1352:
1340:
1328:
893:
225:
1312:
1300:
1288:
735:
248:, it is possible to calculate the band structure analytically by substituting the values for the potential, the lattice spacing and the size of potential well. For two and three-dimensional problems it is more difficult to calculate a band structure based on a similar model with a few parameters accurately. Nevertheless, the properties of the band structure can easily be approximated in most regions by
883:
In three-dimensional space the
Brillouin zone boundaries are planes. The dispersion relations show conics of the free-electron energy dispersion parabolas for all possible reciprocal lattice vectors. This results in a very complicated set intersecting of curves when the dispersion relations are
216:. The energy of the electrons in the "empty lattice" is the same as the energy of free electrons. The model is useful because it clearly illustrates a number of the sometimes very complex features of energy dispersion relations in solids which are fundamental to all electronic band structures.
742:
Though the lattice cells are not spherically symmetric, the dispersion relation still has spherical symmetry from the point of view of a fixed central point in a reciprocal lattice cell if the dispersion relation is extended outside the central
Brillouin zone. The
232:
The periodic potential of the lattice in this free electron model must be weak because otherwise the electrons wouldn't be free. The strength of the scattering mainly depends on the geometry and topology of the system. Topologically defined parameters, like
366:
208:(close to constant). One may also consider an empty irregular lattice, in which the potential is not even periodic. The empty lattice approximation describes a number of properties of energy dispersion relations of non-interacting
244:. For 1-, 2- and 3-dimensional spaces potential wells do always scatter waves, no matter how small their potentials are, what their signs are or how limited their sizes are. For a particle in a one-dimensional lattice, like the
878:
1181:
486:
that determine the bands in an energy interval is limited to two when the energy rises. In two and three dimensional lattices the number of reciprocal lattice vectors that determine the free electron bands
884:
calculated because there is a large number of possible angles between evaluation trajectories, first and higher order
Brillouin zone boundaries and dispersion parabola intersection cones.
1016:
255:
In theory the lattice is infinitely large, so a weak periodic scattering potential will eventually be strong enough to reflect the wave. The scattering process results in the well known
179:
1255:
783:
729:
523:
440:
1213:
1072:
552:
484:
398:
1103:
920:
273:
691:
644:
599:
79:
172:
67:
791:
445:
The figure on the right shows the dispersion relation for three periods in reciprocal space of a one-dimensional lattice with lattice cells of length
101:
525:
increases more rapidly when the length of the wave vector increases and the energy rises. This is because the number of reciprocal lattice vectors
249:
63:
747:
in a three-dimensional lattice will be the same as in the case of the absence of a lattice. For the three-dimensional case the density of states
155:
113:
109:
1398:
165:
1461:
1389:
83:
1432:
1111:
1420:
117:
1215:
between the reciprocal lattice vectors in the
Hamiltonian almost go to zero. As a result, the magnitude of the band gap
1311:
37:
1299:
1287:
935:
143:
131:
45:
958:
197:
124:
94:
71:
1278:
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234:
139:
52:
41:
245:
59:
1456:
237:
1351:
1339:
1327:
105:
952:
strongly reduces the electric field of the ions in the solid. The electrostatic potential is expressed as
1218:
750:
696:
490:
407:
1189:
1048:
528:
460:
374:
1270:
1077:
922:
far outside the first
Brillouin zone are still reflected back into the first Brillouin zone. See the
22:
209:
87:
30:
903:
224:
401:
264:
361:{\displaystyle E_{n}(\mathbf {k} )={\frac {\hbar ^{2}(\mathbf {k} +\mathbf {G} _{n})^{2}}{2m}}}
1394:
1042:
744:
602:
260:
213:
1441:
949:
151:
1429:
892:
1436:
1424:
75:
263:. This is the origin of the periodicity of the dispersion relation and the division of
256:
241:
16:
Theoretical electronic band structure model in which the potential is periodic and weak
649:
608:
557:
1450:
1417:
1026:
135:
873:{\displaystyle D_{3}\left(E\right)=2\pi {\sqrt {\frac {E-E_{0}}{c_{k}^{3}}}}\ .}
734:
1377:
Physics
Lecture Notes. P.Dirac, Feynman, R.,1968. Internet, Amazon,25.03.2014.
267:
in
Brillouin zones. The periodic energy dispersion relation is expressed as:
945:
900:"Free electrons" that move through the lattice of a solid with wave vectors
1041:
is a screening parameter that determines the range of the potential. The
457:
In a one-dimensional lattice the number of reciprocal lattice vectors
1269:
crystallize in three kinds of crystal structures: the BCC and FCC
1266:
941:
891:
733:
1176:{\displaystyle U_{\mathbf {G} }={\frac {4\pi Ze}{q^{2}+G^{2}}}}
240:, depend on the magnitude of the potential and the size of the
693:
in reciprocal space and the slope of the dispersion relation
1257:
collapses and the empty lattice approximation is obtained.
1442:
DoITPoMS Teaching and
Learning Package- "Brillouin Zones"
1418:
Brillouin Zone simple lattice diagrams by Thayer
Watkins
1037:
is the distance to the nucleus of the embedded ion and
1221:
1192:
1114:
1080:
1051:
961:
906:
794:
753:
699:
652:
611:
560:
531:
493:
463:
410:
377:
276:
738:
Figure 3: Free-electron DOS in 3-dimensional k-space
1249:
1207:
1175:
1097:
1066:
1010:
914:
872:
777:
723:
685:
638:
593:
546:
517:
478:
434:
392:
360:
646:depends on the number of states in an interval
228:Free electron bands in a one dimensional lattice
1430:Brillouin Zone 3d lattice diagrams by Technion.
1357:Free electron bands in a HCP crystal structure
1345:Free electron bands in a FCC crystal structure
1333:Free electron bands in a BCC crystal structure
259:of electrons by the periodic potential of the
1186:When the values of the off-diagonal elements
926:section for sites with examples and figures.
173:
8:
1261:The electron bands of common metal crystals
1011:{\displaystyle V(r)={\frac {Ze}{r}}e^{-qr}}
80:Multi-configurational self-consistent field
453:The energy bands and the density of states
180:
166:
18:
1242:
1235:
1234:
1225:
1220:
1198:
1197:
1191:
1164:
1151:
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1120:
1119:
1113:
1087:
1079:
1057:
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1050:
996:
977:
960:
907:
905:
855:
850:
839:
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799:
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713:
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409:
384:
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341:
331:
326:
317:
308:
301:
290:
281:
275:
888:Second, third and higher Brillouin zones
223:
102:Time-dependent density functional theory
64:Semi-empirical quantum chemistry methods
1370:
1283:
305:
123:
93:
51:
29:
21:
114:Linearized augmented-plane-wave method
110:Orbital-free density functional theory
7:
1265:Apart from a few exotic exceptions,
1390:Introduction to Solid State Physics
1250:{\displaystyle 2|U_{\mathbf {G} }|}
923:
778:{\displaystyle D_{3}\left(E\right)}
724:{\displaystyle E_{n}(\mathbf {k} )}
518:{\displaystyle E_{n}(\mathbf {k} )}
435:{\displaystyle E_{n}(\mathbf {k} )}
84:Quantum chemistry composite methods
68:Møller–Plesset perturbation theory
14:
1350:
1338:
1326:
1310:
1298:
1286:
1236:
1208:{\displaystyle U_{\mathbf {G} }}
1199:
1121:
1088:
1067:{\displaystyle U_{\mathbf {G} }}
1058:
908:
714:
676:
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584:
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547:{\displaystyle \mathbf {G} _{n}}
534:
508:
479:{\displaystyle \mathbf {G} _{n}}
466:
425:
393:{\displaystyle \mathbf {G} _{n}}
380:
327:
318:
291:
200:model in which the potential is
1098:{\displaystyle V(\mathbf {r} )}
1033:is the elementary unit charge,
118:Projector augmented wave method
1243:
1226:
1092:
1084:
971:
965:
930:The nearly free electron model
718:
710:
680:
653:
633:
612:
588:
561:
512:
504:
429:
421:
338:
314:
295:
287:
1:
156:Korringa–Kohn–Rostoker method
1074:, of the lattice potential,
915:{\displaystyle \mathbf {k} }
404:vectors to which the bands
194:empty lattice approximation
148:Empty lattice approximation
1478:
1462:Electronic band structures
936:Nearly free electron model
933:
220:Scattering and periodicity
132:Nearly free electron model
46:Modern valence bond theory
198:electronic band structure
125:Electronic band structure
95:Density functional theory
72:Configuration interaction
1271:cubic crystal structures
554:that lie in an interval
140:Muffin-tin approximation
53:Molecular orbital theory
42:Generalized valence bond
1387:C. Kittel (1953–1976).
1305:Face-centered cubic (F)
1293:Body-centered cubic (I)
144:k·p perturbation theory
1317:Hexagonal close-packed
1251:
1209:
1177:
1099:
1068:
1012:
916:
897:
874:
779:
739:
725:
687:
640:
605:in an energy interval
595:
548:
519:
480:
436:
394:
362:
229:
38:Coulson–Fischer theory
1252:
1210:
1178:
1100:
1069:
1013:
917:
895:
875:
780:
737:
726:
688:
641:
596:
549:
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481:
437:
395:
363:
227:
1393:. Wiley & Sons.
1281:crystal structure.
1219:
1190:
1112:
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1049:
959:
904:
792:
751:
697:
650:
609:
558:
529:
491:
461:
408:
375:
274:
250:perturbation methods
212:that move through a
23:Electronic structure
860:
246:Kronig–Penney model
88:Quantum Monte Carlo
60:Hartree–Fock method
31:Valence bond theory
1435:2006-12-05 at the
1423:2006-09-14 at the
1247:
1205:
1173:
1105:, is expressed as
1095:
1064:
1008:
912:
898:
896:FCC Brillouin zone
870:
846:
775:
740:
721:
683:
636:
591:
544:
515:
476:
432:
402:reciprocal lattice
390:
358:
230:
106:Thomas–Fermi model
1400:978-0-471-49024-1
1171:
1043:Fourier transform
990:
866:
862:
861:
745:density of states
603:density of states
356:
261:crystal structure
257:Bragg reflections
196:is a theoretical
190:
189:
1469:
1405:
1404:
1384:
1378:
1375:
1354:
1342:
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1314:
1302:
1290:
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1101:
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1065:
1063:
1062:
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1017:
1015:
1014:
1009:
1007:
1006:
991:
986:
978:
950:screening effect
921:
919:
918:
913:
911:
879:
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776:
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686:{\displaystyle }
684:
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639:{\displaystyle }
637:
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594:{\displaystyle }
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367:
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346:
345:
336:
335:
330:
321:
313:
312:
302:
294:
286:
285:
182:
175:
168:
152:GW approximation
19:
1477:
1476:
1472:
1471:
1470:
1468:
1467:
1466:
1447:
1446:
1437:Wayback Machine
1425:Wayback Machine
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1160:
1147:
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1110:
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1076:
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1052:
1047:
1046:
992:
979:
957:
956:
938:
932:
902:
901:
890:
835:
828:
805:
795:
790:
789:
764:
754:
749:
748:
700:
695:
694:
648:
647:
607:
606:
601:increases. The
556:
555:
532:
527:
526:
494:
489:
488:
464:
459:
458:
455:
411:
406:
405:
378:
373:
372:
348:
337:
325:
304:
303:
277:
272:
271:
222:
214:crystal lattice
186:
154:
150:
146:
142:
138:
134:
116:
112:
108:
104:
86:
82:
78:
76:Coupled cluster
74:
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66:
62:
44:
40:
17:
12:
11:
5:
1475:
1473:
1465:
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1459:
1457:Quantum models
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1412:External links
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1005:
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973:
970:
967:
964:
934:Main article:
931:
928:
924:external links
910:
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280:
242:potential well
238:cross sections
221:
218:
210:free electrons
188:
187:
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1277:close-packed
1276:
1272:
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1260:
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1032:
1028:
1027:atomic number
1024:
1003:
1000:
997:
993:
987:
983:
980:
974:
968:
962:
955:
954:
953:
951:
947:
943:
942:simple metals
937:
929:
927:
925:
894:
887:
885:
867:
856:
851:
847:
840:
836:
832:
829:
822:
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816:
812:
809:
806:
800:
796:
788:
787:
786:
771:
768:
765:
759:
755:
746:
736:
732:
705:
701:
672:
669:
661:
630:
627:
624:
621:
618:
615:
604:
580:
577:
569:
539:
499:
495:
471:
452:
450:
448:
443:
416:
412:
403:
385:
352:
349:
342:
332:
322:
309:
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137:
136:Tight binding
133:
130:
129:
126:
122:
119:
115:
111:
107:
103:
100:
99:
96:
92:
89:
85:
81:
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69:
65:
61:
58:
57:
54:
50:
47:
43:
39:
36:
35:
32:
28:
24:
20:
1388:
1382:
1373:
1264:
1185:
1038:
1034:
1030:
1022:
1020:
939:
899:
882:
741:
456:
446:
444:
370:
254:
231:
205:
201:
193:
191:
147:
1451:Categories
1365:References
235:scattering
1275:hexagonal
1137:π
998:−
946:aluminium
833:−
823:π
306:ℏ
1433:Archived
1421:Archived
1273:and the
940:In most
442:belong.
400:are the
202:periodic
1025:is the
944:, like
265:k-space
25:methods
1397:
1267:metals
1021:where
948:, the
865:
1395:ISBN
785:is;
371:The
206:weak
204:and
192:The
1279:HCP
1453::
1045:,
1029:,
731:.
449:.
252:.
1403:.
1244:|
1237:G
1232:U
1227:|
1223:2
1200:G
1195:U
1166:2
1162:G
1158:+
1153:2
1149:q
1143:e
1140:Z
1134:4
1128:=
1122:G
1117:U
1093:)
1089:r
1085:(
1082:V
1059:G
1054:U
1039:q
1035:r
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1001:q
994:e
988:r
984:e
981:Z
975:=
972:)
969:r
966:(
963:V
909:k
868:.
857:3
852:k
848:c
841:0
837:E
830:E
820:2
817:=
813:)
810:E
807:(
801:3
797:D
772:)
769:E
766:(
760:3
756:D
719:)
715:k
711:(
706:n
702:E
681:]
677:k
673:d
670:+
666:k
662:,
658:k
654:[
634:]
631:E
628:d
625:+
622:E
619:,
616:E
613:[
589:]
585:k
581:d
578:+
574:k
570:,
566:k
562:[
540:n
535:G
513:)
509:k
505:(
500:n
496:E
472:n
467:G
447:a
430:)
426:k
422:(
417:n
413:E
386:n
381:G
353:m
350:2
343:2
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333:n
328:G
323:+
319:k
315:(
310:2
299:=
296:)
292:k
288:(
283:n
279:E
181:e
174:t
167:v
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