1281:
288:
2128:
2828:
2031:; the corresponding probability is directly proportional to the correlation. Nevertheless, both the presence of electron–hole plasma and excitons can equivalently induce the spontaneous emission. A further discussion of the relative weight and nature of plasma vs. exciton sources is presented in connection with the
2958:
3152:
resonance in THz absorption uniquely identifies the presence of excitons as detected experimentally in Ref. As a major difference to atomic spectroscopy, semiconductor resonances contain a strong excitation-induced dephasing that produces much broader resonances than in atomic spectroscopy. In fact,
1395:
trigger spontaneous recombination of electrons and holes (electronic vacancies) via spontaneous emission of photons. At quasiequilibrium, this yields a steady-state photon flux emitted by the semiconductor. By starting from the SLEs, the steady-state photoluminescence (abbreviated as PL) can be cast
1174:
effect. Physically, this can be understood as the elementary property of
Fermions; if a given electronic state is already excited it cannot be excited a second time due to the Pauli exclusion among Fermions. Therefore, the corresponding electronic states can produce only photon emission that is seen
2118:
ground state. This emission peak often remains well below the fundamental bandgap energy even at the high excitations where all states are continuum states. This demonstrates that semiconductors are often subjects to massive
Coulomb-induced renormalizations even when the system appears to have only
267:
range. Technically, the THz investigations are an extension of the ordinary SBEs and/or involve solving the dynamics of two-particle correlations explicitly. Like for the optical absorption and emission problem, one can diagonalize the homogeneous parts that emerge analytically with the help of the
251:
the homogeneous parts of the SBEs and SLEs. Under the steady-state conditions, the resulting equations can be solved analytically when one further approximates dephasing due to higher-order many-body effects. When such effects are fully included, one must resort to a numeric approach. After the
1684:
2447:
2650:
1112:
energy and a continuum of unbound states that appear for energies above the bandgap. Therefore, a typical semiconductor's low-density absorption spectrum shows a series of exciton resonances and then a continuum-absorption tail. For realistic situations,
2205:
state. In several semiconductor systems, one needs THz fields to induce such transitions. By starting from a steady-state configuration of electron–hole correlations, the diagonalization of THz-induced dynamics yields a THz absorption spectrum
2835:
463:
and the semiconductor is initially unexcited. Due to the small dephasing constant used, several excitonic resonances appear (vertical lines) well below the bandgap energy. The magnitude of high-energy resonances are multiplied by 5 for
45:
to describe linear absorption based on properties of a single electron–hole pair. The analysis can be extended to a many-body investigation with full predictive powers when all parameters are computed microscopically using, e.g., the
70:
is provided by the SBEs and SLEs, respectively. Both of them are systematically derived starting from the many-body/quantum-optical system
Hamiltonian and fully describe the resulting quantum dynamics of optical and quantum-optical
1556:
1036:
3143:
In contrast to optical and photoluminescence spectroscopy, THz absorption can directly measure the presence of exciton populations in full analogy to atomic spectroscopy. For example, the presence of a pronounced
2196:
As discussed above, it is often meaningful to tune the electromagnetic field to be resonant with the transitions between two many-body states. For example, one can follow how a bound exciton is excited from its
2215:
1543:
2823:{\displaystyle S^{\nu ,\lambda }(\omega )=\sum _{\beta }{\frac {(E_{\beta }-E_{\nu })J_{\nu \beta }J_{\beta \lambda }}{E_{\beta }-E_{\nu }-\hbar \omega -\mathrm {i} \gamma _{\lambda ,\nu }(\omega )}}}
1991:
1896:
461:
1334:
790:
602:
359:
2636:
2548:
2190:
3089:
156:
106:
solutions and they formally describe
Coulombic binding by oppositely charged electrons and holes. The actual physical meaning of excitonic states is discussed further in connection with the
2486:
1270:
882:
826:
3047:
2997:
271:
All of these derivations rely on the steady-state conditions and the analytic knowledge of the exciton states. Furthermore, the effect of further many-body contributions, such as the
3192:
1085:-like states, following the quantum-number convention of the hydrogen problem. Therefore, optical spectrum of direct-gap semiconductors produces an absorption resonance only for the
3576:
Kaindl, R. A.; Carnahan, M. A.; Hägele, D.; Lövenich, R.; Chemla, D. S. (2003). "Ultrafast terahertz probes of transient conducting and insulating phases in an electron–hole gas".
2119:
electron–hole plasma states as emission resonances. To make an accurate prediction of the exact position and shape at elevated carrier densities, one must resort to the full SLEs.
1781:
1754:
1720:
230:
2029:
1838:
75:
such as optical polarization (SBEs) and photoluminescence intensity (SLEs). All relevant many-body effects can be systematically included by using various techniques such as the
1385:
1138:
757:
706:
2594:
3138:
628:
2953:{\displaystyle J_{\nu \beta }\propto \sum _{\mathbf {k} }\phi _{\nu }^{\star }({\mathbf {k} }){\mathbf {k} }\cdot {\mathbf {k} }_{\rm {THz}}\phi _{\beta }({\mathbf {k} })}
2087:
1940:
1918:
1808:
1234:
1207:
1168:
1063:
959:
909:
655:
203:
2116:
388:
318:
243:
These exciton eigenstates provide valuable insight to SBEs and SLEs, especially, when one analyses the linear semiconductor absorption spectrum or photoluminescence at
2506:
846:
726:
679:
176:
3109:
929:
275:, can be included microscopically to the Wannier solver, which removes the need to introduce phenomenological dephasing constant, energy shifts, or screening of the
2568:
3519:; Gibbs, H.; Hoyer, W.; Kira, M.; Koch, S.; Prineas, J.; Stolz, H. (2004). "Excitonic Photoluminescence in Semiconductor Quantum Wells: Plasma versus Excitons".
2056:
1103:
1083:
408:
1284:
Photoluminescence intensity computed via the
Elliott formula. The population of s-like exciton states follow a Boltzmann distribution at 35 Kelvin, where the 1
2645:
In contrast to optical absorption and photoluminescence, THz absorption may involve all exciton states. This can be seen from the spectral response function
1182:
Even though one can understand the principal behavior of semiconductor absorption on the basis of the
Elliott formula, detailed predictions of the exact
1679:{\displaystyle S_{\lambda }=\sum _{\mathbf {k} }|\phi _{\lambda }({\mathbf {k} })|^{2}f_{\mathbf {k} }^{e}f_{\mathbf {k} }^{h}+\Delta N_{\lambda }\;}
972:
3227:
1140:
increases more rapidly than the exciton-state spacing so that one typically resolves only few lowest exciton resonances in actual experiments.
469:
63:
30:
3503:
3437:
3306:
3257:
3475:
Jahnke, F.; Kira, M.; Koch, S. W.; Tai, K. (1996). "Excitonic
Nonlinearities of Semiconductor Microcavities in the Nonperturbative Regime".
3232:
2032:
111:
87:
51:
34:
1548:
that is very similar to the
Elliott formula for the optical absorption. As a major difference, the numerator has a new contribution – the
2442:{\displaystyle \alpha _{\rm {THz}}(\omega )=\mathrm {Im} \left^{\star }}{\omega (\hbar \omega +\mathrm {i} \gamma (\omega ))}}\right]\;.}
98:
that can be solved analytically in special cases. In particular, the low-density
Wannier equation is analogous to bound solutions of the
3557:
Kira, M.; Hoyer, W.; Stroucken, T.; Koch, S. (2001). "Exciton
Formation in Semiconductors and the Influence of a Photonic Environment".
1280:
3412:
3363:
3344:
3325:
3279:
255:
The same approach can be applied to compute absorption spectrum for fields that are in the terahertz (abbreviated as THz) range of
1405:
1143:
The concentration of charge carriers influence the shape of the absorption spectrum considerably. For high enough densities, all
1108:
In general, the exciton eigen energies consist of a series of bound states that emerge energetically well below the fundamental
3602:
252:
exciton states are obtained, one can eventually express the linear absorption and steady-state photoluminescence analytically.
1945:
1850:
1391:
After the semiconductor becomes electronically excited, the carrier system relaxes into a quasiequilibrium. At the same time,
413:
1291:
762:
3612:
3237:
962:
107:
83:
47:
485:
3242:
323:
2599:
2511:
2154:
3052:
121:
2131:
Terahertz absorption spectrum in bulk GaAs computed using the THz Elliott formula. The vertical lines indicate the n
1170:
energies correspond to continuum states and some of the oscillators strengths may become negative-valued due to the
287:
3607:
3298:
76:
2455:
272:
268:
exciton eigenstates. Once the diagonalization is completed, one can then compute the THz absorption analytically.
1239:
934:
Each of the exciton resonances can produce a peak to the absorption spectrum when the photon energy matches with
851:
795:
256:
3010:
2963:
263:
range, it is mostly resonant with the many-body states, not the interband transitions that are typically in the
3538:
Timusk, T.; Navarro, H.; Lipari, N.O.; Altarelli, M. (1978). "Far-infrared absorption by excitons in silicon".
966:
3164:
2038:
Like for the absorption, a direct-gap semiconductor emits light only at the resonances corresponding to the
1920:. Such a form is expected for a probability of two uncorrelated events to occur simultaneously at a desired
3464:
3456:
Kira, M.; Koch, S.W. (2006). "Many-body correlations and excitonic effects in semiconductor spectroscopy".
2127:
1759:
1725:
1691:
729:
248:
208:
2004:
1813:
1355:
1116:
735:
684:
2642:
that is a state where electrons and holes move with respect to each other without forming bound pairs.
2573:
3114:
3565:
3527:
1549:
1176:
91:
1171:
3546:
1392:
610:
276:
42:
2065:
1923:
1901:
1786:
1212:
1185:
1146:
1041:
937:
887:
633:
320:
of bulk GaAs using two-band SBEs. The decay of polarization is approximated with a decay constant
181:
3483:
2092:
364:
294:
1105:-like state. The width of the resonance is determined by the corresponding dephasing constant.
3499:
3433:
3408:
3359:
3340:
3321:
3302:
3275:
3252:
2491:
2151:-band-gap-transition lies slightly above 4meV, whereas the dephasing constant is chosen to be
1038:
that vanishes for all states except for those that are spherically symmetric. In other words,
831:
711:
664:
161:
67:
3429:
3094:
2058:-like states. As a typical trend, a quasiequilibrium emission is strongly peaked around the 1
914:
94:(SLEs). This homogeneous part yields an eigenvalue problem that can be expressed through the
3247:
2639:
1994:
233:
95:
3516:
3004:
3000:
2553:
90:
contain an identical homogeneous part driven either by a classical field (SBEs) or by a
2041:
1088:
1068:
393:
115:
3596:
3291:
264:
244:
3584:
1993:
is the spontaneous-emission source originating from uncorrelated electron–hole
72:
1272:
requires a full many-body computation already for moderate carrier densities.
2550:
elements formally determine transition amplitudes between two exciton states
473:
26:
1031:{\displaystyle |\sum _{\mathbf {k} }\phi _{\lambda }({\mathbf {k} })|^{2}}
3272:
Quantum Theory of the Optical and Electronic Properties of Semiconductors
1841:
1175:
as negative absorption, i.e., gain that is the prerequisite to realizing
99:
3091:
that generally depends on exciton states involved and the THz frequency
1998:
1288:
population is suppressed to four percent and the dephasing constant is
1109:
658:
103:
2999:
is determined by the direction of the THz field. This leads to dipole
1898:
term defines the probability to find an electron and a hole with same
25:
describes analytically, or with few adjustable parameters such as the
38:
2126:
1279:
286:
237:
3140:
that stems from the decay constant of macroscopic THz currents.
3391:
3049:
and the resonance width is determined by a dephasing constant
792:. However, a full microscopic computation generally produces
260:
1810:
contains also a direct contribution from exciton populations
102:
problem of quantum mechanics. These are often referred to as
1538:{\displaystyle \mathrm {PL} (\omega )=\mathrm {Im} \left\,,}
965:, the oscillator strength is proportional to the product of
708:
is the dephasing constant associated with the exciton state
410:. The energy is shifted with respect to the band-gap energy
247:
conditions. One simply uses the constructed eigenstates to
3003:
among exciton states, in full analog to the atomic dipole
390:
is computed as function of the pump field's photon energy
1986:{\displaystyle f_{\mathbf {k} }^{e}f_{\mathbf {k} }^{h}}
1891:{\displaystyle f_{\mathbf {k} }^{e}f_{\mathbf {k} }^{h}}
456:{\displaystyle E_{\mathrm {gap} }=1.490\,\mathrm {meV} }
2001:
pairs is defined by a two-particle exciton correlation
1329:{\displaystyle \hbar \gamma \approx 1/,\mathrm {meV} }
3167:
3117:
3097:
3055:
3013:
2966:
2838:
2653:
2602:
2576:
2556:
2514:
2494:
2458:
2218:
2157:
2095:
2068:
2044:
2007:
1948:
1926:
1904:
1853:
1816:
1789:
1762:
1728:
1694:
1559:
1408:
1358:
1294:
1242:
1215:
1188:
1149:
1119:
1091:
1071:
1044:
975:
940:
917:
890:
854:
834:
798:
785:{\displaystyle \gamma _{\lambda }\rightarrow \gamma }
765:
738:
714:
687:
667:
636:
613:
488:
416:
396:
367:
326:
297:
211:
184:
164:
124:
2638:
build up spontaneously and they describe correlated
597:{\displaystyle \alpha (\omega )=\mathrm {Im} \left}
62:One of the most accurate theories of semiconductor
3290:
3186:
3132:
3103:
3083:
3041:
3007:. Each allowed transition produces a resonance in
2991:
2952:
2822:
2630:
2588:
2562:
2542:
2500:
2480:
2441:
2184:
2110:
2081:
2050:
2023:
1985:
1934:
1912:
1890:
1832:
1802:
1775:
1748:
1714:
1678:
1537:
1379:
1328:
1264:
1228:
1201:
1162:
1132:
1097:
1077:
1057:
1030:
953:
923:
903:
876:
840:
820:
784:
751:
720:
700:
673:
649:
622:
596:
455:
402:
382:
354:{\displaystyle \hbar \gamma =0.13\,\mathrm {meV} }
353:
312:
224:
197:
170:
150:
3356:Optics of Semiconductors and Their Nanostructures
2631:{\displaystyle \Delta N_{\nu ,\lambda \neq \nu }}
2543:{\displaystyle \Delta N_{\nu ,\lambda \neq \nu }}
2185:{\displaystyle \hbar \gamma =1.7\,\mathrm {meV} }
3084:{\displaystyle \gamma _{\lambda ,\nu }(\omega )}
151:{\displaystyle \phi _{\lambda }({\mathbf {k} })}
3274:(5th ed.). World Scientific. p. 216.
3426:Principles of Terahertz Science and Technology
1688:that contains electron and hole distributions
1336:. The vertical lines indicate the position of
848:and photon frequency. As a general tendency,
259:. Since the THz-photon energy lies within the
2960:between two exciton states. The unit vector
2452:In this notation, the diagonal contributions
2139:transition energies of which the first one (2
759:can be used as a single fit parameter, i.e.,
8:
3289:Ashcroft, Neil W.; Mermin, N. David (1976).
2481:{\displaystyle \Delta N_{\lambda ,\lambda }}
476:weak optical probe can then be expressed as
3218:resonances merge into one asymmetric tail.
1265:{\displaystyle \gamma _{\lambda }(\omega )}
877:{\displaystyle \gamma _{\lambda }(\omega )}
821:{\displaystyle \gamma _{\lambda }(\omega )}
3452:
3450:
3448:
3446:
3042:{\displaystyle S^{\nu ,\lambda }(\omega )}
2992:{\displaystyle {\mathbf {k} }_{\rm {THz}}}
2832:that contains the current-matrix elements
2435:
1675:
291:Characteristic linear absorption spectrum
178:labels the exciton state with eigenenergy
3172:
3166:
3161:resonance because the dephasing constant
3116:
3096:
3060:
3054:
3018:
3012:
2976:
2975:
2969:
2968:
2965:
2941:
2940:
2931:
2914:
2913:
2907:
2906:
2896:
2895:
2886:
2885:
2876:
2871:
2860:
2859:
2843:
2837:
2796:
2787:
2769:
2756:
2741:
2728:
2715:
2702:
2692:
2686:
2658:
2652:
2610:
2601:
2575:
2555:
2522:
2513:
2493:
2466:
2457:
2408:
2385:
2368:
2337:
2312:
2284:
2268:
2261:
2249:
2224:
2223:
2217:
2171:
2170:
2156:
2094:
2073:
2067:
2043:
2015:
2006:
1977:
1971:
1970:
1960:
1954:
1953:
1947:
1927:
1925:
1905:
1903:
1882:
1876:
1875:
1865:
1859:
1858:
1852:
1824:
1815:
1794:
1788:
1767:
1766:
1761:
1740:
1734:
1733:
1727:
1706:
1700:
1699:
1693:
1669:
1653:
1647:
1646:
1636:
1630:
1629:
1619:
1614:
1604:
1603:
1594:
1585:
1578:
1577:
1564:
1558:
1531:
1508:
1499:
1481:
1469:
1459:
1452:
1446:
1429:
1409:
1407:
1364:
1363:
1357:
1315:
1307:
1293:
1247:
1241:
1220:
1214:
1193:
1187:
1154:
1148:
1124:
1118:
1090:
1070:
1049:
1043:
1022:
1017:
1007:
1006:
997:
986:
985:
976:
974:
945:
939:
916:
895:
889:
859:
853:
833:
803:
797:
770:
764:
743:
737:
713:
692:
686:
666:
641:
635:
612:
571:
562:
544:
533:
527:
521:
504:
487:
442:
441:
422:
421:
415:
395:
366:
340:
339:
325:
296:
216:
215:
210:
189:
183:
163:
139:
138:
129:
123:
18:Formula for solid emission and absorption
1352:, etc. The bandgap energy is denoted by
3386:Kuper, C. G.; Whitfield, G. D. (1963).
3379:
3194:is broader than energetic spacing of n-
3187:{\displaystyle \gamma _{\nu ,\lambda }}
2778:
2399:
2158:
1783:is the carrier momentum. Additionally,
1763:
1490:
1295:
614:
553:
327:
212:
3494:Walls, D. F.; Milburn, G. J. (2008).
3258:Terahertz spectroscopy and technology
1387:and 'arb. u.' means arbitrary units.
7:
3515:Chatterjee, S.; Ell, C.; Mosor, S.;
3465:doi:10.1016/j.pquantelec.2006.12.002
3233:Semiconductor luminescence equations
3153:one typically can resolve only one 1
1776:{\displaystyle \hbar {\mathbf {k} }}
1749:{\displaystyle f_{\mathbf {k} }^{h}}
1715:{\displaystyle f_{\mathbf {k} }^{e}}
225:{\displaystyle \hbar {\mathbf {k} }}
52:semiconductor luminescence equations
2024:{\displaystyle \Delta N_{\lambda }}
1833:{\displaystyle \Delta N_{\lambda }}
1340:-like excitonic resonances, i.e., 1
828:that depends on both exciton index
2983:
2980:
2977:
2921:
2918:
2915:
2788:
2603:
2515:
2459:
2409:
2361:
2305:
2253:
2250:
2231:
2228:
2225:
2178:
2175:
2172:
2008:
1817:
1662:
1500:
1433:
1430:
1413:
1410:
1380:{\displaystyle E_{\mathrm {gap} }}
1371:
1368:
1365:
1322:
1319:
1316:
1133:{\displaystyle \gamma _{\lambda }}
752:{\displaystyle \gamma _{\lambda }}
701:{\displaystyle \gamma _{\lambda }}
657:is the oscillator strength of the
563:
508:
505:
449:
446:
443:
429:
426:
423:
347:
344:
341:
14:
3566:doi:10.1103/PhysRevLett.87.176401
3528:doi:10.1103/PhysRevLett.92.067402
3111:. The THz response also contains
2589:{\displaystyle \lambda \neq \nu }
1400:Photoluminescence Elliott formula
1276:Photoluminescence Elliott formula
3547:doi:10.1016/0038-1098(78)90216-8
3354:Kalt, H.; Hetterich, M. (2004).
3133:{\displaystyle \gamma (\omega )}
2970:
2942:
2908:
2897:
2887:
2861:
1997:. The possibility to have truly
1972:
1955:
1928:
1906:
1877:
1860:
1768:
1735:
1701:
1648:
1631:
1605:
1579:
1008:
987:
217:
140:
3484:doi:10.1103/PhysRevLett.77.5257
3458:Progress in Quantum Electronics
41:. It was originally derived by
3407:. Cambridge University Press.
3403:Kira, M.; Koch, S. W. (2011).
3320:. Cambridge University Press.
3316:Kira, M.; Koch, S. W. (2011).
3270:Haug, H.; Koch, S. W. (2009).
3127:
3121:
3078:
3072:
3036:
3030:
2947:
2937:
2892:
2882:
2814:
2808:
2721:
2695:
2676:
2670:
2425:
2422:
2416:
2396:
2358:
2349:
2302:
2296:
2243:
2237:
2147:transition) is dominant. The 1
1615:
1610:
1600:
1586:
1520:
1514:
1423:
1417:
1259:
1253:
1018:
1013:
1003:
977:
871:
865:
815:
809:
776:
583:
577:
498:
492:
377:
371:
307:
301:
145:
135:
1:
3430:doi:10.1007/978-0-387-09540-0
3238:Semiconductor Bloch equations
1065:is nonvanishing only for the
623:{\displaystyle \hbar \omega }
50:(abbreviated as SBEs) or the
48:semiconductor Bloch equations
3405:Semiconductor Quantum Optics
3318:Semiconductor Quantum Optics
3243:Quantum-optical spectroscopy
2488:determine the population of
2082:{\displaystyle S_{\lambda }}
1935:{\displaystyle \mathbf {k} }
1913:{\displaystyle \mathbf {k} }
1803:{\displaystyle S_{\lambda }}
1229:{\displaystyle F_{\lambda }}
1202:{\displaystyle E_{\lambda }}
1163:{\displaystyle E_{\lambda }}
1058:{\displaystyle F_{\lambda }}
954:{\displaystyle E_{\lambda }}
904:{\displaystyle E_{\lambda }}
650:{\displaystyle F_{\lambda }}
630:is the probe-photon energy,
273:excitation-induced dephasing
198:{\displaystyle E_{\lambda }}
96:generalized Wannier equation
2508:excitons. The off-diagonal
2111:{\displaystyle \lambda =1s}
2089:is usually largest for the
1840:that describes truly bound
1550:spontaneous-emission source
92:spontaneous-emission source
3629:
3540:Solid State Communications
3335:Klingshirn, C. F. (2006).
3299:Holt, Rinehart and Winston
2596:. For elevated densities,
931:dependence is often weak.
383:{\displaystyle \alpha (E)}
313:{\displaystyle \alpha (E)}
236:of charge carriers in the
77:cluster-expansion approach
2123:Terahertz Elliott formula
1393:vacuum-field fluctuations
963:direct-gap semiconductors
283:Linear optical absorption
257:electromagnetic radiation
2501:{\displaystyle \lambda }
1999:correlated electron–hole
841:{\displaystyle \lambda }
721:{\displaystyle \lambda }
674:{\displaystyle \lambda }
171:{\displaystyle \lambda }
3585:doi:10.1038/nature01676
3559:Physical Review Letters
3521:Physical Review Letters
3477:Physical Review Letters
3104:{\displaystyle \omega }
924:{\displaystyle \omega }
884:increases for elevated
54:(abbreviated as SLEs).
3603:Semiconductor analysis
3188:
3134:
3105:
3085:
3043:
2993:
2954:
2824:
2632:
2590:
2564:
2544:
2502:
2482:
2443:
2193:
2186:
2112:
2083:
2052:
2025:
1987:
1936:
1914:
1892:
1834:
1804:
1777:
1756:, respectively, where
1750:
1716:
1680:
1539:
1388:
1381:
1330:
1266:
1230:
1203:
1164:
1134:
1099:
1079:
1059:
1032:
955:
925:
905:
878:
842:
822:
786:
753:
722:
702:
675:
651:
624:
598:
480:Linear Elliott formula
465:
457:
404:
384:
355:
314:
226:
199:
172:
152:
3390:. Plenum Press. LCCN
3388:Polarons and Excitons
3189:
3135:
3106:
3086:
3044:
2994:
2955:
2825:
2633:
2591:
2565:
2545:
2503:
2483:
2444:
2187:
2130:
2113:
2084:
2053:
2026:
1988:
1937:
1915:
1893:
1835:
1805:
1778:
1751:
1717:
1681:
1540:
1382:
1331:
1283:
1267:
1231:
1204:
1165:
1135:
1100:
1080:
1060:
1033:
967:dipole-matrix element
956:
926:
906:
879:
843:
823:
787:
754:
723:
703:
676:
652:
625:
599:
458:
405:
385:
356:
315:
290:
227:
200:
173:
153:
3613:Equations of physics
3337:Semiconductor Optics
3165:
3115:
3095:
3053:
3011:
2964:
2836:
2651:
2640:electron–hole plasma
2600:
2574:
2563:{\displaystyle \nu }
2554:
2512:
2492:
2456:
2216:
2155:
2093:
2066:
2042:
2005:
1946:
1924:
1902:
1851:
1814:
1787:
1760:
1726:
1692:
1557:
1406:
1356:
1292:
1240:
1213:
1186:
1177:semiconductor lasers
1147:
1117:
1089:
1069:
1042:
973:
938:
915:
888:
852:
832:
796:
763:
736:
712:
685:
665:
634:
611:
486:
414:
394:
365:
324:
295:
209:
182:
162:
122:
29:constant, the light
3424:Lee, Y.-S. (2009).
3293:Solid State Physics
2881:
2210:THz Elliott formula
2201:ground state to a 2
1982:
1965:
1887:
1870:
1745:
1711:
1658:
1641:
277:Coulomb interaction
43:Roger James Elliott
3184:
3130:
3101:
3081:
3039:
2989:
2950:
2867:
2866:
2820:
2691:
2628:
2586:
2560:
2540:
2498:
2478:
2439:
2279:
2194:
2182:
2108:
2079:
2062:resonance because
2048:
2021:
1983:
1966:
1949:
1942:value. Therefore,
1932:
1910:
1888:
1871:
1854:
1830:
1800:
1773:
1746:
1729:
1712:
1695:
1676:
1642:
1625:
1584:
1535:
1451:
1389:
1377:
1326:
1262:
1226:
1199:
1160:
1130:
1095:
1075:
1055:
1028:
992:
951:
921:
901:
874:
838:
818:
782:
749:
718:
698:
671:
647:
620:
594:
526:
466:
453:
400:
380:
351:
310:
222:
195:
168:
148:
3608:Quantum mechanics
3583:(6941): 734–738.
3504:978-3-540-28574-8
3482:(26): 5257–5260.
3438:978-0-387-09539-4
3308:978-0-03-083993-1
3253:Photoluminescence
2855:
2818:
2682:
2429:
2264:
2051:{\displaystyle s}
1573:
1524:
1442:
1098:{\displaystyle s}
1078:{\displaystyle s}
981:
587:
517:
403:{\displaystyle E}
68:photoluminescence
3620:
3587:
3574:
3568:
3555:
3549:
3536:
3530:
3513:
3507:
3492:
3486:
3473:
3467:
3454:
3441:
3422:
3416:
3401:
3395:
3384:
3369:
3350:
3331:
3312:
3296:
3285:
3248:Wannier equation
3193:
3191:
3190:
3185:
3183:
3182:
3139:
3137:
3136:
3131:
3110:
3108:
3107:
3102:
3090:
3088:
3087:
3082:
3071:
3070:
3048:
3046:
3045:
3040:
3029:
3028:
2998:
2996:
2995:
2990:
2988:
2987:
2986:
2974:
2973:
2959:
2957:
2956:
2951:
2946:
2945:
2936:
2935:
2926:
2925:
2924:
2912:
2911:
2901:
2900:
2891:
2890:
2880:
2875:
2865:
2864:
2851:
2850:
2829:
2827:
2826:
2821:
2819:
2817:
2807:
2806:
2791:
2774:
2773:
2761:
2760:
2750:
2749:
2748:
2736:
2735:
2720:
2719:
2707:
2706:
2693:
2690:
2669:
2668:
2637:
2635:
2634:
2629:
2627:
2626:
2595:
2593:
2592:
2587:
2569:
2567:
2566:
2561:
2549:
2547:
2546:
2541:
2539:
2538:
2507:
2505:
2504:
2499:
2487:
2485:
2484:
2479:
2477:
2476:
2448:
2446:
2445:
2440:
2434:
2430:
2428:
2412:
2391:
2390:
2389:
2384:
2380:
2379:
2378:
2348:
2347:
2323:
2322:
2295:
2294:
2278:
2262:
2256:
2236:
2235:
2234:
2191:
2189:
2188:
2183:
2181:
2117:
2115:
2114:
2109:
2088:
2086:
2085:
2080:
2078:
2077:
2057:
2055:
2054:
2049:
2030:
2028:
2027:
2022:
2020:
2019:
1992:
1990:
1989:
1984:
1981:
1976:
1975:
1964:
1959:
1958:
1941:
1939:
1938:
1933:
1931:
1919:
1917:
1916:
1911:
1909:
1897:
1895:
1894:
1889:
1886:
1881:
1880:
1869:
1864:
1863:
1839:
1837:
1836:
1831:
1829:
1828:
1809:
1807:
1806:
1801:
1799:
1798:
1782:
1780:
1779:
1774:
1772:
1771:
1755:
1753:
1752:
1747:
1744:
1739:
1738:
1721:
1719:
1718:
1713:
1710:
1705:
1704:
1685:
1683:
1682:
1677:
1674:
1673:
1657:
1652:
1651:
1640:
1635:
1634:
1624:
1623:
1618:
1609:
1608:
1599:
1598:
1589:
1583:
1582:
1569:
1568:
1544:
1542:
1541:
1536:
1530:
1526:
1525:
1523:
1513:
1512:
1503:
1486:
1485:
1475:
1474:
1473:
1464:
1463:
1453:
1450:
1436:
1416:
1386:
1384:
1383:
1378:
1376:
1375:
1374:
1335:
1333:
1332:
1327:
1325:
1311:
1271:
1269:
1268:
1263:
1252:
1251:
1235:
1233:
1232:
1227:
1225:
1224:
1208:
1206:
1205:
1200:
1198:
1197:
1169:
1167:
1166:
1161:
1159:
1158:
1139:
1137:
1136:
1131:
1129:
1128:
1104:
1102:
1101:
1096:
1084:
1082:
1081:
1076:
1064:
1062:
1061:
1056:
1054:
1053:
1037:
1035:
1034:
1029:
1027:
1026:
1021:
1012:
1011:
1002:
1001:
991:
990:
980:
960:
958:
957:
952:
950:
949:
930:
928:
927:
922:
910:
908:
907:
902:
900:
899:
883:
881:
880:
875:
864:
863:
847:
845:
844:
839:
827:
825:
824:
819:
808:
807:
791:
789:
788:
783:
775:
774:
758:
756:
755:
750:
748:
747:
730:phenomenological
727:
725:
724:
719:
707:
705:
704:
699:
697:
696:
680:
678:
677:
672:
656:
654:
653:
648:
646:
645:
629:
627:
626:
621:
603:
601:
600:
595:
593:
589:
588:
586:
576:
575:
566:
549:
548:
538:
537:
528:
525:
511:
462:
460:
459:
454:
452:
434:
433:
432:
409:
407:
406:
401:
389:
387:
386:
381:
360:
358:
357:
352:
350:
319:
317:
316:
311:
234:crystal momentum
231:
229:
228:
223:
221:
220:
204:
202:
201:
196:
194:
193:
177:
175:
174:
169:
157:
155:
154:
149:
144:
143:
134:
133:
3628:
3627:
3623:
3622:
3621:
3619:
3618:
3617:
3593:
3592:
3591:
3590:
3575:
3571:
3556:
3552:
3537:
3533:
3514:
3510:
3493:
3489:
3474:
3470:
3455:
3444:
3423:
3419:
3402:
3398:
3385:
3381:
3376:
3366:
3353:
3347:
3334:
3328:
3315:
3309:
3288:
3282:
3269:
3266:
3264:Further reading
3224:
3202:states making 1
3168:
3163:
3162:
3113:
3112:
3093:
3092:
3056:
3051:
3050:
3014:
3009:
3008:
3005:selection rules
3001:selection rules
2967:
2962:
2961:
2927:
2905:
2839:
2834:
2833:
2792:
2765:
2752:
2751:
2737:
2724:
2711:
2698:
2694:
2654:
2649:
2648:
2606:
2598:
2597:
2572:
2571:
2552:
2551:
2518:
2510:
2509:
2490:
2489:
2462:
2454:
2453:
2450:
2392:
2364:
2333:
2332:
2328:
2327:
2308:
2280:
2263:
2257:
2219:
2214:
2213:
2153:
2152:
2125:
2091:
2090:
2069:
2064:
2063:
2040:
2039:
2011:
2003:
2002:
1944:
1943:
1922:
1921:
1900:
1899:
1849:
1848:
1820:
1812:
1811:
1790:
1785:
1784:
1758:
1757:
1724:
1723:
1690:
1689:
1665:
1613:
1590:
1560:
1555:
1554:
1546:
1504:
1477:
1476:
1465:
1455:
1454:
1441:
1437:
1404:
1403:
1359:
1354:
1353:
1290:
1289:
1278:
1243:
1238:
1237:
1216:
1211:
1210:
1189:
1184:
1183:
1150:
1145:
1144:
1120:
1115:
1114:
1087:
1086:
1067:
1066:
1045:
1040:
1039:
1016:
993:
971:
970:
941:
936:
935:
913:
912:
891:
886:
885:
855:
850:
849:
830:
829:
799:
794:
793:
766:
761:
760:
739:
734:
733:
710:
709:
688:
683:
682:
663:
662:
637:
632:
631:
609:
608:
605:
567:
540:
539:
529:
516:
512:
484:
483:
417:
412:
411:
392:
391:
363:
362:
322:
321:
293:
292:
285:
207:
206:
185:
180:
179:
160:
159:
125:
120:
119:
118:are denoted by
60:
23:Elliott formula
19:
12:
11:
5:
3626:
3624:
3616:
3615:
3610:
3605:
3595:
3594:
3589:
3588:
3569:
3550:
3545:(4): 217–219.
3531:
3508:
3496:Quantum Optics
3487:
3468:
3463:(5): 155–296.
3442:
3417:
3413:978-0521875097
3396:
3378:
3377:
3375:
3372:
3371:
3370:
3365:978-3540383451
3364:
3351:
3346:978-3540383451
3345:
3332:
3327:978-0521875097
3326:
3313:
3307:
3286:
3281:978-9812838841
3280:
3265:
3262:
3261:
3260:
3255:
3250:
3245:
3240:
3235:
3230:
3223:
3220:
3181:
3178:
3175:
3171:
3129:
3126:
3123:
3120:
3100:
3080:
3077:
3074:
3069:
3066:
3063:
3059:
3038:
3035:
3032:
3027:
3024:
3021:
3017:
2985:
2982:
2979:
2972:
2949:
2944:
2939:
2934:
2930:
2923:
2920:
2917:
2910:
2904:
2899:
2894:
2889:
2884:
2879:
2874:
2870:
2863:
2858:
2854:
2849:
2846:
2842:
2816:
2813:
2810:
2805:
2802:
2799:
2795:
2790:
2786:
2783:
2780:
2777:
2772:
2768:
2764:
2759:
2755:
2747:
2744:
2740:
2734:
2731:
2727:
2723:
2718:
2714:
2710:
2705:
2701:
2697:
2689:
2685:
2681:
2678:
2675:
2672:
2667:
2664:
2661:
2657:
2625:
2622:
2619:
2616:
2613:
2609:
2605:
2585:
2582:
2579:
2559:
2537:
2534:
2531:
2528:
2525:
2521:
2517:
2497:
2475:
2472:
2469:
2465:
2461:
2438:
2433:
2427:
2424:
2421:
2418:
2415:
2411:
2407:
2404:
2401:
2398:
2395:
2388:
2383:
2377:
2374:
2371:
2367:
2363:
2360:
2357:
2354:
2351:
2346:
2343:
2340:
2336:
2331:
2326:
2321:
2318:
2315:
2311:
2307:
2304:
2301:
2298:
2293:
2290:
2287:
2283:
2277:
2274:
2271:
2267:
2260:
2255:
2252:
2248:
2245:
2242:
2239:
2233:
2230:
2227:
2222:
2208:
2180:
2177:
2174:
2169:
2166:
2163:
2160:
2124:
2121:
2107:
2104:
2101:
2098:
2076:
2072:
2047:
2018:
2014:
2010:
1980:
1974:
1969:
1963:
1957:
1952:
1930:
1908:
1885:
1879:
1874:
1868:
1862:
1857:
1827:
1823:
1819:
1797:
1793:
1770:
1765:
1743:
1737:
1732:
1709:
1703:
1698:
1672:
1668:
1664:
1661:
1656:
1650:
1645:
1639:
1633:
1628:
1622:
1617:
1612:
1607:
1602:
1597:
1593:
1588:
1581:
1576:
1572:
1567:
1563:
1534:
1529:
1522:
1519:
1516:
1511:
1507:
1502:
1498:
1495:
1492:
1489:
1484:
1480:
1472:
1468:
1462:
1458:
1449:
1445:
1440:
1435:
1432:
1428:
1425:
1422:
1419:
1415:
1412:
1398:
1396:into the form
1373:
1370:
1367:
1362:
1324:
1321:
1318:
1314:
1310:
1306:
1303:
1300:
1297:
1277:
1274:
1261:
1258:
1255:
1250:
1246:
1223:
1219:
1196:
1192:
1172:Pauli-blocking
1157:
1153:
1127:
1123:
1094:
1074:
1052:
1048:
1025:
1020:
1015:
1010:
1005:
1000:
996:
989:
984:
979:
948:
944:
920:
898:
894:
873:
870:
867:
862:
858:
837:
817:
814:
811:
806:
802:
781:
778:
773:
769:
746:
742:
717:
695:
691:
670:
644:
640:
619:
616:
592:
585:
582:
579:
574:
570:
565:
561:
558:
555:
552:
547:
543:
536:
532:
524:
520:
515:
510:
507:
503:
500:
497:
494:
491:
478:
451:
448:
445:
440:
437:
431:
428:
425:
420:
399:
379:
376:
373:
370:
349:
346:
343:
338:
335:
332:
329:
309:
306:
303:
300:
284:
281:
219:
214:
192:
188:
167:
147:
142:
137:
132:
128:
116:eigenfunctions
114:. The exciton
59:
56:
17:
13:
10:
9:
6:
4:
3:
2:
3625:
3614:
3611:
3609:
3606:
3604:
3601:
3600:
3598:
3586:
3582:
3579:
3573:
3570:
3567:
3563:
3560:
3554:
3551:
3548:
3544:
3541:
3535:
3532:
3529:
3525:
3522:
3518:
3512:
3509:
3505:
3501:
3497:
3491:
3488:
3485:
3481:
3478:
3472:
3469:
3466:
3462:
3459:
3453:
3451:
3449:
3447:
3443:
3439:
3435:
3431:
3427:
3421:
3418:
3414:
3410:
3406:
3400:
3397:
3393:
3389:
3383:
3380:
3373:
3367:
3361:
3357:
3352:
3348:
3342:
3338:
3333:
3329:
3323:
3319:
3314:
3310:
3304:
3300:
3295:
3294:
3287:
3283:
3277:
3273:
3268:
3267:
3263:
3259:
3256:
3254:
3251:
3249:
3246:
3244:
3241:
3239:
3236:
3234:
3231:
3229:
3226:
3225:
3221:
3219:
3217:
3213:
3209:
3205:
3201:
3197:
3179:
3176:
3173:
3169:
3160:
3156:
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3358:. Springer.
3355:
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3498:(2nd ed.).
464:visibility.
249:diagonalize
73:observables
37:spectra of
3597:Categories
3374:References
3228:Absorption
3198:and (n+1)-
911:while the
470:absorption
64:absorption
58:Background
31:absorption
3214:-to-(n+1)
3180:λ
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27:dephasing
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3222:See also
728:. For a
100:hydrogen
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1844:pairs.
1110:bandgap
659:exciton
468:Linear
232:is the
104:exciton
3578:Nature
3564:(17).
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158:where
39:solids
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3210:and 1
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238:solid
3500:ISBN
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3409:ISBN
3360:ISBN
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2570:and
2033:SLEs
1847:The
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361:and
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