33 |pages=130–133
 |year=1930
 |doi=
}}
* {{cite journal
 |first1=L. |last1=Schubnikow
 |first2=W.J. |last2=De Haas
 |title=Neue Erscheinungen bei der Widerstandsänderung von Wismuthkristallen im Magnetfeld bei der Temperatur von flüssigem Wasserstoff (I)
 |trans-title=New phenomena in the change in resistance of bismuth crystals in a magnetic field at the temperature of liquid hydrogen (I)
 |url=http://www.dwc.knaw.nl/DL/publications/PU00015898.pdf
 |journal=Proceedings of the Royal Netherlands Academy of Arts and Science
 |volume=33 |pages=363–378
 |year=1930
 |doi=
}}
* {{cite journal
 |first1=L. |last1=Schubnikow
 |first2=W.J. |last2=De Haas
 |title=Neue Erscheinungen bei der Widerstandsänderung von Wismuthkristallen im Magnetfeld bei der Temperatur von flüssigem Wasserstoff (II)
 |url=http://www.dwc.knaw.nl/DL/publications/PU00015905.pdf
 |journal=Proceedings of the Royal Netherlands Academy of Arts and Science
 |volume=33 |pages=418–432
 |year=1930
 |doi=
}}
* {{cite journal
 |first1=L. |last1=Schubnikow
 |first2=W.J. |last2=De Haas
 |title=Die Widerstandsänderung von Wismuthkristallen im Magnetfeld bei der Temperatur von flüssigem Stickstoff
 |trans-title=The change in resistance of bismuth crystals in a magnetic field at the temperature of liquid nitrogen
 |url=http://www.dwc.knaw.nl/DL/publications/PU00015906.pdf
 |journal=Proceedings of the Royal Netherlands Academy of Arts and Science
 |volume=33 |pages=433–439
 |year=1930
 |doi=
}}

== External links ==
* The article uses text from [http://www.lanl.gov/orgs/mpa/nhmfl/users/pages/Shubnikov.htm Shubnikov effect on Lang.gov] {{Webarchive|url=https://web.archive.org/web/20170922102355/http://www.lanl.gov/orgs/mpa/nhmfl/users/pages/Shubnikov.htm |date=2017-09-22 }} that is a Public Domain as a work of a US government agency.
* [http://www.physics.montana.edu/magnet/highmag/index.htm Material behavior in strong magnetic fields] {{Webarchive|url=https://web.archive.org/web/20060903142736/http://www.physics.montana.edu/magnet/highmag/index.htm |date=2006-09-03 }}

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[[Category:Cryogenics]]
[[Category:Magnetism]]