Zitterbewegung of nearly-free and tightly-bound electrons in semiconductors
Open accessAbstract
We show theoretically that non-relativistic nearly-free electrons in solids should experience a trembling motion (Zitterbewegung, ZB) in the absence of external fields, similarly to relativistic electrons in a vacuum. The ZB is directly related to the influence of the periodic potential on the free electron motion. The frequency of the ZB is , where E g is the energy gap. The amplitude of the ZB is determined by the strength of periodic potential and the lattice period, and it can be of the order of nanometres. We show that the amplitude of the ZB does not depend much on the width of the wavepacket representing an electron in real space. An analogue of the Foldy–Wouthuysen transformation, known from relativistic quantum mechanics, is introduced in order to decouple electron states in various bands. We demonstrate that after the bands are decoupled electrons should be treated as particles of a finite size. In contrast to nearly-free electrons we consider a two-band model of tightly-bound electrons. We show that in this case also the electrons should experience the trembling motion. It is concluded that the phenomenon of ZB of electrons in crystalline solids is the rule rather than the exception.
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Physically, the transient character of Zitterbewegung comes about as a result of waning interference of the two sub-packets belonging to positive and negative energies as they go apart because of different speeds.
“Physically, the transient character of ZB comes about as a result of waning interference of the two sub-packets belonging to positive and negative energies as they go apart because of different speeds, see (24) and (25) and References [17, 84].”
From Zitterbewegung (trembling motion) of electrons in semiconductors: a review (2011) · cites this paper as [17]
The Introduction (page unavailable) says Zitterbewegung arises from interference between positive- and negative-energy states, and §4, “Wavepacket” (page unavailable), analyzes electron wavepackets. Neither passage says the oscillation is transient because two sub-packets separate at different speeds and their interference wanes; that mechanism is not supported by our copy. (LLM, conf=0.84)
Experimental — accuracy not yet measuredIf an electron is represented by a wave packet, its ZB has a transient character, i.e. it disappears with time.
“Lock [6] showed that, if an electron is represented by a wave packet, its ZB has a transient character, i.e. it disappears with time.”
From Zitterbewegung (trembling motion) of electrons in semiconductors: a review (2011) · cites this paper as [17]
Section 4 (Wavepacket; page unavailable) models an electron as a Gaussian wavepacket, but discusses how ZB amplitude varies with packet width and wavevector, not whether it disappears over time. Searches for transient, decaying, or disappearing oscillations found no passage in our copy establishing the claim or attributing it to Lock. (LLM, conf=0.76)
Experimental — accuracy not yet measuredIn a carbon nanotube, after the transient ZB oscillations disappear, a constant shift ̄y_sh remains.
“It is seen that, after the transient ZB oscillations disappear, there remains a shift ¯ y sh .”
From Zitterbewegung (trembling motion) of electrons in semiconductors: a review (2011) · cites this paper as [17]
The Introduction mentions Zitterbewegung in carbon nanotubes, but does not describe its oscillations fading or a residual shift. The Wavepacket section discusses oscillation amplitudes rather than a lasting displacement; searches for the transient oscillations and y_sh found no passage addressing the claimed shift. Our copy therefore does not verify this claim. (LLM, conf=0.77)
Experimental — accuracy not yet measuredThe decay time of Zitterbewegung is usually much longer in one-dimensional systems.
“The decay time is usually much longer in one-dimensional systems, see Figure 4 and References [17, 81].”
From Zitterbewegung (trembling motion) of electrons in semiconductors: a review (2011) · cites this paper as [17]
Section 4 (Wavepacket; page unavailable) discusses Zitterbewegung oscillations and how their amplitude varies with wavepacket properties, but does not compare decay times across dimensions. Searches for decay time, damping, and longer-lasting oscillations in one-dimensional systems found no passage supporting the claimed comparison. (LLM, conf=0.70)
Experimental — accuracy not yet measuredThe decay time of Zitterbewegung is inversely proportional to the momentum spread Δk of the wave packet.
“One can show that the decay time is inversely proportional to the momentum spread ∆ k of the wave packet, see (16) and References [17, 81].”
From Zitterbewegung (trembling motion) of electrons in semiconductors: a review (2011) · cites this paper as [17]
Section 4 (Wavepacket; page unavailable) defines the packet’s momentum width and averages the Zitterbewegung oscillation over it, but does not state how decay time depends on that width. I searched for decay time, damping, and an inverse-width relationship; the retrieved passages discuss oscillation amplitude and frequency instead. (LLM, conf=0.72)
Experimental — accuracy not yet measured
ZB-like motion should occur in other two-band situations, both in solids and in other systems.
“It was further shown that the ZB-like motion should occur in other two-band situations, both in solids [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] and in other systems [19, 20, 21, 22, 23, 24, 25, 26].”
From Theory of electron Zitterbewegung in graphene probed by femtosecond laser pulses (2009) · cites this paper as [12]
All recent theoretical work on ZB assumed initially Gaussian electron wave packets.
“All the recent theoretical work on ZB assumed that initially the electrons are represented by Gaussian wave packets [6, 12, 15, 21, 22, 23, 24, 28, 30].”
From Theory of electron Zitterbewegung in graphene probed by femtosecond laser pulses (2009) · cites this paper as [12]
Decay times of ZB are of the order of femtoseconds to microseconds depending on the system.
“This was confirmed by very recent calculations which predicted that the decay times of ZB are of the order of femtoseconds to microseconds depending on the system in question [12, 16, 21].”
From Theory of electron Zitterbewegung in graphene probed by femtosecond laser pulses (2009) · cites this paper as [12]
The presence of a magnetic field is essential for a successful experiment since in the absence of a magnetic field the ZB oscillations have a transient character with the decay time of tenths of femtoseconds and the detection of such oscillations is difficult.
“The presence of a magnetic field is essential for a successful experiment since, as we said above, in the absence of magnetic field the ZB oscillations have a transient character with the decay time of tenths of femtoseconds [12, 16, 18] and the detection of such oscillations is difficult.”
From Theory of electron Zitterbewegung in graphene probed by femtosecond laser pulses (2009) · cites this paper as [12]
Positive and negative energies are a necessary condition to create ZB motion.
“This wave packet contains states with positive and negative energies, which is a necessary condition to create the ZB motion [6, 12].”
From Theory of electron Zitterbewegung in graphene probed by femtosecond laser pulses (2009) · cites this paper as [12]
The existence of ZB is related to a nonzero momentum of the electron.
“It was shown in the previous work on ZB that its existence is related to a nonzero momentum of the electron [6, 12].”
From Theory of electron Zitterbewegung in graphene probed by femtosecond laser pulses (2009) · cites this paper as [12]
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- Authors
- Rusin, T. M., & Zawadzki, W.
- Affiliations
- Institute of Physics (PL)
- ORCID
- Rusin, T M
- Venue
- Journal of Physics: Condensed Matter
- Publisher
- IOP Publishing
- Published
- 2007-03-15
- Volume
- 19
- Issue
- 13
- Pages
- 136219
- ISSN
- 0953-8984 (ISSN-L)
- 1361-648X
- OpenAlex
- W2039545505
- Cited by
- 27
- References
- 34
- Citations by year
- 2026: 1 · 2023: 2 · 2022: 1 · 2021: 2 · 2020: 1 · 2019: 1 · 2015: 1
- FWCI
- 2.86
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Journal of Physics Condensed Matter (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- arXiv (Cornell University)
- 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 › Topological Materials and Phenomena
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Semiconductor Quantum Structures and Devices
- Keywords
- Zitterbewegung, nearly free electrons, periodic potential, band gap, Foldy-Wouthuysen transformation, two-band model, electron size, semiconductor band structure
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