891:
540:
247:
1634:. Therefore, even in the case of a perfect ballistic transport, there is a fundamental ballistic conductance which saturates the current of the device with a resistance of approximately 12.9 kΩ per mode (spin degeneracy included). There is, however, a generalization of the Landauer–Büttiker formalism of transport applicable to time-dependent problems in the presence of
535:{\displaystyle {\frac {1}{\lambda _{\mathrm {MFP} }}}={\frac {1}{\lambda _{\mathrm {el-el} }}}+{\frac {1}{\lambda _{\mathrm {ap} }}}+{\frac {1}{\lambda _{\mathrm {op,ems} }}}+{\frac {1}{\lambda _{\mathrm {op,abs} }}}+{\frac {1}{\lambda _{\mathrm {impurity} }}}+{\frac {1}{\lambda _{\mathrm {defect} }}}+{\frac {1}{\lambda _{\mathrm {boundary} }}}}
1685:
or ECR, arises as an electric current flowing through a rough interface is restricted to a limited number of contact spots. The size and distribution of these contact spots is governed by the topological structures of the contacting surfaces forming the electrical contact. In particular, for surfaces
1878:
Consider a coherent source of electrons connected to a conductor. Over a limited distance, the electron wave function will remain coherent. You still can deterministically predict its behavior (and use it for computation theoretically). After some greater distance, scattering causes each electron to
74:
of a particle can be described as the average length that the particle can travel freely, i.e., before a collision, which could change its momentum. The mean free path can be increased by reducing the number of impurities in a crystal or by lowering its temperature. Ballistic transport is observed
1874:
Electrons can be scattered several ways in a conductor. Electrons have several properties: wavelength (energy), direction, phase, and spin orientation. Different materials have different scattering probabilities which cause different incoherence rates (stochasticity). Some kinds of scattering can
2008:
m). So a nanotube or graphene nanoribbon could be a good ballistic conductor if the electrons in transit don't scatter with too many phonons and if the device is about 100 nm long. Such a transport regime has been found to depend on the nanoribbon edge structure and the electron energy.
1153:
1963:
The dominant scattering mechanism at room temperature is that of electrons emitting optical phonons. If electrons don't scatter with enough phonons (for example if the scattering rate is low), the mean free path tends to be very long
874:
emission normally dominates, depending on the material and transport conditions. There are also other scattering mechanisms which apply to different carriers that are not considered here (e.g. remote interface phonon scattering,
1792:
105:. It is theoretically possible for ballistic conduction to be extended to other quasi-particles, but this has not been experimentally verified. For a specific example, ballistic transport can be observed in a metal
1540:. For example, electrons in carbon nanotubes have two intervalley modes and two spin modes. Since the contacts and the GNR channel are connected by leads, the transmission probability is smaller at contacts
2266:
Koswatta, Siyuranga O.; Hasan, Sayed; Lundstrom, Mark S.; Anantram, M. P.; Nikonov, Dmitri E. (2006-07-10). "Ballisticity of nanotube field-effect transistors: Role of phonon energy and gate bias".
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2006:
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when the mean free path of the particle is (much) longer than the dimension of the medium through which the particle travels. The particle alters its motion only upon collision with the
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in the material. The presence of resistance implies that the heat is dissipated in the leads outside of the "ballistic" conductor, where inelastic scattering effects can take place.
1710:, the resistance is dominated by the Sharvin mechanism, in which electrons travel ballistically through these micro-contacts with resistance that can be described by the following
1899:. From the resistance point of view, stochastic (not oriented) movement of electrons is useless even if they carry the same energy – they move thermally. If the electrons undergo
235:
1903:
interactions too, they lose energy and the result is a second mechanism of resistance. Electrons which undergo inelastic interaction are then similar to non-monochromatic light.
83:
reflecting the electrons and preventing them from exiting toward the empty space/open air. This is because there is an energy to be paid to extract the electron from the medium (
1955:
or graphene nanoribbons are often considered ballistic, but these devices only very closely resemble ballistic conduction. Their ballisticity is nearly 0.9 at room temperature.
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967:
931:
1849:
1822:
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1708:
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196:
1952:
2173:
Pastawski, Horacio M. (1992-08-15). "Classical and quantum transport from generalized
Landauer-B\"uttiker equations. II. Time-dependent resonant tunneling".
101:, because of extreme size quantization effects in these materials. Ballistic conduction is not limited to electrons (or holes) but can also apply to
1622:(which means the length of the active channel is less than the phase-breaking mean free path) and the transmission functions can be calculated from
2217:
1871:
or a high-quality optical assembly. Non-ballistic electrons behave like light diffused in milk or reflected off a white wall or a piece of paper.
1942:
electron interactions with the environment, each other, and other particles are generally stronger than interactions with and between photons.
1716:
1855:
of the two contacting surfaces, is known as
Sharvin resistance. Electrical contacts resulting in ballistic electron conduction are known as
1867:
A comparison with light provides an analogy between ballistic and non-ballistic conduction. Ballistic electrons behave like light in a
1859:. When the radius of a contact spot is larger than the mean free path of electrons, the contact resistance can be treated classically.
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contact spots may be very small. In such cases, when the radius of the contact spot is smaller than the mean free path of electrons
2104:
2114:
1554:
2073:
Takayanagi, Kunio; Kondo, Yukihito; Ohnishi, Hideaki (2001). "Suspended gold nanowires: ballistic transport of electrons".
59:, or, generally, by any freely-moving atom/molecule composing a gas or liquid. Without scattering, electrons simply obey
1928:
between electrons thus this analogy is good only for single-electron conduction because electron processes are strongly
1332:
1148:{\displaystyle I_{\rm {AB}}={\frac {g_{\text{s}}e}{h}}\int _{E_{\rm {F_{B}}}}^{E_{\rm {F_{A}}}}M(E)f^{\prime }(E)T(E)dE}
2130:
Pastawski, Horacio M. (1991-09-15). "Classical and quantum transport from generalized
Landauer-BĂĽttiker equations".
819:
720:
1967:
1528:. The contacts have a multiplicity of modes due to their larger size in comparison to the channel. Conversely, the
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it is more likely that an electron would lose more energy than a photon would, because of the electron's non-zero
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995:(GNR-FET) on the right (where the channel is assumed to be ballistic), the current from A to B, given by the
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interactions. Information about the state of the electrons at the input is then lost. Transport becomes
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2017:
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in the 1D GNR channel constricts the number of modes to carrier degeneracy and restrictions from the
1221:
1185:
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900:
2109:. Haroon Ahmad, Alec Broers, Michael Pepper. New York: Cambridge University Press. pp. 57–111.
894:
A graphene nanoribbon field-effect transistor (GNR-FET). Here contacts A and B are at two different
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of a material exists because an electron, while moving inside a medium, is scattered by impurities,
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43:), or energy-carrying particles, over relatively long distances in a material. In general, the
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proposed that conduction in a 1D system could be viewed as a transmission problem. For the 1D
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79:. In the case of a wire suspended in air/vacuum the surface of the wire plays the role of the
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113:-scale or 10 meters scale) and the mean free path which can be longer than that in a metal.
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1670:-like effects) could be exploited in electronic systems at nanoscale in systems including
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Thus the quantum conductance is approximately the same if measured at A and B or C and D.
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Koch, Matthias; Ample, Francisco; Joachim, Christian; Grill, Leonhard (2012-10-14).
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2036: – Low-power electronic circuits which use reversible logic to conserve energy
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1278:) is the deviation from the equilibrium electron distribution (perturbation), and
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For correct usage of this analogy consideration of several facts is needed:
879:). To get these characteristic scattering rates, one would need to derive a
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Ballistic conduction is typically observed in quasi-1D structures, such as
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only cause a change in electron direction, others can cause energy loss.
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is the number of modes in the transmission channel and spin is included.
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44:
2457:(1994). "Universal Quantum Signatures of Chaos in Ballistic Transport".
1787:{\displaystyle R_{\rm {S}}={\frac {\lambda (\rho _{1}+\rho _{2})}{2a}}.}
2390:(2008-07-20). "Approaching ballistic transport in suspended graphene".
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and the voltage separation between the Fermi levels is approximately
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is the length of the active part of the device (e.g., a channel in a
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is the acoustic phonon (emission and absorption) scattering length,
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The
Landauer–Büttiker formalism holds as long as the carriers are
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due to 1) a finite, non-zero resistance and 2) the absence of the
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and/or direction. But there is still almost no energy loss. Like
95:
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2329:"Voltage-dependent conductance of a single graphene nanoribbon"
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2218:"Interfacial electro-mechanical behaviour at rough surfaces"
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is the mean free path for the carrier which can be given by
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In general, carriers will exhibit ballistic conduction when
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can have a significantly higher thermal conductivity. See
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1604:{\displaystyle T\approx {\frac {M}{M_{\rm {contact}}}}}
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Movement of charge carriers with negligible scattering
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Pages displaying wikidata descriptions as a fallback
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is the electron scattering length with the boundary.
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is the optical phonon absorption scattering length,
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2012:
1286:= 1 for ballistic). Based on the definition of
1887:light passing through milk, electrons undergo
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859:{\displaystyle \lambda _{\mathrm {boundary} }}
760:{\displaystyle \lambda _{\mathrm {impurity} }}
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2001:{\displaystyle \lambda _{MFP}\approx 1{\mu }}
806:{\displaystyle \lambda _{\mathrm {defect} }}
708:{\displaystyle \lambda _{\mathrm {op,abs} }}
662:{\displaystyle \lambda _{\mathrm {op,ems} }}
767:is the electron-impurity scattering length,
589:is the electron-electron scattering length,
582:{\displaystyle \lambda _{\mathrm {el-el} }}
2106:Electronic Transport in Mesoscopic Systems
171:{\displaystyle L\leq \lambda _{\rm {MFP}}}
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1483:{\displaystyle G_{0}={\frac {2e^{2}}{h}}}
813:is the electron-defect scattering length,
2386:Du, Xu; Skachko, Ivan; Barker, Anthony;
1959:Carbon nanotubes and graphene nanoribbon
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616:{\displaystyle \lambda _{\mathrm {ap} }}
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1951:As mentioned, nanostructures such as
1681:The widely encountered phenomenon of
1654:. Ballistic transport is coherent in
109:: due to the small size of the wire (
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1646:Ballistic conduction enables use of
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2453:Jalabert, R. A.; Pichard, J.-L.;
1932:and dependent on other electrons;
1282:is the transmission probability (
1318:{\displaystyle G={\frac {I}{V}}}
2045:Ballistic deflection transistor
2040:Ballistic collection transistor
1247:{\displaystyle E_{\rm {F_{B}}}}
1211:{\displaystyle E_{\rm {F_{A}}}}
962:{\displaystyle E_{\rm {F_{B}}}}
926:{\displaystyle E_{\rm {F_{A}}}}
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2022:List of thermal conductivities
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1534:energy dispersion relationship
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241:, written here for electrons:
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2216:Zhai, C; et al. (2016).
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2013:Isotopically enriched diamond
1683:electrical contact resistance
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61:Newton's second law of motion
1628:semiclassical approximations
997:Boltzmann transport equation
887:for the system in question.
31:) is the unimpeded flow (or
7:
2027:
1946:
1851:correspond to the specific
974:Landauer–Büttiker formalism
10:
2548:
2489:10.1209/0295-5075/27/4/001
1879:have a slightly different
977:
51:, thermal fluctuations of
2459:EPL (Europhysics Letters)
2245:10.1016/j.eml.2016.03.021
2225:Extreme Mechanics Letters
2018:Isotopically pure diamond
1844:{\displaystyle \rho _{2}}
1817:{\displaystyle \rho _{1}}
1431:{\displaystyle G=G_{0}MT}
127:
2195:10.1103/PhysRevB.46.4053
2152:10.1103/PhysRevB.44.6329
1703:{\displaystyle \lambda }
1660:double-slit interference
1254:are the Fermi levels of
1174:is the electron charge,
2268:Applied Physics Letters
1650:properties of electron
993:field effect transistor
65:non-relativistic speeds
2422:10.1038/nnano.2008.199
2353:10.1038/nnano.2012.169
2002:
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1666:(and other optical or
1658:terms. Phenomena like
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2333:Nature Nanotechnology
2253:1959.4/unsworks_60452
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25:ballistic conduction
2481:1994EL.....27..255J
2455:Beenakker, C. W. J.
2414:2008NatNa...3..491D
2345:2012NatNa...7..713K
2290:2006ApPhL..89b3125K
2237:2016ExML....9..422Z
2187:1992PhRvB..46.4053P
2144:1991PhRvB..44.6329P
1926:coulombic repulsion
1626:or approximated by
1530:quantum confinement
1526:conductance quantum
1095:
990:graphene nanoribbon
885:Fermi's golden rule
29:ballistic transport
2532:Mesoscopic physics
2075:JSAP International
2054:Velocity overshoot
1998:
1917:and electrons are
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239:Matthiessen's rule
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21:mesoscopic physics
2298:10.1063/1.2218322
2175:Physical Review B
2138:(12): 6329–6339.
2132:Physical Review B
2116:978-0-521-59943-6
2034:Adiabatic circuit
1863:Optical analogies
1797:This term, where
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1688:fractal dimension
1632:WKB approximation
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191:{\displaystyle L}
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57:crystalline solid
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2339:(11): 713–717.
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1508:
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1502:
1499:
1498:
1468:
1464:
1460:
1458:
1449:
1445:
1443:
1440:
1439:
1416:
1412:
1404:
1401:
1400:
1377:
1373:
1372:
1371:
1367:
1355:
1351:
1350:
1349:
1345:
1334:
1331:
1330:
1305:
1297:
1294:
1293:
1235:
1231:
1230:
1229:
1225:
1223:
1220:
1219:
1199:
1195:
1194:
1193:
1189:
1187:
1184:
1183:
1180:Planck constant
1168:spin degeneracy
1165:
1112:
1108:
1085:
1081:
1080:
1079:
1075:
1074:
1064:
1060:
1059:
1058:
1054:
1053:
1034:
1030:
1029:
1027:
1014:
1013:
1009:
1007:
1004:
1003:
982:
976:
950:
946:
945:
944:
940:
938:
935:
934:
914:
910:
909:
908:
904:
902:
899:
898:
828:
827:
823:
821:
818:
817:
781:
780:
776:
774:
771:
770:
729:
728:
724:
722:
719:
718:
683:
682:
678:
676:
673:
672:
637:
636:
632:
630:
627:
626:
603:
602:
598:
596:
593:
592:
560:
559:
555:
553:
550:
549:
502:
501:
497:
492:
465:
464:
460:
455:
422:
421:
417:
412:
385:
384:
380:
375:
348:
347:
343:
338:
323:
322:
318:
313:
289:
288:
284:
279:
261:
260:
256:
251:
249:
246:
245:
214:
213:
209:
207:
204:
203:
183:
180:
179:
155:
154:
150:
142:
139:
138:
135:
130:
122:Meissner effect
37:charge carriers
17:
12:
11:
5:
2545:
2535:
2534:
2529:
2524:
2510:
2509:
2450:
2398:(8): 491–495.
2388:Andrei, Eva Y.
2381:
2378:
2375:
2374:
2319:
2258:
2208:
2165:
2122:
2115:
2092:
2064:
2063:
2061:
2058:
2057:
2056:
2051:
2042:
2037:
2029:
2026:
2014:
2011:
1996:
1992:
1989:
1984:
1981:
1978:
1974:
1960:
1957:
1948:
1945:
1944:
1943:
1940:
1933:
1922:
1864:
1861:
1838:
1834:
1811:
1807:
1795:
1794:
1783:
1777:
1774:
1769:
1764:
1760:
1756:
1751:
1747:
1743:
1740:
1734:
1728:
1723:
1699:
1656:wave mechanics
1652:wave functions
1643:
1640:
1613:
1612:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1572:
1568:
1563:
1560:
1538:Brillouin zone
1511:
1507:
1491:
1490:
1477:
1471:
1467:
1463:
1457:
1452:
1448:
1427:
1424:
1419:
1415:
1411:
1408:
1380:
1376:
1370:
1366:
1358:
1354:
1348:
1344:
1341:
1338:
1327:
1326:
1312:
1309:
1304:
1301:
1238:
1234:
1228:
1202:
1198:
1192:
1163:
1157:
1156:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1115:
1111:
1107:
1104:
1101:
1098:
1088:
1084:
1078:
1067:
1063:
1057:
1052:
1046:
1042:
1033:
1026:
1020:
1017:
1012:
978:Main article:
975:
972:
953:
949:
943:
917:
913:
907:
872:optical phonon
868:
867:
852:
849:
846:
843:
840:
837:
834:
831:
826:
814:
799:
796:
793:
790:
787:
784:
779:
768:
753:
750:
747:
744:
741:
738:
735:
732:
727:
716:
701:
698:
695:
692:
689:
686:
681:
670:
655:
652:
649:
646:
643:
640:
635:
624:
609:
606:
601:
590:
575:
572:
569:
566:
563:
558:
543:
542:
526:
523:
520:
517:
514:
511:
508:
505:
500:
496:
491:
483:
480:
477:
474:
471:
468:
463:
459:
454:
446:
443:
440:
437:
434:
431:
428:
425:
420:
416:
411:
403:
400:
397:
394:
391:
388:
383:
379:
374:
366:
363:
360:
357:
354:
351:
346:
342:
337:
329:
326:
321:
317:
312:
304:
301:
298:
295:
292:
287:
283:
278:
270:
267:
264:
259:
255:
223:
220:
217:
212:
187:
164:
161:
158:
153:
149:
146:
134:
131:
129:
126:
72:mean free path
15:
9:
6:
4:
3:
2:
2544:
2533:
2530:
2528:
2525:
2523:
2520:
2519:
2517:
2506:
2502:
2498:
2494:
2490:
2486:
2482:
2478:
2473:
2468:
2464:
2460:
2456:
2451:
2447:
2443:
2439:
2435:
2431:
2427:
2423:
2419:
2415:
2411:
2406:
2401:
2397:
2393:
2389:
2384:
2383:
2370:
2366:
2362:
2358:
2354:
2350:
2346:
2342:
2338:
2334:
2330:
2323:
2315:
2311:
2307:
2303:
2299:
2295:
2291:
2287:
2282:
2277:
2274:(2): 023125.
2273:
2269:
2262:
2254:
2250:
2246:
2242:
2238:
2234:
2230:
2226:
2219:
2212:
2204:
2200:
2196:
2192:
2188:
2184:
2180:
2176:
2169:
2161:
2157:
2153:
2149:
2145:
2141:
2137:
2133:
2126:
2118:
2112:
2108:
2107:
2102:
2101:Supriyo Datta
2096:
2088:
2084:
2080:
2076:
2069:
2065:
2055:
2052:
2046:
2043:
2041:
2038:
2035:
2032:
2031:
2025:
2023:
2019:
2010:
1994:
1990:
1987:
1982:
1979:
1976:
1972:
1956:
1954:
1941:
1938:
1934:
1931:
1927:
1923:
1920:
1916:
1912:
1909:
1908:
1907:
1904:
1902:
1898:
1894:
1890:
1886:
1885:monochromatic
1882:
1876:
1872:
1870:
1860:
1858:
1854:
1836:
1832:
1809:
1805:
1781:
1775:
1772:
1762:
1758:
1754:
1749:
1745:
1738:
1732:
1721:
1713:
1712:
1711:
1697:
1689:
1684:
1679:
1677:
1673:
1669:
1665:
1661:
1657:
1653:
1649:
1639:
1637:
1633:
1629:
1625:
1621:
1616:
1570:
1566:
1561:
1558:
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1550:
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1475:
1469:
1465:
1461:
1455:
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1292:
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1257:
1226:
1190:
1181:
1177:
1173:
1169:
1162:
1142:
1139:
1133:
1127:
1121:
1109:
1102:
1096:
1076:
1055:
1050:
1044:
1040:
1031:
1024:
1010:
1002:
1001:
1000:
998:
994:
991:
987:
986:Rolf Landauer
981:
941:
905:
897:
892:
888:
886:
882:
878:
873:
824:
815:
777:
769:
725:
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690:
679:
671:
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625:
599:
591:
567:
556:
548:
547:
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498:
494:
489:
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392:
381:
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340:
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319:
315:
310:
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281:
276:
257:
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210:
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185:
151:
147:
144:
125:
123:
119:
114:
112:
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104:
100:
97:
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85:work function
82:
78:
73:
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38:
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26:
22:
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1962:
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1267:
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1175:
1171:
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1160:
1158:
983:
896:Fermi levels
869:
544:
136:
115:
89:
80:
76:
69:
28:
24:
18:
2231:: 422–429.
1893:statistical
1853:resistivity
1636:dissipation
1630:, like the
1288:conductance
881:Hamiltonian
45:resistivity
2516:Categories
2465:(4): 255.
2060:References
1897:stochastic
1686:with high
1662:, spatial
1642:Importance
883:and solve
2497:0295-5075
2446:118441080
2430:1748-3387
2405:0802.2933
2361:1748-3387
2306:0003-6951
1995:μ
1988:≈
1973:λ
1937:rest mass
1930:nonlinear
1924:there is
1901:inelastic
1869:waveguide
1833:ρ
1806:ρ
1759:ρ
1746:ρ
1739:λ
1698:λ
1676:nanotubes
1672:nanowires
1668:microwave
1664:resonance
1562:≈
1365:−
1114:′
1051:∫
984:In 1957,
825:λ
778:λ
726:λ
680:λ
634:λ
600:λ
568:−
557:λ
499:λ
462:λ
419:λ
382:λ
345:λ
320:λ
297:−
286:λ
258:λ
211:λ
152:λ
148:≤
111:nanometer
99:nanowires
41:electrons
39:(usually
33:transport
2505:55864480
2438:18685637
2369:23064554
2314:44232115
2203:10004135
2103:(1997).
2087:28636503
2028:See also
1947:Examples
1919:fermions
1620:coherent
1536:and the
107:nanowire
2477:Bibcode
2410:Bibcode
2341:Bibcode
2286:Bibcode
2233:Bibcode
2183:Bibcode
2160:9998497
2140:Bibcode
1911:photons
1889:elastic
1438:, with
1178:is the
103:phonons
96:silicon
49:defects
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1915:bosons
1493:where
1159:where
545:where
200:MOSFET
178:where
128:Theory
2501:S2CID
2467:arXiv
2442:S2CID
2400:arXiv
2310:S2CID
2276:arXiv
2221:(PDF)
2083:S2CID
2081:(9).
1881:phase
999:, is
77:walls
55:in a
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2493:ISSN
2434:PMID
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2199:PMID
2156:PMID
2111:ISBN
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1895:and
1824:and
1674:and
1544:and
1280:T(E)
1258:and
1218:and
933:and
816:and
70:The
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2485:doi
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