17:
25:
96:
each other exactly and these (otherwise random in sign) quantum interference terms survive disorder averaging. Since it is much more likely to find a self-crossing trajectory in low dimensions, the weak localization effect manifests itself much more strongly in low-dimensional systems (films and wires).
95:
The weak localization correction can be shown to come mostly from quantum interference between self-crossing paths in which an electron can propagate in the clock-wise and counter-clockwise direction around a loop. Due to the identical length of the two paths along a loop, the quantum phases cancel
84:
part (individual probabilities of diffusive paths) and a number of interference terms (products of the amplitudes corresponding to different paths). These interference terms effectively make it more likely that a carrier will "wander around in a circle" than it would otherwise, which leads to an
83:
tells us that to find the total probability we have to sum up the quantum-mechanical amplitudes of the paths rather than the probabilities themselves. Therefore, the correct (quantum-mechanical) formula for the probability for an electron to move from a point A to a point B includes the classical
108:, the spin of a carrier is coupled to its momentum. The spin of the carrier rotates as it goes around a self-intersecting path, and the direction of this rotation is opposite for the two directions about the loop. Because of this, the two paths along any loop interfere
78:
of the system is related to the probability of an electron to propagate between two given points in space. Classical physics assumes that the total probability is just the sum of the probabilities of the paths connecting the two points. However
2297:
1405:
807:
415:
1119:
972:
2330:
The strength of either weak localization or weak anti-localization falls off quickly in the presence of a magnetic field, which causes carriers to acquire an additional phase as they move around paths.
1821:
523:
1586:
124:
In two dimensions the change in conductivity from applying a magnetic field, due to either weak localization or weak anti-localization can be described by the Hikami-Larkin-Nagaoka equation:
2125:
2001:
882:
67:
motion is diffusive rather than ballistic. That is, an electron does not move along a straight line, but experiences a series of random scatterings off impurities which results in a
1029:
92:) corresponds to the former classical terms, while the weak localization correction corresponds to the latter quantum interference terms averaged over disorder realizations.
464:
2055:
1913:
2028:
1886:
2320:
1233:
2082:
1940:
1855:
1203:
1176:
1149:
1942:
is the elastic characteristic field. The characteristic fields are better understood in terms of their corresponding characteristic lengths which are deduced from
529:. This theoretically derived equation was soon restated in terms of characteristic fields, which are more directly experimentally relevant quantities:
2360:
Altshuler, B. L.; D. Khmel'nitzkii; A. I. Larkin; P. A. Lee (1980). "Magnetoresistance and Hall effect in a disordered two-dimensional electron gas".
2133:
2558:
1241:
535:
130:
2057:
can be thought of as the distance traveled before the spin of the electron undergoes the effect of the spin–orbit interaction, and finally
2464:
Poole, D A; Pepper, M; Hughes, A (1982-11-20). "Spin-orbit coupling and weak localisation in the 2D inversion layer of indium phosphide".
1035:
888:
35:
is a physical effect which occurs in disordered electronic systems at very low temperatures. The effect manifests itself as a
1915:
is the spin–orbit characteristic field which can be considered a measure of the strength of the spin–orbit interaction and
1618:
469:
1425:
2404:
1888:
is the phase coherence characteristic field, which is roughly the magnetic field required to destroy phase coherence,
2553:
2090:
1945:
63:
The effect is quantum-mechanical in nature and has the following origin: In a disordered electronic system, the
818:
2548:
978:
423:
2033:
1891:
16:
2340:
105:
2030:
can then be understood as the distance traveled by an electron before it loses phase coherence,
2006:
1864:
2305:
52:
1208:
2508:
2436:
2369:
2060:
1918:
1840:
1181:
1154:
1127:
526:
8:
88:
in the net resistivity. The usual formula for the conductivity of a metal (the so-called
2512:
2440:
2373:
2477:
2524:
2499:
Bergman, Gerd (1982-04-12). "Influence of Spin-Orbit
Coupling on Weak Localization".
2481:
2400:
80:
2516:
2473:
2444:
2425:"Spin–Orbit Interaction and Magnetoresistance in the Two-Dimensional Random System"
2377:
1858:
2359:
2520:
2292:{\displaystyle \sigma (B)-\sigma (0)=\alpha {e^{2} \over 2\pi ^{2}\hbar }\left}
24:
2542:
2528:
2485:
2381:
89:
48:
1400:{\displaystyle \sigma (B)-\sigma (0)=+{e^{2} \over 2\pi ^{2}\hbar }\left}
802:{\displaystyle \sigma (B)-\sigma (0)=-{e^{2} \over 2\pi ^{2}\hbar }\left}
410:{\displaystyle \sigma (B)-\sigma (0)=-{e^{2} \over 2\pi ^{2}\hbar }\left}
75:
68:
40:
51:. The name emphasizes the fact that weak localization is a precursor of
28:
Weak localization is due primarily to self-intersecting scattering paths
2449:
2424:
1235:
is spin-orbit scattering. Under some condition, this can be rewritten:
64:
2322:
is -1 for weak antilocalization and +1/2 for weak localization.
20:
There are many possible scattering paths in a disordered system
1114:{\displaystyle H_{4}={2 \over 3}H_{S}+{4 \over 3}H_{SO}+H_{i}}
967:{\displaystyle H_{2}={4 \over 3}H_{SO}+{2 \over 3}H_{S}+H_{i}}
44:
2422:
2308:
2136:
2093:
2063:
2036:
2009:
1948:
1921:
1894:
1867:
1843:
1621:
1428:
1244:
1211:
1184:
1157:
1130:
1038:
981:
891:
821:
538:
472:
426:
133:
1816:{\displaystyle -{3e^{2} \over 2\pi ^{2}\hbar }\left}
518:{\displaystyle \tau ,\tau _{1},\tau _{2},\tau _{3}}
2507:(15). American Physical Society (APS): 1046–1049.
2314:
2291:
2119:
2076:
2049:
2022:
1995:
1934:
1907:
1880:
1849:
1815:
1581:{\displaystyle +{e^{2} \over \pi ^{2}\hbar }\left}
1580:
1399:
1227:
1197:
1170:
1143:
1113:
1023:
966:
876:
801:
517:
458:
409:
2463:
2540:
2423:Hikami, S.; A. I Larkin; Y. Nagaoka (1980).
2325:
2087:In the limit of strong spin–orbit coupling
2397:Electronic Transport in Mesoscopic Systems
2120:{\displaystyle B_{\text{SO}}\gg B_{\phi }}
1996:{\displaystyle {B_{i}=\hbar /4el_{i}^{2}}}
2466:Journal of Physics C: Solid State Physics
2448:
99:
2418:
2416:
877:{\displaystyle H_{1}=H_{0}+H_{SO}+H_{s}}
23:
15:
2498:
2559:Electric and magnetic fields in matter
2541:
2413:
2394:
812:Where the characteristic fields are:
119:
58:
2472:(32). IOP Publishing: L1137–L1145.
55:, which occurs at strong disorder.
13:
1024:{\displaystyle H_{3}=2H_{S}+H_{i}}
14:
2570:
2196:
2127:, the equation above reduces to:
1963:
1656:
1455:
1304:
598:
450:
193:
2429:Progress of Theoretical Physics
2492:
2457:
2399:. Cambridge University Press.
2388:
2353:
2161:
2155:
2146:
2140:
1771:
1757:
1694:
1680:
1269:
1263:
1254:
1248:
791:
758:
739:
706:
687:
654:
645:
612:
563:
557:
548:
542:
459:{\displaystyle a=4DeH/\hbar c}
399:
361:
342:
304:
285:
247:
238:
207:
158:
152:
143:
137:
1:
2346:
2050:{\displaystyle l_{\text{SO}}}
1908:{\displaystyle B_{\text{SO}}}
1205:is magnetic scattering, and
7:
2521:10.1103/physrevlett.48.1046
2478:10.1088/0022-3719/15/32/005
2334:
10:
2575:
2084:is the mean free path.
2326:Magnetic field dependence
2023:{\displaystyle l_{\phi }}
1881:{\displaystyle B_{\phi }}
1178:is inelastic scattering,
1151:is potential scattering,
2554:Condensed matter physics
2382:10.1103/PhysRevB.22.5142
2501:Physical Review Letters
2341:Coherent backscattering
2315:{\displaystyle \alpha }
2316:
2293:
2121:
2078:
2051:
2024:
1997:
1936:
1909:
1882:
1851:
1817:
1582:
1401:
1229:
1228:{\displaystyle H_{SO}}
1199:
1172:
1145:
1115:
1025:
968:
878:
803:
519:
460:
411:
100:Weak anti-localization
29:
21:
2317:
2294:
2122:
2079:
2077:{\displaystyle l_{e}}
2052:
2025:
1998:
1937:
1935:{\displaystyle B_{e}}
1910:
1883:
1852:
1850:{\displaystyle \psi }
1818:
1583:
1402:
1230:
1200:
1198:{\displaystyle H_{S}}
1173:
1171:{\displaystyle H_{i}}
1146:
1144:{\displaystyle H_{0}}
1116:
1026:
969:
879:
804:
520:
461:
412:
53:Anderson localization
27:
19:
2306:
2134:
2091:
2061:
2034:
2007:
1946:
1919:
1892:
1865:
1841:
1619:
1426:
1242:
1209:
1182:
1155:
1128:
1036:
979:
889:
819:
536:
470:
424:
131:
2513:1982PhRvL..48.1046B
2441:1980PThPh..63..707H
2374:1980PhRvB..22.5142A
1991:
106:spin–orbit coupling
2549:Mesoscopic physics
2450:10.1143/PTP.63.707
2395:Datta, S. (1995).
2312:
2289:
2117:
2074:
2047:
2020:
1993:
1977:
1932:
1905:
1878:
1847:
1813:
1578:
1397:
1225:
1195:
1168:
1141:
1111:
1021:
964:
874:
799:
515:
456:
407:
116:net resistivity.
39:correction to the
30:
22:
2302:In this equation
2277:
2257:
2232:
2200:
2101:
2044:
1902:
1801:
1781:
1749:
1724:
1704:
1660:
1566:
1546:
1531:
1506:
1486:
1459:
1385:
1365:
1340:
1308:
1083:
1060:
939:
913:
789:
769:
753:
737:
717:
701:
685:
665:
643:
623:
602:
397:
372:
356:
340:
315:
299:
283:
258:
236:
218:
197:
120:In two dimensions
112:which leads to a
104:In a system with
81:quantum mechanics
59:General principle
33:Weak localization
2566:
2533:
2532:
2496:
2490:
2489:
2461:
2455:
2454:
2452:
2420:
2411:
2410:
2392:
2386:
2385:
2357:
2321:
2319:
2318:
2313:
2298:
2296:
2295:
2290:
2288:
2284:
2283:
2279:
2278:
2273:
2272:
2263:
2258:
2250:
2237:
2233:
2228:
2227:
2218:
2201:
2199:
2195:
2194:
2181:
2180:
2171:
2126:
2124:
2123:
2118:
2116:
2115:
2103:
2102:
2099:
2083:
2081:
2080:
2075:
2073:
2072:
2056:
2054:
2053:
2048:
2046:
2045:
2042:
2029:
2027:
2026:
2021:
2019:
2018:
2002:
2000:
1999:
1994:
1992:
1990:
1985:
1970:
1959:
1958:
1941:
1939:
1938:
1933:
1931:
1930:
1914:
1912:
1911:
1906:
1904:
1903:
1900:
1887:
1885:
1884:
1879:
1877:
1876:
1859:digamma function
1856:
1854:
1853:
1848:
1822:
1820:
1819:
1814:
1812:
1808:
1807:
1803:
1802:
1797:
1796:
1795:
1783:
1782:
1779:
1767:
1755:
1750:
1742:
1729:
1725:
1720:
1719:
1718:
1706:
1705:
1702:
1690:
1678:
1661:
1659:
1655:
1654:
1641:
1640:
1639:
1626:
1587:
1585:
1584:
1579:
1577:
1573:
1572:
1568:
1567:
1562:
1561:
1560:
1548:
1547:
1544:
1537:
1532:
1524:
1511:
1507:
1502:
1501:
1500:
1488:
1487:
1484:
1477:
1460:
1458:
1454:
1453:
1443:
1442:
1433:
1406:
1404:
1403:
1398:
1396:
1392:
1391:
1387:
1386:
1381:
1380:
1371:
1366:
1358:
1345:
1341:
1336:
1335:
1326:
1309:
1307:
1303:
1302:
1289:
1288:
1279:
1234:
1232:
1231:
1226:
1224:
1223:
1204:
1202:
1201:
1196:
1194:
1193:
1177:
1175:
1174:
1169:
1167:
1166:
1150:
1148:
1147:
1142:
1140:
1139:
1120:
1118:
1117:
1112:
1110:
1109:
1097:
1096:
1084:
1076:
1071:
1070:
1061:
1053:
1048:
1047:
1030:
1028:
1027:
1022:
1020:
1019:
1007:
1006:
991:
990:
973:
971:
970:
965:
963:
962:
950:
949:
940:
932:
927:
926:
914:
906:
901:
900:
883:
881:
880:
875:
873:
872:
860:
859:
844:
843:
831:
830:
808:
806:
805:
800:
798:
794:
790:
785:
784:
775:
770:
762:
754:
746:
738:
733:
732:
723:
718:
710:
702:
694:
686:
681:
680:
671:
666:
658:
644:
639:
638:
629:
624:
616:
603:
601:
597:
596:
583:
582:
573:
527:relaxation times
524:
522:
521:
516:
514:
513:
501:
500:
488:
487:
465:
463:
462:
457:
449:
416:
414:
413:
408:
406:
402:
398:
396:
392:
391:
378:
373:
365:
357:
349:
341:
339:
335:
334:
321:
316:
308:
300:
292:
284:
282:
278:
277:
264:
259:
251:
237:
235:
224:
219:
211:
198:
196:
192:
191:
178:
177:
168:
2574:
2573:
2569:
2568:
2567:
2565:
2564:
2563:
2539:
2538:
2537:
2536:
2497:
2493:
2462:
2458:
2421:
2414:
2407:
2393:
2389:
2358:
2354:
2349:
2337:
2328:
2307:
2304:
2303:
2268:
2264:
2262:
2249:
2248:
2244:
2223:
2219:
2217:
2213:
2206:
2202:
2190:
2186:
2182:
2176:
2172:
2170:
2135:
2132:
2131:
2111:
2107:
2098:
2094:
2092:
2089:
2088:
2068:
2064:
2062:
2059:
2058:
2041:
2037:
2035:
2032:
2031:
2014:
2010:
2008:
2005:
2004:
1986:
1981:
1966:
1954:
1950:
1949:
1947:
1944:
1943:
1926:
1922:
1920:
1917:
1916:
1899:
1895:
1893:
1890:
1889:
1872:
1868:
1866:
1863:
1862:
1842:
1839:
1838:
1791:
1787:
1778:
1774:
1763:
1756:
1754:
1741:
1740:
1736:
1714:
1710:
1701:
1697:
1686:
1679:
1677:
1673:
1666:
1662:
1650:
1646:
1642:
1635:
1631:
1627:
1625:
1620:
1617:
1616:
1556:
1552:
1543:
1539:
1538:
1536:
1523:
1522:
1518:
1496:
1492:
1483:
1479:
1478:
1476:
1472:
1465:
1461:
1449:
1445:
1444:
1438:
1434:
1432:
1427:
1424:
1423:
1376:
1372:
1370:
1357:
1356:
1352:
1331:
1327:
1325:
1321:
1314:
1310:
1298:
1294:
1290:
1284:
1280:
1278:
1243:
1240:
1239:
1216:
1212:
1210:
1207:
1206:
1189:
1185:
1183:
1180:
1179:
1162:
1158:
1156:
1153:
1152:
1135:
1131:
1129:
1126:
1125:
1105:
1101:
1089:
1085:
1075:
1066:
1062:
1052:
1043:
1039:
1037:
1034:
1033:
1015:
1011:
1002:
998:
986:
982:
980:
977:
976:
958:
954:
945:
941:
931:
919:
915:
905:
896:
892:
890:
887:
886:
868:
864:
852:
848:
839:
835:
826:
822:
820:
817:
816:
780:
776:
774:
761:
745:
728:
724:
722:
709:
693:
676:
672:
670:
657:
634:
630:
628:
615:
608:
604:
592:
588:
584:
578:
574:
572:
537:
534:
533:
509:
505:
496:
492:
483:
479:
471:
468:
467:
445:
425:
422:
421:
387:
383:
382:
377:
364:
348:
330:
326:
325:
320:
307:
291:
273:
269:
268:
263:
250:
228:
223:
210:
203:
199:
187:
183:
179:
173:
169:
167:
132:
129:
128:
122:
102:
61:
12:
11:
5:
2572:
2562:
2561:
2556:
2551:
2535:
2534:
2491:
2456:
2435:(2): 707–710.
2412:
2406:978-0521599436
2405:
2387:
2351:
2350:
2348:
2345:
2344:
2343:
2336:
2333:
2327:
2324:
2311:
2300:
2299:
2287:
2282:
2276:
2271:
2267:
2261:
2256:
2253:
2247:
2243:
2240:
2236:
2231:
2226:
2222:
2216:
2212:
2209:
2205:
2198:
2193:
2189:
2185:
2179:
2175:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2114:
2110:
2106:
2097:
2071:
2067:
2040:
2017:
2013:
1989:
1984:
1980:
1976:
1973:
1969:
1965:
1962:
1957:
1953:
1929:
1925:
1898:
1875:
1871:
1846:
1836:
1835:
1834:
1833:
1832:
1831:
1830:
1829:
1828:
1827:
1826:
1825:
1824:
1823:
1811:
1806:
1800:
1794:
1790:
1786:
1777:
1773:
1770:
1766:
1762:
1759:
1753:
1748:
1745:
1739:
1735:
1732:
1728:
1723:
1717:
1713:
1709:
1700:
1696:
1693:
1689:
1685:
1682:
1676:
1672:
1669:
1665:
1658:
1653:
1649:
1645:
1638:
1634:
1630:
1624:
1601:
1600:
1599:
1598:
1597:
1596:
1595:
1594:
1593:
1592:
1591:
1590:
1589:
1588:
1576:
1571:
1565:
1559:
1555:
1551:
1542:
1535:
1530:
1527:
1521:
1517:
1514:
1510:
1505:
1499:
1495:
1491:
1482:
1475:
1471:
1468:
1464:
1457:
1452:
1448:
1441:
1437:
1431:
1408:
1407:
1395:
1390:
1384:
1379:
1375:
1369:
1364:
1361:
1355:
1351:
1348:
1344:
1339:
1334:
1330:
1324:
1320:
1317:
1313:
1306:
1301:
1297:
1293:
1287:
1283:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1222:
1219:
1215:
1192:
1188:
1165:
1161:
1138:
1134:
1122:
1121:
1108:
1104:
1100:
1095:
1092:
1088:
1082:
1079:
1074:
1069:
1065:
1059:
1056:
1051:
1046:
1042:
1031:
1018:
1014:
1010:
1005:
1001:
997:
994:
989:
985:
974:
961:
957:
953:
948:
944:
938:
935:
930:
925:
922:
918:
912:
909:
904:
899:
895:
884:
871:
867:
863:
858:
855:
851:
847:
842:
838:
834:
829:
825:
810:
809:
797:
793:
788:
783:
779:
773:
768:
765:
760:
757:
752:
749:
744:
741:
736:
731:
727:
721:
716:
713:
708:
705:
700:
697:
692:
689:
684:
679:
675:
669:
664:
661:
656:
653:
650:
647:
642:
637:
633:
627:
622:
619:
614:
611:
607:
600:
595:
591:
587:
581:
577:
571:
568:
565:
562:
559:
556:
553:
550:
547:
544:
541:
512:
508:
504:
499:
495:
491:
486:
482:
478:
475:
455:
452:
448:
444:
441:
438:
435:
432:
429:
418:
417:
405:
401:
395:
390:
386:
381:
376:
371:
368:
363:
360:
355:
352:
347:
344:
338:
333:
329:
324:
319:
314:
311:
306:
303:
298:
295:
290:
287:
281:
276:
272:
267:
262:
257:
254:
249:
246:
243:
240:
234:
231:
227:
222:
217:
214:
209:
206:
202:
195:
190:
186:
182:
176:
172:
166:
163:
160:
157:
154:
151:
148:
145:
142:
139:
136:
121:
118:
101:
98:
60:
57:
9:
6:
4:
3:
2:
2571:
2560:
2557:
2555:
2552:
2550:
2547:
2546:
2544:
2530:
2526:
2522:
2518:
2514:
2510:
2506:
2502:
2495:
2487:
2483:
2479:
2475:
2471:
2467:
2460:
2451:
2446:
2442:
2438:
2434:
2430:
2426:
2419:
2417:
2408:
2402:
2398:
2391:
2383:
2379:
2375:
2371:
2367:
2363:
2356:
2352:
2342:
2339:
2338:
2332:
2323:
2309:
2285:
2280:
2274:
2269:
2265:
2259:
2254:
2251:
2245:
2241:
2238:
2234:
2229:
2224:
2220:
2214:
2210:
2207:
2203:
2191:
2187:
2183:
2177:
2173:
2167:
2164:
2158:
2152:
2149:
2143:
2137:
2130:
2129:
2128:
2112:
2108:
2104:
2095:
2085:
2069:
2065:
2038:
2015:
2011:
1987:
1982:
1978:
1974:
1971:
1967:
1960:
1955:
1951:
1927:
1923:
1896:
1873:
1869:
1860:
1844:
1809:
1804:
1798:
1792:
1788:
1784:
1775:
1768:
1764:
1760:
1751:
1746:
1743:
1737:
1733:
1730:
1726:
1721:
1715:
1711:
1707:
1698:
1691:
1687:
1683:
1674:
1670:
1667:
1663:
1651:
1647:
1643:
1636:
1632:
1628:
1622:
1615:
1614:
1613:
1612:
1611:
1610:
1609:
1608:
1607:
1606:
1605:
1604:
1603:
1602:
1574:
1569:
1563:
1557:
1553:
1549:
1540:
1533:
1528:
1525:
1519:
1515:
1512:
1508:
1503:
1497:
1493:
1489:
1480:
1473:
1469:
1466:
1462:
1450:
1446:
1439:
1435:
1429:
1422:
1421:
1420:
1419:
1418:
1417:
1416:
1415:
1414:
1413:
1412:
1411:
1410:
1409:
1393:
1388:
1382:
1377:
1373:
1367:
1362:
1359:
1353:
1349:
1346:
1342:
1337:
1332:
1328:
1322:
1318:
1315:
1311:
1299:
1295:
1291:
1285:
1281:
1275:
1272:
1266:
1260:
1257:
1251:
1245:
1238:
1237:
1236:
1220:
1217:
1213:
1190:
1186:
1163:
1159:
1136:
1132:
1106:
1102:
1098:
1093:
1090:
1086:
1080:
1077:
1072:
1067:
1063:
1057:
1054:
1049:
1044:
1040:
1032:
1016:
1012:
1008:
1003:
999:
995:
992:
987:
983:
975:
959:
955:
951:
946:
942:
936:
933:
928:
923:
920:
916:
910:
907:
902:
897:
893:
885:
869:
865:
861:
856:
853:
849:
845:
840:
836:
832:
827:
823:
815:
814:
813:
795:
786:
781:
777:
771:
766:
763:
755:
750:
747:
742:
734:
729:
725:
719:
714:
711:
703:
698:
695:
690:
682:
677:
673:
667:
662:
659:
651:
648:
640:
635:
631:
625:
620:
617:
609:
605:
593:
589:
585:
579:
575:
569:
566:
560:
554:
551:
545:
539:
532:
531:
530:
528:
510:
506:
502:
497:
493:
489:
484:
480:
476:
473:
453:
446:
442:
439:
436:
433:
430:
427:
403:
393:
388:
384:
379:
374:
369:
366:
358:
353:
350:
345:
336:
331:
327:
322:
317:
312:
309:
301:
296:
293:
288:
279:
274:
270:
265:
260:
255:
252:
244:
241:
232:
229:
225:
220:
215:
212:
204:
200:
188:
184:
180:
174:
170:
164:
161:
155:
149:
146:
140:
134:
127:
126:
125:
117:
115:
111:
110:destructively
107:
97:
93:
91:
90:Drude formula
87:
82:
77:
72:
70:
66:
56:
54:
50:
49:semiconductor
46:
42:
38:
34:
26:
18:
2504:
2500:
2494:
2469:
2465:
2459:
2432:
2428:
2396:
2390:
2368:(11): 5142.
2365:
2362:Phys. Rev. B
2361:
2355:
2329:
2301:
2086:
1837:
1123:
811:
525:are various
419:
123:
113:
109:
103:
94:
85:
73:
62:
36:
32:
31:
76:resistivity
69:random walk
41:resistivity
2543:Categories
2347:References
2529:0031-9007
2486:0022-3719
2310:α
2270:ϕ
2242:ψ
2239:−
2225:ϕ
2211:
2197:ℏ
2188:π
2168:α
2153:σ
2150:−
2138:σ
2113:ϕ
2105:≫
2016:ϕ
1964:ℏ
1874:ϕ
1845:ψ
1793:ϕ
1734:ψ
1731:−
1716:ϕ
1671:
1657:ℏ
1648:π
1623:−
1516:ψ
1513:−
1470:
1456:ℏ
1447:π
1378:ϕ
1350:ψ
1347:−
1333:ϕ
1319:
1305:ℏ
1296:π
1261:σ
1258:−
1246:σ
756:ψ
743:−
704:ψ
652:ψ
649:−
610:ψ
599:ℏ
590:π
570:−
555:σ
552:−
540:σ
507:τ
494:τ
481:τ
474:τ
451:ℏ
385:τ
359:ψ
346:−
328:τ
302:ψ
271:τ
245:ψ
242:−
230:τ
205:ψ
194:ℏ
185:π
165:−
150:σ
147:−
135:σ
2335:See also
86:increase
65:electron
37:positive
2509:Bibcode
2437:Bibcode
2370:Bibcode
1857:is the
2527:
2484:
2403:
1124:Where
466:, and
420:Where
114:lower
45:metal
43:of a
2525:ISSN
2482:ISSN
2401:ISBN
74:The
2517:doi
2474:doi
2445:doi
2378:doi
47:or
2545::
2523:.
2515:.
2505:48
2503:.
2480:.
2470:15
2468:.
2443:.
2433:63
2431:.
2427:.
2415:^
2376:.
2366:22
2364:.
2208:ln
2100:SO
2043:SO
2003:.
1901:SO
1861:.
1780:SO
1703:SO
1668:ln
1545:SO
1485:SO
1467:ln
1316:ln
71:.
2531:.
2519::
2511::
2488:.
2476::
2453:.
2447::
2439::
2409:.
2384:.
2380::
2372::
2286:]
2281:)
2275:B
2266:B
2260:+
2255:2
2252:1
2246:(
2235:)
2230:B
2221:B
2215:(
2204:[
2192:2
2184:2
2178:2
2174:e
2165:=
2162:)
2159:0
2156:(
2147:)
2144:B
2141:(
2109:B
2096:B
2070:e
2066:l
2039:l
2012:l
1988:2
1983:i
1979:l
1975:e
1972:4
1968:/
1961:=
1956:i
1952:B
1928:e
1924:B
1897:B
1870:B
1810:]
1805:)
1799:B
1789:B
1785:+
1776:B
1772:)
1769:3
1765:/
1761:4
1758:(
1752:+
1747:2
1744:1
1738:(
1727:)
1722:B
1712:B
1708:+
1699:B
1695:)
1692:3
1688:/
1684:4
1681:(
1675:(
1664:[
1652:2
1644:2
1637:2
1633:e
1629:3
1575:]
1570:)
1564:B
1558:e
1554:B
1550:+
1541:B
1534:+
1529:2
1526:1
1520:(
1509:)
1504:B
1498:e
1494:B
1490:+
1481:B
1474:(
1463:[
1451:2
1440:2
1436:e
1430:+
1394:]
1389:)
1383:B
1374:B
1368:+
1363:2
1360:1
1354:(
1343:)
1338:B
1329:B
1323:(
1312:[
1300:2
1292:2
1286:2
1282:e
1276:+
1273:=
1270:)
1267:0
1264:(
1255:)
1252:B
1249:(
1221:O
1218:S
1214:H
1191:S
1187:H
1164:i
1160:H
1137:0
1133:H
1107:i
1103:H
1099:+
1094:O
1091:S
1087:H
1081:3
1078:4
1073:+
1068:S
1064:H
1058:3
1055:2
1050:=
1045:4
1041:H
1017:i
1013:H
1009:+
1004:S
1000:H
996:2
993:=
988:3
984:H
960:i
956:H
952:+
947:S
943:H
937:3
934:2
929:+
924:O
921:S
917:H
911:3
908:4
903:=
898:2
894:H
870:s
866:H
862:+
857:O
854:S
850:H
846:+
841:0
837:H
833:=
828:1
824:H
796:]
792:)
787:H
782:4
778:H
772:+
767:2
764:1
759:(
751:2
748:1
740:)
735:H
730:3
726:H
720:+
715:2
712:1
707:(
699:2
696:1
691:+
688:)
683:H
678:2
674:H
668:+
663:2
660:1
655:(
646:)
641:H
636:1
632:H
626:+
621:2
618:1
613:(
606:[
594:2
586:2
580:2
576:e
567:=
564:)
561:0
558:(
549:)
546:B
543:(
511:3
503:,
498:2
490:,
485:1
477:,
454:c
447:/
443:H
440:e
437:D
434:4
431:=
428:a
404:]
400:)
394:a
389:3
380:1
375:+
370:2
367:1
362:(
354:2
351:1
343:)
337:a
332:2
323:1
318:+
313:2
310:1
305:(
297:2
294:1
289:+
286:)
280:a
275:1
266:1
261:+
256:2
253:1
248:(
239:)
233:a
226:1
221:+
216:2
213:1
208:(
201:[
189:2
181:2
175:2
171:e
162:=
159:)
156:0
153:(
144:)
141:B
138:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
