CoChat
Sign Up
Cited by 8Quantum and electron transport phenomenaSemiconductor Quantum Structures and DevicesAdvancements in Semiconductor Devices and Circuit Design

Quantum Oscillatory Phenomena in a p-n Junction of Quantum Wells

Open access
Shmueli, N, Yacoby, A, Imry, Y·1995·Europhysics Letters (EPL)
DOI 10.1209/0295-5075/29/9/010
Open this paper in CoChat — free accountOpen PDF

Abstract

We consider tunnelling across a p-n junction (a tunnelling diode) where the motion on both sides is constrained by quantum wells (QW) in the direction perpendicular to the interface to form 2D n -type and p -type electron gases. Coulomb interactions are neglected. The Hamiltonian near an intersection of the dispersion curves, E(k) , of the n side and the p side resembles that of a relativistic Dirac electron. An electron, prepared in one of the QWs, will oscillate periodically between the two QWs with a frequency proportional to the tunnelling amplitude. In the tunnelling process, the velocity parallel to the interface is reversed resulting in periodic velocity oscillations similar to the Zitterbewegung where in the present case the opposite directions of the velocity are separated in space. We also find that the eigenstates have finite "persistent" circular currents which create magnetic moments of several Bohr magnetons, typically. The noise spectrum, and the linear response of the system are also obtained.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper theoretically studies tunneling across a p-n junction made from two quantum wells, where the band dispersion near the crossing mimics a relativistic Dirac electron. It predicts periodic velocity oscillations analogous to Zitterbewegung, persistent circular currents with magnetic moments of a few Bohr magnetons, and specific noise/linear-response signatures.

Key findings

  • An electron initially placed in one quantum well oscillates between the n and p wells at a frequency tied to the tunneling amplitude; the velocity parallel to the interface reverses during tunneling, producing a solid-state analogue of Zitterbewegung.
  • Eigenstates carry finite circulating currents, yielding magnetic moments typically on the order of a few Bohr magnetons, with possible enhancement for small effective masses.
  • The dynamic structure factor and conductivity show a feature near \(\omega = 2\Delta\), with singular behavior in the 1D case and a related 2D result; for Fermi levels in either band, \(2E_F\) replaces \(2\Delta\) in the condition on \(\omega\).
  • The authors propose that the effect could be observed using high-frequency transport, noise, or electromagnetic absorption measurements in coupled electron-hole quantum-well systems.

Methods

  • Theoretical model of a p-n junction formed by n-type and p-type 2D quantum wells coupled by tunneling in the x-direction, with an ideal interface conserving lateral momentum.
  • Near the band intersection, the Hamiltonian is reduced to a Dirac-like 2×2 form; spin-orbit coupling is neglected, giving two uncoupled spin problems.
  • The analysis derives the velocity operator, its Heisenberg evolution, the dynamic structure factor at \(T=0\), and the linear response, considering both 1D and 2D bands.
  • Two experimental realizations are discussed: an n-type AlSb-InAs-AlSb quantum well coupled to a p-type AlSb-GaSb-AlSb quantum well, and an AlGaAs p-n junction with an intrinsic GaAs layer.

Limitations

  • The paper explicitly neglects Coulomb interactions and spin-orbit coupling, and assumes an ideal interface with conserved lateral momentum and equal absolute effective masses in the two bands.
  • The treatment is theoretical; no experimental data are reported, and the predicted magnetic moments and noise features are not yet confirmed.
  • Evident from the text: several parameters, such as the tunneling matrix element and well separation, are treated as controllable but not fixed to specific measured values, so quantitative experimental predictions depend on system choices.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1209/0295-5075/29/9/010

Cites

5 of 5 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 11 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning

    For non-interacting electrons, the excited states relevant to the dynamic structure factor are electron-hole excitations with conserved spin S.

    “For non-interacting electrons the excited states are electron-hole excitations, i.e. the transitions: I + I + S ) , where S is conserved.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Coulomb interactions are neglected in the model.
    • The authors state this as an assumption: Spin-orbit coupling is neglected.
  • Findingnone given

    For a 2D band with the Fermi level outside the gap, the Fermi energy surfaces have an annular shape.

    “For a 2D band the Fermi energy surfaces for E F > 4 or E F c -A have a very interesting annular shape.”
    • Also stated in: § Abstract · plain
      “The interesting annular shape of the Fermi surface should also have other consequences (for example in Shubnikov de Haas and the de Haas Van Alphen effects).”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Coulomb interactions are neglected in the model.
    • The authors state this as an assumption: Spin-orbit coupling is neglected.
  • Findingreasoning

    For times much longer than t0, the overlap is small and four non-overlapping wave packets travel away in the two sides.

    “For times >> t o , the overlap is relatively small and we get only four non-overlapping wave packets travelling away in the two sides..”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Coulomb interactions are neglected in the model.
    • The authors state this as an assumption: The junction is ideal, so the lateral component of k is conserved during tunnelling.
    • The authors state this as an assumption: The effective mass has the same absolute value in the n and p bands.
    • The authors state this as an assumption: Spin-orbit coupling is neglected.
    • The authors state this as an assumption: The interband tunnelling amplitude Δ is taken to be independent of θ.
  • Findingreasoning

    For ω much larger than 2Δ, the dynamic structure factor drops to zero because mixing between the states is negligible.

    “For w much larger than 24, S, ( w ) drops to zero due to the negligible mixing between #: and $: in the electron states.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Coulomb interactions are neglected in the model.
    • The authors state this as an assumption: Spin-orbit coupling is neglected.
  • Findingreasoning

    The eigenstates carry finite persistent circular currents that produce magnetic moments of order several Bohr magnetons for typical parameters.

    “We also find that the eigenstates have finite <<persistent>> circular currents which create magnetic moments of several Bohr magnetons, typically.”
    • Also stated in: § Abstract · plain
      “For typical values of s and d we find that the magnetic moment is of the order of few Bohr magnetons”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Coulomb interactions are neglected in the model.
    • The authors state this as an assumption: The junction is ideal, so the lateral component of k is conserved during tunnelling.
    • The authors state this as an assumption: The effective mass has the same absolute value in the n and p bands.
    • The authors state this as an assumption: Spin-orbit coupling is neglected.
  • May combine two claims. 10
  • The quote only partly states this claim. 4
  • Classified as probably not a substantial claim. 6

  • May combine two claims.

    Findingreasoning

    Tunnelling reverses the velocity parallel to the interface, producing periodic velocity oscillations analogous to Zitterbewegung, with the opposite velocity directions spatially separated between the p and n wells.

    “In the tunnelling process, the velocity parallel to the interface is reversed resulting in periodic velocity oscillations similar to the Zitterbewegung where in the present case the opposite directions of the velocity are separated in space.”
    • Also stated in: § Abstract · plain
      “The two components of the wave function that have opposite velocities are separated in space (i.e. they are localized in the p and n wells).”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Coulomb interactions are neglected in the model.
    • The authors state this as an assumption: The junction is ideal, so the lateral component of k is conserved during tunnelling.
    • The authors state this as an assumption: The effective mass has the same absolute value in the n and p bands.
    • The authors state this as an assumption: Spin-orbit coupling is neglected.

  • May combine two claims. The quote only partly states this claim.

    Findingreasoning

    The spatial extent of the zero-point motion in this system is given by ħs/Δ and may reach a few hundreds of nanometers.

    “In the case described here the spatial extent is given by hsld, and may reach a few hundreds nm.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Coulomb interactions are neglected in the model.
    • The authors state this as an assumption: Spin-orbit coupling is neglected.

  • May combine two claims.

    Findingreasoning

    Near Δ=0, the occupations of the n and p sides are each 1/2, and the lateral currents on the two sides are opposite and close a current loop.

    “For the most interesting case, near d = 0, ( N e ) = (N,) = 1/2 and the current in the n side of the junction is given by e(Gc)s = es/2L, where L is the length of the junction.Since the velocity in the p side is in the opposite direction, the current there is given byes/2L.These two opposite lateral currents, with the appropriate tunnelling currents, close a current loop of strength I .”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Coulomb interactions are neglected in the model.
    • The authors state this as an assumption: The junction is ideal, so the lateral component of k is conserved during tunnelling.
    • The authors state this as an assumption: The effective mass has the same absolute value in the n and p bands.
    • The authors state this as an assumption: Spin-orbit coupling is neglected.

  • The quote only partly states this claim.

    Findingreasoning

    For a 1D band with the Fermi level in the gap and ω close to 2Δ, the dynamic structure factor has a singularity at ω=2Δ due to the singular 1D density of states.

    “A calculation of S, (U) for a 1D band with the Fermi level in the gap (we measure E F from the gap centre, therefore, its range is -4 < E F < 4) and for w close to 24 yields”
    • Also stated in: § Abstract · plain
      “The singularity in S, (w) at w = 24 for both the 1D and 2D cases is a result of the singular nature of the 1D density of states.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Coulomb interactions are neglected in the model.
    • The authors state this as an assumption: Spin-orbit coupling is neglected.

  • May combine two claims.

    Findingnone given

    For times around t0, the wave packet on each side splits into two Gaussians moving in opposite directions with group velocity ±vg and width of order s^2 t/(2Δ).

    “For times B to the wave packet on each side effectively splits into two Gaussians moving in opposite directions with a group velocity ?vg and having the usual width of order s2 t/23.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Coulomb interactions are neglected in the model.
    • The authors state this as an assumption: The junction is ideal, so the lateral component of k is conserved during tunnelling.
    • The authors state this as an assumption: The effective mass has the same absolute value in the n and p bands.
    • The authors state this as an assumption: Spin-orbit coupling is neglected.
    • The authors state this as an assumption: The interband tunnelling amplitude Δ is taken to be independent of θ.

  • The quote only partly states this claim.

    Findingreasoning

    For times shorter than t0, the density oscillates between the two sides with angular frequency approximately 2Δ/ħ.

    “For times less than to = 3x/v, = = E ( d ) / s 2 0 d >> l / d the density oscillates between the two sides with angular frequency of 2E(d)/tl B Zd/tl.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Coulomb interactions are neglected in the model.
    • The authors state this as an assumption: The junction is ideal, so the lateral component of k is conserved during tunnelling.
    • The authors state this as an assumption: The effective mass has the same absolute value in the n and p bands.
    • The authors state this as an assumption: Spin-orbit coupling is neglected.
    • The authors state this as an assumption: The interband tunnelling amplitude Δ is taken to be independent of θ.

  • The quote only partly states this claim.

    Interpretationreasoning

    The spatial separation of the two opposite-velocity components makes Zitterbewegung effects observable by regular solid-state techniques such as electromagnetic absorption, transport and noise measurements.

    “The former makes Zitterbewegung effects observable by regular solid-state physics techniques such as electromagnetic absorption, transport and noise measurements.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Coulomb interactions are neglected in the model.
    • The authors state this as an assumption: The junction is ideal, so the lateral component of k is conserved during tunnelling.
    • The authors state this as an assumption: The effective mass has the same absolute value in the n and p bands.
    • The authors state this as an assumption: Spin-orbit coupling is neglected.

  • May combine two claims.

    Premisereasoning

    In Dirac theory, because the velocity operator eigenvalues are ±c, any free electron eigenstate can be written as a superposition of the two velocity eigenstates, making the velocity probability distribution bimodal.

    “Since the eigenvalues of the velocity operator are & c (c being the velocity of light) any free electron eigenstate, having an arbitrary small expectation value of the velocity, U , can be written as a superposition of these two velocity eigenstates.Thus the probability distribution of the velocity is bimodal.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [5] Dirac P A M, 1958, The Principles of Quantum Mechanics

      Not checked in the free run.

    • cites [6] Bjorken J D, 1965, Relativistic Quantum Mechanics

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    The analogy between the semiconducting gap and relativistic quantum mechanics has been exploited by Kane and Blount and by Zawadzki for tunnelling across p-n junctions while neglecting spin coupling.

    “It has been exploited by Kane and Blount [2] and by Zawadzki [3] for the tunnelling across p-n junction (neglecting the coupling to the spin).”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [2] Kane E O, 1969, Tunnelling Phenomena in Solids

      Not checked in the free run.

    • cites [3] Zawadzki W, 1969, Tunneling Phenomena in Solids

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Aronov and Pikus used the full Dirac equation including spin in tunnelling calculations for the p-n junction.

    “Aronov and Pikus [4] used the full Dirac equation including the spin in the tunnelling calculations for the p-n junction.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [4] Aronov A G, 1967, Sov. Phys. JETP

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationreasoning

    An AlGaAs p-n junction sandwiching an intrinsic GaAs layer is another generic system; the intrinsic electric field creates two triangular wells, one for holes and one for electrons.

    “Another generic system is a AlGaAs p-n junction sandwiching an intrinsic GaAs layer.Due to the intrinsic electric field in the junction one obtains two triangular wells, one for holes and one for electrons.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationcitation

    Applying a magnetic field breaks the symmetry between the two populations, creating a total magnetization that may be significantly enhanced using multiple InAs-GaSb junctions.

    “Applying magnetic field will break the symmetry between the two populations therefore, creating a total magnetization which may be enhanced significantly using for example multiple junctions of InAs-GaSb [9].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [9] Yacoby A

      Not checked in the free run.

    • The authors state this as an assumption: Coulomb interactions are neglected in the model.
    • The authors state this as an assumption: Spin-orbit coupling is neglected.

  • May combine two claims.

    Interpretationnone given

    The tunnelling matrix element Δ can have low values and may be controlled in principle.

    “The tunnelling matrix element d can have low values, and may be controlled in principle.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: Coulomb interactions are neglected in the model.
    • The authors state this as an assumption: Spin-orbit coupling is neglected.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given

    These other examples do not have spatial separation of electrons and holes.

    “However, these examples do not have the spatial separation of the electrons and the holes.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    In that system, the separation between wells can be controlled accurately, enabling control of the gap energy, and the crossing points can be varied by an external electric field, enabling continuous control over k0.

    “The separation between wells can be controlled very accurately enabling direct control of the gap energy (typically few MeV) and the crossing points of the two bands can also be continuously varied by an externally applied electric field thereby enabling a continuous control over k o .”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [8] Mikhailova M P, 1993, Semicond. Sci. Technol.

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation

    A similar energy spectrum to that of relativistic electrons exists around any band gap at least in the nearly free-electron picture and for Bogolyubov quasiparticles in a superconductor.

    “It has to be mentioned that a similar energy spectrum, analogous to that of relativistic electrons exists also around any band gap at least in the nearly free-electron picture and also for the Bogolyubov quasi-particles in a superconductor.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [7] source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given

    For kB T much less than Δ, ordinary Nyquist noise should be added to the calculated noise.

    “For kB T << A, the ordinary Nyquist noise should be added to the above.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: Coulomb interactions are neglected in the model.
    • The authors state this as an assumption: Spin-orbit coupling is neglected.
17 of 17

  • The authors state this as an assumption.

    Coulomb interactions are neglected in the model.

    “Coulomb interactions are neglected.”

  • The authors state this as an assumption.

    The junction is ideal, so the lateral component of k is conserved during tunnelling.

    “The junction is assumed ideal, so that the lateral component (in the (x, y)-plane) of k is conserved upon tunnelling.”

  • The authors state this as an assumption.

    The effective mass has the same absolute value in the n and p bands.

    “m is the effective mass (which is assumed to have the same absolute value in the two bands)”

  • The authors state this as an assumption.

    Spin-orbit coupling is neglected.

    “Neglecting the spin-orbit coupling, the dynamics near each intersection is in a one-to-one correspondence with the same limit of the Dirac problem.”

  • The authors state this as an assumption.

    The interband tunnelling amplitude Δ is taken to be independent of θ.

    “A, taken independently of 8, is the interband tunnelling amplitude across the interface”
5 of 5

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • Jiang et al. found that time-dependence of hole trajectories and their spin in degenerate Γ8 bands described by the Luttinger Hamiltonian in a constant electric field contain rapid oscillations reminiscent of ZB, with frequencies given by differences of light- and heavy-hole energies ħω(t)=E_L(t)-E_H(t), increasing in time as holes are accelerated.

    “Jiang et al. [33] studied numerically the dynamics of holes in degenerate Γ 8 bands described by the Luttinger Hamiltonian in the presence of a constant electric field. It was found that the time-dependence of hole trajectories and their spin contain rapid oscillations reminiscent of the Zitterbewegung. Frequencies of the oscillations are given by the differences of light- and heavy-holes energies”

    From Zitterbewegung (trembling motion) of electrons in semiconductors: a review (2011) · cites this paper as [33]

    Split into 2 atoms — 0 confirmed, 1 partially supported, 1 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

    Checked by a reasoning model — may be wrong.

    Experimental — accuracy not yet measured

  • The Zitterbewegung of holes in the valence bands of Γ8 symmetry has its Zitterbewegung frequency given by the energy difference of light and heavy hole bands.

    “Another illustration is the ZB of holes in the valence bands of Γ 8 symmetry [35], where the ZB frequency is given by the energy difference of light and heavy hole bands [15, 33, 36].”

    From Zitterbewegung (trembling motion) of electrons in semiconductors: a review (2011) · cites this paper as [33]

    The passage on the p–n quantum-well junction (indexed as Abstract; page unavailable) describes Zitterbewegung involving a valence band on the p-side and a conduction band on the n-side. It does not identify Γ8 holes or light- and heavy-hole bands. In the citing sentence, this source is cited for the example of hole Zitterbewegung; the frequency statement has separate citations. (LLM, conf=0.82)

    Experimental — accuracy not yet measured

  • For B≠0 both interband and intraband (cyclotron) frequencies contribute to ZB; for B=0 there are no intraband frequencies.

    “For B = 0 both interband and intraband (cyclotron) frequencies appear in ZB; for B = 0 there are no intraband frequencies.”

    From Zitterbewegung of electrons in graphene in a magnetic field (2008) · cites this paper as [5]

Cites

Flagged first, then by citations

Research works printed but not linked: 3

  1. Kane E O, 1969, Tunnelling Phenomena in Solids
  2. Aronov A G, 1967, Sov. Phys. JETP, 24, 188
  3. Mikhailova M P, 1993, Semicond. Sci. Technol., 7, 1

Cited by

Sorted by citations

Related

Authors
Shmueli, N., Yacoby, A., & Imry, Y.
Affiliations
Weizmann Institute of Science (IL)
ORCID
Yacoby, A
Venue
Europhysics Letters (EPL)
Publisher
IOP Publishing
Published
1995-03-20
Volume
29
Issue
9
Pages
711-716
ISSN
  • 0295-5075 (ISSN-L)
  • 1286-4854
DOI
10.1209/0295-5075/29/9/010
OpenAlex
W2079559080
Cited by
8
References
5
Citations by year
2021: 1 · 2013: 1
FWCI
0.00
Type
journal article
Language
en
Open access
Locations
Europhysics Letters (EPL) (publishedVersion)
Funders
U.S. Department of Energy
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and electron transport phenomena
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Semiconductor Quantum Structures and Devices
  • Physical Sciences › Engineering › Electrical and Electronic Engineering › Advancements in Semiconductor Devices and Circuit Design
Keywords
quantum wells, p-n junction, quantum oscillations, Zitterbewegung, two-dimensional electron gas, velocity oscillations, persistent currents, magnetic moment, noise spectrum, linear response

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-01.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-01.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review