989:
Other treatments of the problem include a mean-field numerical approach, as well as more recent treatments such as, also verifying the upper bound suggested above is a tight bound. Many Monte Carlo simulations were also performed, some of them in disagreement with the result quoted above. Few works
1056:
dependence is obtained, for any dimension. The observation of this is expected to occur below a certain temperature, such that the optimal energy of hopping would be smaller than the width of the
Coulomb gap. The transition from Mott to so-called Efros–Shklovskii variable-range hopping has been
998:
Direct experimental confirmation of the gap has been done via tunneling experiments, which probed the single-particle DOS in two and three dimensions. The experiments clearly showed a linear gap in two dimensions, and a parabolic gap in three dimensions. Another experimental consequence of the
553:
of these would be occupied, and the others unoccupied. Of all pairs of occupied and unoccupied sites, let us choose the one where the two are closest to each other. If we assume the sites are randomly distributed in space, we find that the distance between these two sites is of order:
307:
such that all sites with energies above it are empty, and below it are full (this is the Fermi energy, but since we are dealing with a system with interactions it is not obvious a-priori that it is still well-defined). Assume we have a finite single-particle DOS at the Fermi energy,
986:. This is an upper bound for the Coulomb gap. Efros considered single electron excitations, and obtained an integro-differential equation for the DOS, showing the Coulomb gap in fact follows the above equation (i.e., the upper bound is a tight bound).
31:(DOS) of a system of interacting localized electrons. Due to the long-range Coulomb interactions, the single-particle DOS vanishes at the chemical potential, at low enough temperatures, such that thermal excitations do not wash out the gap.
990:
deal with the quantum aspect of the problem. Classical
Coulomb gap in clean system without disorder is well captured within Extended Dynamical Mean Field Theory (EDMFT) supported by Metropolis Monte Carlo simulations.
106:, due to the disorder and the Coulomb interaction with all other electrons (we define this both for occupied and unoccupied sites), it is easy to see that the energy needed to move an electron from an occupied site
763:
999:
Coulomb gap is found in the conductivity of samples in the localized regime. The existence of a gap in the spectrum of excitations would result in a lowered conductivity than that predicted by
227:
832:
523:
952:
1100:
610:
984:
1054:
411:
1057:
observed experimentally for various systems. Nevertheless, no rigorous derivation of the Efros–Shklovskii conductivity formula has been put forth, and in some experiments
852:
342:
1120:
906:
879:
305:
258:
80:
551:
783:
650:
630:
472:
382:
362:
278:
144:
124:
104:
1646:
1434:
1377:
1275:
1391:
A. Mobius, M. Richter, and B. Drittler (1992). "Coulomb gap in two- and three-dimensional systems: Simulation results for large samples".
655:
280:, but after moving the electron this term should not be considered. It is easy to see from this that there exists an energy
1542:
J. G. Massey and M. Lee (1995). "Direct
Observation of the Coulomb Correlation Gap in a Nonmetallic Semiconductor, Si: B".
1726:
152:
1151:
M. Pollak (1970). "Effect of carrier-carrier interactions on some transport properties in disordered semiconductors".
384:, the energy invested should be positive, since we are assuming we are in the ground state of the system, i.e.,
1731:
788:
1736:
1248:
M. Grunewald, B. Pohlmann, L. Schweitzer, and D.Wurtz (1982). "Mean field approach to the electron glass".
1178:
A L Efros and B I Shklovskii (1975). "Coulomb gap and low temperature conductivity of disordered systems".
477:
1659:
B. Shklovskii and A. Efros, Electronic properties of doped semiconductors (Springer-Verlag, Berlin, 1984).
1585:
V. Y. Butko, J. F. Ditusa, and P. W. Adams (2000). "Coulomb Gap: How a Metal Film
Becomes an Insulator".
911:
1669:
Rogatchev, A.Yu.; Mizutani, U. (2000). "Hopping conductivity and specific heat in insulating amorphousTi
1060:
557:
1003:. If one uses the analytical expression of the single-particle DOS in the Mott derivation, a universal
957:
1006:
387:
39:
At zero temperature, a classical treatment of a system gives an upper bound for the DOS near the
1000:
881:
led to a contradiction. Repeating the above calculation under the assumption that the DOS near
837:
311:
1640:
1428:
1371:
1289:
M. Muller and S. Pankov (2007). "Mean-field theory for the three-dimensional
Coulomb glass".
1269:
1105:
1686:
1604:
1551:
1514:
1457:
1400:
1351:
1308:
1222:
1187:
884:
857:
283:
236:
58:
1491:
Pramudya, Y.; Terletska, H.; Pankov, S.; Manousakis, E.; Dobrosavljević, V. (2011-09-12).
8:
528:
1690:
1608:
1555:
1518:
1461:
1404:
1355:
1312:
1226:
1191:
413:. Assuming we have a large system, consider all the sites with energies in the interval
1628:
1594:
1504:
1324:
1298:
768:
635:
615:
416:
367:
347:
263:
129:
109:
89:
1261:
1702:
1620:
1567:
1473:
1416:
1328:
1247:
1234:
28:
1632:
1199:
1694:
1612:
1559:
1522:
1465:
1408:
1359:
1316:
1257:
1230:
1195:
1160:
1131:
48:
44:
1616:
1563:
1527:
1492:
1363:
1320:
1698:
1412:
1720:
1706:
1469:
1624:
1571:
1420:
52:
40:
1584:
1477:
1599:
1303:
1164:
260:
contains a term due to the interaction with the electron present at site
1341:
24:
1390:
344:. For every possible transfer of an electron from an occupied site
83:
1509:
1342:
J. H. Davies, P. A. Lee, and T. M. Rice (1982). "Electron Glass".
758:{\displaystyle E_{j}-E_{i}-Ce^{2}(\epsilon g(E_{f})/V)^{1/d}>0}
1490:
834:, this inequality will necessarily be violated for small enough
1122:
that fits neither the Mott nor the Efros–Shklovskii theories.
1177:
233:
The subtraction of the last term accounts for the fact that
1213:
A. L. Efros (1976). "Coulomb gap in disordered systems".
1541:
1288:
632:
is the dimension of space. Plugging the expression for
652:
into the previous equation, we obtain the inequality:
1662:
1108:
1063:
1009:
960:
914:
887:
860:
840:
791:
771:
658:
638:
618:
560:
531:
480:
419:
390:
370:
350:
314:
286:
266:
239:
155:
132:
112:
92:
61:
1114:
1094:
1048:
978:
946:
900:
873:
846:
826:
777:
757:
644:
624:
604:
545:
517:
466:
405:
376:
356:
336:
299:
272:
252:
221:
138:
118:
98:
74:
1668:
222:{\displaystyle \Delta E=E_{j}-E_{i}-e^{2}/r_{ij}}
51:. The argument is as follows: Let us look at the
1718:
1171:
1448:G. Vignale (1987). "Quantum electron glass".
1645:: CS1 maint: multiple names: authors list (
1433:: CS1 maint: multiple names: authors list (
1376:: CS1 maint: multiple names: authors list (
1274:: CS1 maint: multiple names: authors list (
993:
1212:
1447:
1598:
1526:
1508:
1302:
1250:Journal of Physics C: Solid State Physics
1215:Journal of Physics C: Solid State Physics
1150:
827:{\displaystyle E_{j}-E_{i}<2\epsilon }
785:is a coefficient of order unity. Since
474:The number of these, by assumption, is
1719:
1102:behavior is observed, with a value of
55:configuration of the system. Defining
518:{\displaystyle N=2\epsilon g(E_{f}).}
947:{\displaystyle (E-E_{f})^{\alpha }}
19:First introduced by M. Pollak, the
13:
1153:Discussions of the Faraday Society
1095:{\displaystyle e^{-1/T^{\alpha }}}
854:. Hence, assuming a finite DOS at
605:{\displaystyle R\sim (N/V)^{-1/d}}
391:
156:
14:
1748:
1653:
1578:
1535:
1493:"Nearly frozen Coulomb liquids"
979:{\displaystyle \alpha >=d-1}
1484:
1441:
1384:
1335:
1282:
1241:
1206:
1144:
1049:{\displaystyle e^{-1/T^{1/2}}}
935:
915:
732:
720:
707:
698:
582:
567:
509:
496:
458:
420:
406:{\displaystyle \Delta E>=0}
331:
318:
1:
1137:
146:is given by the expression:
7:
1617:10.1103/PhysRevLett.84.1543
1564:10.1103/PhysRevLett.75.4266
1262:10.1088/0022-3719/15/32/007
1125:
1001:Mott variable-range hopping
10:
1753:
1727:Electronic band structures
1528:10.1103/PhysRevB.84.125120
1364:10.1103/PhysRevLett.49.758
1321:10.1103/PhysRevB.75.144201
1235:10.1088/0022-3719/9/11/012
1699:10.1103/PhysRevB.61.15550
1413:10.1103/PhysRevB.45.11568
1200:10.1088/0022-3719/8/4/003
994:Experimental observations
847:{\displaystyle \epsilon }
34:
1470:10.1103/PhysRevB.36.8192
337:{\displaystyle g(E_{f})}
1587:Physical Review Letters
1544:Physical Review Letters
1344:Physical Review Letters
1115:{\displaystyle \alpha }
27:in the single-particle
1116:
1096:
1050:
980:
948:
902:
875:
848:
828:
779:
759:
646:
626:
606:
547:
519:
468:
407:
378:
364:to an unoccupied site
358:
338:
301:
274:
254:
223:
140:
126:to an unoccupied site
120:
100:
76:
1732:Statistical mechanics
1117:
1097:
1051:
981:
949:
903:
901:{\displaystyle E_{f}}
876:
874:{\displaystyle E_{f}}
849:
829:
780:
760:
647:
627:
607:
548:
520:
469:
408:
379:
359:
339:
302:
300:{\displaystyle E_{f}}
275:
255:
253:{\displaystyle E_{j}}
224:
141:
121:
101:
77:
75:{\displaystyle E_{i}}
43:, first suggested by
1180:Journal of Physics C
1165:10.1039/DF9705000013
1106:
1061:
1007:
958:
912:
885:
858:
838:
789:
769:
656:
636:
616:
558:
529:
478:
417:
388:
368:
348:
312:
284:
264:
237:
153:
130:
110:
90:
82:as the energy of an
59:
1737:Physical quantities
1691:2000PhRvB..6115550R
1685:(23): 15550–15553.
1609:2000PhRvL..84.1543B
1556:1995PhRvL..75.4266M
1519:2011PhRvB..84l5120P
1462:1987PhRvB..36.8192V
1405:1992PhRvB..4511568M
1399:(20): 11568–11579.
1356:1982PhRvL..49..758D
1313:2007PhRvB..75n4201M
1227:1976JPhC....9.2021E
1192:1975JPhC....8L..49E
908:is proportional to
546:{\displaystyle N/2}
16:Physical phenomenon
1112:
1092:
1046:
976:
944:
898:
871:
844:
824:
775:
755:
642:
622:
602:
543:
515:
464:
403:
374:
354:
334:
297:
270:
250:
219:
136:
116:
96:
72:
1679:Physical Review B
1550:(23): 4266–4269.
1497:Physical Review B
1456:(15): 8192–8195.
1450:Physical Review B
1393:Physical Review B
1291:Physical Review B
778:{\displaystyle C}
645:{\displaystyle N}
625:{\displaystyle d}
467:{\displaystyle .}
377:{\displaystyle j}
357:{\displaystyle i}
273:{\displaystyle i}
139:{\displaystyle j}
119:{\displaystyle i}
99:{\displaystyle i}
29:density of states
1744:
1711:
1710:
1666:
1660:
1657:
1651:
1650:
1644:
1636:
1602:
1600:cond-mat/0006025
1582:
1576:
1575:
1539:
1533:
1532:
1530:
1512:
1488:
1482:
1481:
1445:
1439:
1438:
1432:
1424:
1388:
1382:
1381:
1375:
1367:
1339:
1333:
1332:
1306:
1304:cond-mat/0611021
1286:
1280:
1279:
1273:
1265:
1245:
1239:
1238:
1210:
1204:
1203:
1175:
1169:
1168:
1148:
1121:
1119:
1118:
1113:
1101:
1099:
1098:
1093:
1091:
1090:
1089:
1088:
1079:
1055:
1053:
1052:
1047:
1045:
1044:
1043:
1042:
1038:
1025:
985:
983:
982:
977:
953:
951:
950:
945:
943:
942:
933:
932:
907:
905:
904:
899:
897:
896:
880:
878:
877:
872:
870:
869:
853:
851:
850:
845:
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831:
830:
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814:
813:
801:
800:
784:
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781:
776:
764:
762:
761:
756:
748:
747:
743:
727:
719:
718:
697:
696:
681:
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668:
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651:
649:
648:
643:
631:
629:
628:
623:
611:
609:
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603:
601:
600:
596:
577:
552:
550:
549:
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539:
524:
522:
521:
516:
508:
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473:
471:
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451:
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432:
431:
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404:
383:
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363:
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343:
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335:
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329:
306:
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279:
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259:
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228:
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205:
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199:
187:
186:
174:
173:
145:
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142:
137:
125:
123:
122:
117:
105:
103:
102:
97:
81:
79:
78:
73:
71:
70:
1752:
1751:
1747:
1746:
1745:
1743:
1742:
1741:
1717:
1716:
1715:
1714:
1676:
1672:
1667:
1663:
1658:
1654:
1638:
1637:
1583:
1579:
1540:
1536:
1489:
1485:
1446:
1442:
1426:
1425:
1389:
1385:
1369:
1368:
1350:(10): 758-761.
1340:
1336:
1287:
1283:
1267:
1266:
1246:
1242:
1211:
1207:
1176:
1172:
1149:
1145:
1140:
1128:
1107:
1104:
1103:
1084:
1080:
1075:
1068:
1064:
1062:
1059:
1058:
1034:
1030:
1026:
1021:
1014:
1010:
1008:
1005:
1004:
996:
959:
956:
955:
938:
934:
928:
924:
913:
910:
909:
892:
888:
886:
883:
882:
865:
861:
859:
856:
855:
839:
836:
835:
809:
805:
796:
792:
790:
787:
786:
770:
767:
766:
739:
735:
731:
723:
714:
710:
692:
688:
676:
672:
663:
659:
657:
654:
653:
637:
634:
633:
617:
614:
613:
592:
585:
581:
573:
559:
556:
555:
535:
530:
527:
526:
503:
499:
479:
476:
475:
446:
442:
427:
423:
418:
415:
414:
389:
386:
385:
369:
366:
365:
349:
346:
345:
325:
321:
313:
310:
309:
291:
287:
285:
282:
281:
265:
262:
261:
244:
240:
238:
235:
234:
210:
206:
201:
195:
191:
182:
178:
169:
165:
154:
151:
150:
131:
128:
127:
111:
108:
107:
91:
88:
87:
66:
62:
60:
57:
56:
37:
17:
12:
11:
5:
1750:
1740:
1739:
1734:
1729:
1713:
1712:
1674:
1670:
1661:
1652:
1577:
1534:
1503:(12): 125120.
1483:
1440:
1383:
1334:
1297:(14): 144201.
1281:
1240:
1205:
1170:
1142:
1141:
1139:
1136:
1135:
1134:
1127:
1124:
1111:
1087:
1083:
1078:
1074:
1071:
1067:
1041:
1037:
1033:
1029:
1024:
1020:
1017:
1013:
995:
992:
975:
972:
969:
966:
963:
941:
937:
931:
927:
923:
920:
917:
895:
891:
868:
864:
843:
823:
820:
817:
812:
808:
804:
799:
795:
774:
754:
751:
746:
742:
738:
734:
730:
726:
722:
717:
713:
709:
706:
703:
700:
695:
691:
687:
684:
679:
675:
671:
666:
662:
641:
621:
599:
595:
591:
588:
584:
580:
576:
572:
569:
566:
563:
542:
538:
534:
525:As explained,
514:
511:
506:
502:
498:
495:
492:
489:
486:
483:
463:
460:
457:
454:
449:
445:
441:
438:
435:
430:
426:
422:
402:
399:
396:
393:
373:
353:
333:
328:
324:
320:
317:
294:
290:
269:
247:
243:
231:
230:
216:
213:
209:
204:
198:
194:
190:
185:
181:
177:
172:
168:
164:
161:
158:
135:
115:
95:
69:
65:
36:
33:
15:
9:
6:
4:
3:
2:
1749:
1738:
1735:
1733:
1730:
1728:
1725:
1724:
1722:
1708:
1704:
1700:
1696:
1692:
1688:
1684:
1680:
1665:
1656:
1648:
1642:
1634:
1630:
1626:
1622:
1618:
1614:
1610:
1606:
1601:
1596:
1593:(7): 1543–6.
1592:
1588:
1581:
1573:
1569:
1565:
1561:
1557:
1553:
1549:
1545:
1538:
1529:
1524:
1520:
1516:
1511:
1506:
1502:
1498:
1494:
1487:
1479:
1475:
1471:
1467:
1463:
1459:
1455:
1451:
1444:
1436:
1430:
1422:
1418:
1414:
1410:
1406:
1402:
1398:
1394:
1387:
1379:
1373:
1365:
1361:
1357:
1353:
1349:
1345:
1338:
1330:
1326:
1322:
1318:
1314:
1310:
1305:
1300:
1296:
1292:
1285:
1277:
1271:
1263:
1259:
1256:(32): L1153.
1255:
1251:
1244:
1236:
1232:
1228:
1224:
1220:
1216:
1209:
1201:
1197:
1193:
1189:
1185:
1181:
1174:
1166:
1162:
1158:
1154:
1147:
1143:
1133:
1132:Coulomb's law
1130:
1129:
1123:
1109:
1085:
1081:
1076:
1072:
1069:
1065:
1039:
1035:
1031:
1027:
1022:
1018:
1015:
1011:
1002:
991:
987:
973:
970:
967:
964:
961:
939:
929:
925:
921:
918:
893:
889:
866:
862:
841:
821:
818:
815:
810:
806:
802:
797:
793:
772:
752:
749:
744:
740:
736:
728:
724:
715:
711:
704:
701:
693:
689:
685:
682:
677:
673:
669:
664:
660:
639:
619:
597:
593:
589:
586:
578:
574:
570:
564:
561:
540:
536:
532:
512:
504:
500:
493:
490:
487:
484:
481:
461:
455:
452:
447:
443:
439:
436:
433:
428:
424:
400:
397:
394:
371:
351:
326:
322:
315:
292:
288:
267:
245:
241:
214:
211:
207:
202:
196:
192:
188:
183:
179:
175:
170:
166:
162:
159:
149:
148:
147:
133:
113:
93:
85:
67:
63:
54:
50:
46:
42:
32:
30:
26:
22:
1682:
1678:
1664:
1655:
1641:cite journal
1590:
1586:
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1707:0163-1829
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