3972:
4400:
3045:
3201:, which is just proportional to the heat capacity, so the Drude model predicts a constant that is hundred times larger than the value of the free electron model. While the latter get as coefficient that is linear in temperature and provides much more accurate absolute values in the order of a few tens of ÎŒV/K at room temperature. However this models fails to predict the sign change of the thermopower in
691:
4424:
4436:
4412:
2887:
3220:
The free electron model presents several physical quantities that have the wrong temperature dependence, or no dependence at all like the electrical conductivity. The thermal conductivity and specific heat are well predicted for alkali metals at low temperatures, but fails to predict high temperature
3264:
The conductivity of some metals can depend of the orientation of the sample with respect to the electric field. Sometimes even the electrical current is not parallel to the field. This possibility is not described because the model does not integrate the crystallinity of metals, i.e. the existence
3092:
V/K while the Drude prediction is off by about half the value, which is not a large difference. The close prediction to the Lorenz number in the Drude model was a result of the classical kinetic energy of electron being about 100 smaller than the quantum version, compensating the large value of the
2502:
are constants related to the material. The linear term comes from the electronic contribution while the cubic term comes from Debye model. At high temperature this expression is no longer correct, the electronic heat capacity can be neglected, and the total heat capacity of the metal tends to a
2179:
Nevertheless, such a large additional contribution to the heat capacity of metals was never measured, raising suspicions about the argument above. By using
Sommerfeld's expansion one can obtain corrections of the energy density at finite temperature and obtain the volumetric heat capacity of an
860:
1549:, which is null at zero temperature. For an ideal gas to have the same energy as the electron gas, the temperatures would need to be of the order of the Fermi temperature. Thermodynamically, this energy of the electron gas corresponds to a zero-temperature pressure given by
2300:
2032:) coming from the orbital motion of the electrons in the presence of a magnetic field, and a paramagnetic contribution (Pauli's paramagnetism). The latter contribution is three times larger in absolute value than the diamagnetic contribution and comes from the electron
3040:{\displaystyle L=\left\{{\begin{matrix}\displaystyle {\frac {3}{2}}\left({\frac {k_{\rm {B}}}{e}}\right)^{2}\;,&{\text{Drude}}\\\displaystyle {\frac {\pi ^{2}}{3}}\left({\frac {k_{\rm {B}}}{e}}\right)^{2}\;,&{\text{free electron model.}}\end{matrix}}\right.}
154:
Free electron approximation: The interaction between the ions and the valence electrons is mostly neglected, except in boundary conditions. The ions only keep the charge neutrality in the metal. Unlike in the Drude model, the ions are not necessarily the source of
2395:, i.e. the observation that the heat capacity of a metal is still constant at high temperatures. The free electron model can be improved in this sense by adding the contribution of the vibrations of the crystal lattice. Two famous quantum corrections include the
1849:
This expression gives the right order of magnitude for the bulk modulus for alkali metals and noble metals, which show that this pressure is as important as other effects inside the metal. For other metals the crystalline structure has to be taken into account.
1844:
542:
Many properties of the free electron model follow directly from equations related to the Fermi gas, as the independent electron approximation leads to an ensemble of non-interacting electrons. For a three-dimensional electron gas we can define the
2856:
3257:
that have a strong magnetic field dependence. The free electron model also predicts that the traverse magnetoresistance, the resistance in the direction of the current, does not depend on the strength of the field. In almost all the cases it
650:
1645:
129:
The free electron model solved many of the inconsistencies related to the Drude model and gave insight into several other properties of metals. The free electron model considers that metals are composed of a quantum electron gas where
1862:, a classical system at thermodynamic equilibrium cannot have a magnetic response. The magnetic properties of matter in terms of a microscopic theory are purely quantum mechanical. For an electron gas, the total magnetic response is
1955:
1721:
and does not come from repulsion or motion of the electrons but from the restriction that no more than two electrons (due to the two values of spin) can occupy the same energy level. This pressure defines the compressibility or
2627:, at least one order of magnitude larger than any possible classical calculation. The mean free path is then not a result of electronâion collisions but instead is related to imperfections in the material, either due to
721:
2170:
388:
2106:
at large temperatures, it did correctly predict its behavior at low temperatures. In the case of metals that are good conductors, it was expected that the electrons contributed also the heat capacity.
2700:
2186:
329:
2621:
1192:
2750:
is the mean (square) speed of the electrons or the Fermi speed in the case of the free electron model). This implies that the ratio between thermal and electric conductivity is given by the
1502:
1276:
3339:
showed that a Fermi gas under repulsive interactions, can be seen as a gas of equivalent quasiparticles that slightly modify the properties of the metal. Landau's model is now known as the
3886:
1732:
2456:
2748:
3199:
1547:
2555:
3245:
in Drude's model and in the free electron model. This value is independent of temperature and the strength of the magnetic field. The Hall coefficient is actually dependent on the
3090:
3136:
2515:
should be infinite. The Drude model considered the mean free path of electrons to be close to the distance between ions in the material, implying the earlier conclusion that the
1401:
2366:
1357:
2760:
2104:
439:
413:
168:
Relaxation-time approximation: There is some unknown scattering mechanism such that the electron probability of collision is inversely proportional to the relaxation time
2018:
1309:
961:
2480:
1715:
2639:
While Drude's model predicts a similar value for the electric conductivity as the free electron model, the models predict slightly different thermal conductivities.
2334:
991:
673:
553:
1985:
483:
186:
1555:
924:
889:
3879:
2879:
2500:
1215:
507:
459:
1435:
891:
is the energy of a given electron. This formula takes into account the spin degeneracy but does not consider a possible energy shift due to the bottom of the
3306:
More exact values for the electrical conductivity and
WiedemannâFranz law can be obtained by softening the relaxation-time approximation by appealing to the
2386:
1876:
1668:
1315:). The perturbative approach is justified as the Fermi temperature is usually of about 10 K for a metal, hence at room temperature or lower the Fermi energy
221:
The crystal lattice is not explicitly taken into account in the free electron model, but a quantum-mechanical justification was given a year later (1928) by
2511:
Notice that without the relaxation time approximation, there is no reason for the electrons to deflect their motion, as there are no interactions, thus the
3303:
Other inadequacies are present in the
WiedemannâFranz law at intermediate temperatures and the frequency-dependence of metals in the optical spectrum.
194:: Each quantum state of the system can only be occupied by a single electron. This restriction of available electron states is taken into account by
3872:
855:{\displaystyle g(E)={\frac {m_{e}}{\pi ^{2}\hbar ^{3}}}{\sqrt {2m_{e}E}}={\frac {3}{2}}{\frac {n}{E_{\rm {F}}}}{\sqrt {\frac {E}{E_{\rm {F}}}}},}
3299:
carrying positive electric charge. Conduction of holes leads to an opposite sign for the Hall and
Seebeck coefficients predicted by the model.
3810:
3780:
2403:. With the addition of the latter, the volumetric heat capacity of a metal at low temperatures can be more precisely written in the form,
236:
3213:
The free electron model presents several inadequacies that are contradicted by experimental observation. We list some inaccuracies below:
4472:
3771:
276:
3836:
158:
4416:
2115:
4467:
4404:
3855:
3287:, with narrow conduction bands also exist. This diversity is not predicted by the model and can only by explained by analysing the
3694:
2295:{\displaystyle c_{V}=\left({\frac {\partial u}{\partial T}}\right)_{n}={\frac {\pi ^{2}}{2}}{\frac {T}{T_{\rm {F}}}}nk_{\rm {B}}}
334:
4440:
2645:
683:
defines the energy of the highest energy electron at zero temperature. For metals the Fermi energy is in the order of units of
300:
999:
895:. For 2D the density of states is constant and for 1D is inversely proportional to the square root of the electron energy.
4385:
3996:
2560:
2519:
of the electrons was due to collisions with the ions. The mean free paths in the free electron model are instead given by
1717:
is the total energy, the derivative performed at temperature and chemical potential constant. This pressure is called the
3931:
3376:
3100:(thermopower), which relates the generation of a potential difference by applying a temperature gradient across a sample
1859:
3371:
3288:
1718:
892:
3325:
1839:{\displaystyle B=-V\left({\frac {\partial P}{\partial V}}\right)_{T,\mu }={\frac {5}{3}}P={\frac {2}{3}}nE_{\rm {F}}.}
1447:
1220:
195:
188:, which represents the average time between collisions. The collisions do not depend on the electronic configuration.
58:
516:
Other quantities that remain the same under the free electron model as under Drude's are the AC susceptibility, the
4482:
4176:
4085:
3329:
2109:
The classical calculation using Drude's model, based on an ideal gas, provides a volumetric heat capacity given by
2409:
4125:
4100:
2705:
2702:
for free particles, which is proportional to the heat capacity and the mean free path which depend on the model (
191:
4049:
4295:
4039:
3895:
2064:
2054:
2037:
2751:
73:
3141:
161:: The interactions between electrons are ignored. The electrostatic fields in metals are weak because of the
4462:
4141:
3276:
2628:
2176:
If this was the case, the heat capacity of a metals should be 1.5 of that obtained by the DulongâPetit law.
1867:
1510:
122:
77:
2522:
3053:
88:
68:
Given its simplicity, it is surprisingly successful in explaining many experimental phenomena, especially
3103:
1362:
4192:
4171:
4105:
112:
2392:
2068:
2036:, an intrinsic quantum degree of freedom that can take two discrete values and it is associated to the
3335:
Adding repulsive interactions between electrons does not change very much the picture presented here.
2851:{\displaystyle {\frac {\kappa }{\sigma }}={\frac {m_{\rm {e}}c_{V}\langle v^{2}\rangle }{3ne^{2}}}=LT}
2339:
2059:
One open problem in solid-state physics before the arrival of quantum mechanics was to understand the
1318:
4161:
3936:
3634:
3314:
3272:
964:
203:
81:
4115:
4095:
4029:
3340:
3097:
2074:
1988:
118:
108:
27:
4423:
422:
396:
19:
This article is about the solid-state model for metals. For the model of a free electron gas, see
4262:
4252:
4001:
3734:
3361:
3307:
1312:
1285:
937:
904:
517:
462:
225:: an unbound electron moves in a periodic potential as a free electron in vacuum, except for the
4090:
3356:
3317:
is totally excluded from this model and its inclusion can lead to other magnetic responses like
1994:
715:(number of energy states, per energy per volume) of a non-interacting electron gas is given by:
286:
Mainly, the free electron model and the Drude model predict the same DC electrical conductivity
222:
3669:
645:{\displaystyle E_{\rm {F}}={\frac {\hbar ^{2}}{2m_{e}}}\left(3\pi ^{2}n\right)^{\frac {2}{3}},}
275:, as some equations do not depend on the statistical distribution of the particles. Taking the
214:
The name of the model comes from the first two assumptions, as each electron can be treated as
4477:
4370:
4335:
4242:
4166:
3851:
3832:
3820:
3806:
3776:
3650:
3622:
3344:
1279:
712:
695:
521:
510:
95:
51:
47:
35:
16:
Simple model for the behaviour of valence electrons in a crystal structure of a metallic solid
3291:. Additionally, electrons are not the only charge carriers in a metal, electron vacancies or
2465:
1640:{\displaystyle P=-\left({\frac {\partial U}{\partial V}}\right)_{T,\mu }={\frac {2}{3}}u(0),}
4340:
4232:
4212:
4207:
4202:
4197:
4054:
4034:
3991:
3956:
3926:
3744:
3642:
2309:
1673:
1507:
which does not depend on temperature. Compare with the energy per electron of an ideal gas:
970:
658:
162:
468:
171:
4365:
3986:
3903:
3794:
3722:
1963:
909:
676:
416:
4320:
868:
3638:
4428:
4375:
4257:
4146:
3971:
3766:
3318:
3246:
2864:
2624:
2512:
2485:
2396:
2033:
1950:{\displaystyle \chi ={\frac {2}{3}}\mu _{0}\mu _{\mathrm {B} }^{2}g(E_{\mathrm {F} }),}
1200:
492:
486:
444:
254:
39:
3625:(1928-01-01). "Zur Elektronentheorie der Metalle auf Grund der Fermischen Statistik".
4456:
4300:
4282:
4267:
4247:
4151:
4120:
3951:
3799:
3790:
3381:
3366:
3324:
An immediate continuation to the free electron model can be obtained by assuming the
3296:
3292:
3280:
2060:
2021:
1863:
1414:
257:
computations that were not originally taken into account in the free electron model.
250:
226:
215:
3249:
and the difference with the model can be quite dramatic when studying elements like
4237:
4059:
3961:
3748:
3673:
2371:
2029:
2025:
1723:
1653:
931:
684:
680:
544:
135:
291:
4357:
4075:
4044:
4024:
3386:
2400:
1438:
927:
690:
525:
272:
266:
207:
139:
54:
4272:
4110:
3946:
3824:
3336:
3723:"First Principles Explanation of the Positive Seebeck Coefficient of Lithium"
3654:
4330:
4156:
3941:
3284:
3254:
3250:
2516:
537:
280:
199:
20:
3864:
702:
is proportional to the square root of the kinetic energy of the particles.
150:
In the free electron model four main assumptions are taken into account:
2368:, about 100 times smaller at room temperature and much smaller at lower
4380:
4347:
4325:
4305:
3646:
3391:
3347:, where interactions can be attractive, require a more refined theory.
3202:
699:
134:
play almost no role. The model can be very predictive when applied to
4315:
4310:
3916:
3222:
3096:
However, Drude's mode predicts the wrong order of magnitude for the
249:(one can even use negative effective mass to describe conduction by
4290:
3911:
3739:
2391:
Evidently, the electronic contribution alone does not predict the
2024:. This value results from the competition of two contributions: a
689:
218:
with a respective quadratic relation between energy and momentum.
202:). Main predictions of the free-electron model are derived by the
43:
3328:, which forms the basis of the band structure model known as the
283:
only changes the results related to the speed of the electrons.
3868:
2165:{\displaystyle c_{V}^{\text{Drude}}={\frac {3}{2}}nk_{\rm {B}}}
3921:
131:
2631:
and impurities in the metal, or due to thermal fluctuations.
3050:
The free electron model is closer to the measured value of
3034:
383:{\displaystyle \quad \sigma ={\frac {ne^{2}\tau }{m_{e}}},}
3670:"Fermi Energies, Fermi Temperatures, and Fermi Velocities"
2695:{\displaystyle \kappa =c_{V}\tau \langle v^{2}\rangle /3}
2623:
is the Fermi speed) and are in the order of hundreds of
2902:
2563:
2525:
2374:
2342:
1997:
1966:
1676:
1656:
1513:
1417:
1223:
871:
324:{\displaystyle \mathbf {J} =\sigma \mathbf {E} \quad }
3144:
3106:
3056:
2966:
2905:
2890:
2867:
2763:
2708:
2648:
2616:{\textstyle v_{\rm {F}}={\sqrt {2E_{\rm {F}}/m_{e}}}}
2488:
2468:
2412:
2312:
2189:
2118:
2077:
1879:
1735:
1558:
1450:
1365:
1321:
1288:
1203:
1187:{\displaystyle E_{\rm {F}}(T)=E_{\rm {F}}(T=0)\left,}
1002:
973:
940:
912:
724:
661:
556:
495:
471:
447:
425:
399:
337:
303:
206:
of the FermiâDirac occupancy for energies around the
174:
4356:
4281:
4225:
4185:
4134:
4068:
4017:
4010:
3979:
3902:
3279:), some can conduct when impurities are added like
3798:
3193:
3130:
3084:
3039:
2873:
2850:
2742:
2694:
2615:
2549:
2494:
2474:
2450:
2380:
2360:
2328:
2294:
2164:
2098:
2012:
1979:
1949:
1838:
1709:
1662:
1639:
1541:
1496:
1429:
1395:
1351:
1303:
1270:
1209:
1186:
985:
955:
918:
883:
854:
667:
644:
501:
477:
453:
433:
407:
382:
323:
279:of an ideal gas or the velocity distribution of a
271:Many physical properties follow directly from the
180:
1437:) can also be calculated by integrating over the
1407:Compressibility of metals and degeneracy pressure
1497:{\displaystyle u(0)={\frac {3}{5}}nE_{\rm {F}},}
1271:{\textstyle T_{\rm {F}}=E_{\rm {F}}/k_{\rm {B}}}
46:solid. It was developed in 1927, principally by
3721:Xu, Bin; Verstraete, Matthieu J. (2014-05-14).
3583:
3562:
3547:
3535:
3518:
3506:
3494:
3477:
3465:
3453:
3441:
3426:
3414:
3275:, some do not conduct electricity very well (
2336:is considerably smaller than the 3/2 found in
3880:
926:of electrons in a solid is also known as the
687:above the free electron band minimum energy.
8:
3850:(2nd ed.). Cambridge university press.
3775:. University of Michigan: Wiley & Sons.
2815:
2802:
2723:
2709:
2681:
2668:
2451:{\displaystyle c_{V}\approx \gamma T+AT^{3}}
2063:of metals. While most solids had a constant
2743:{\displaystyle \langle v^{2}\rangle ^{1/2}}
4014:
3970:
3887:
3873:
3865:
3231:The Hall coefficient has a constant value
3018:
2950:
967:can be used to calculate the Fermi level (
524:, and the Hall coefficient related to the
3738:
3185:
3174:
3173:
3168:
3160:
3159:
3154:
3143:
3105:
3073:
3055:
3025:
3012:
2996:
2995:
2989:
2973:
2967:
2957:
2944:
2928:
2927:
2921:
2906:
2901:
2889:
2866:
2830:
2809:
2796:
2785:
2784:
2777:
2764:
2762:
2730:
2726:
2716:
2707:
2684:
2675:
2659:
2647:
2605:
2596:
2589:
2588:
2579:
2569:
2568:
2562:
2537:
2536:
2524:
2487:
2467:
2442:
2417:
2411:
2373:
2352:
2347:
2341:
2320:
2311:
2285:
2284:
2268:
2267:
2258:
2247:
2241:
2232:
2208:
2194:
2188:
2155:
2154:
2137:
2128:
2123:
2117:
2089:
2088:
2076:
2003:
2002:
1996:
1971:
1965:
1934:
1933:
1917:
1911:
1910:
1900:
1886:
1878:
1826:
1825:
1808:
1792:
1777:
1753:
1734:
1675:
1655:
1612:
1597:
1573:
1557:
1529:
1528:
1514:
1512:
1484:
1483:
1466:
1449:
1416:
1371:
1370:
1364:
1327:
1326:
1320:
1294:
1293:
1287:
1261:
1260:
1251:
1244:
1243:
1229:
1228:
1222:
1202:
1164:
1151:
1150:
1141:
1125:
1119:
1110:
1097:
1096:
1087:
1071:
1065:
1032:
1031:
1008:
1007:
1001:
972:
946:
945:
939:
911:
870:
839:
838:
828:
819:
818:
809:
799:
785:
776:
767:
757:
746:
740:
723:
660:
628:
614:
592:
578:
572:
562:
561:
555:
494:
470:
446:
426:
424:
400:
398:
369:
355:
345:
336:
315:
304:
302:
173:
3805:. New York: Holt, Rinehart and Winston.
2503:constant given by the Dulongâpetit law.
253:). Effective masses can be derived from
3614:
3592:
3407:
3205:and noble metals like gold and silver.
3194:{\displaystyle S=-{c_{\rm {V}}}/{|ne|}}
3138:. This coefficient can be showed to be
1542:{\textstyle {\frac {3}{2}}k_{\rm {B}}T}
764:
662:
575:
3831:. Berlin Heidelberg: Springer Verlag.
3599:
3550:, p. 23 and 52(Eq. 1.53 and 2.93)
2550:{\textstyle \lambda =v_{\rm {F}}\tau }
3579:
3577:
3575:
3573:
3571:
3558:
3556:
3531:
3529:
3527:
3490:
3488:
3486:
3221:behaviour coming from ion motion and
2642:The thermal conductivity is given by
1411:The total energy per unit volume (at
7:
4411:
3437:
3435:
3085:{\displaystyle L=2.44\times 10^{-8}}
2635:Thermal conductivity and thermopower
242:which may deviate considerably from
4435:
3772:Introduction to Solid State Physics
3131:{\displaystyle \nabla V=-S\nabla T}
1396:{\displaystyle E_{\rm {F}}(T>0)}
101:the range of binding energy values;
3848:Principles of the theory of solids
3161:
3122:
3107:
2997:
2929:
2786:
2590:
2570:
2538:
2286:
2269:
2219:
2211:
2156:
2090:
2004:
1935:
1912:
1827:
1764:
1756:
1584:
1576:
1530:
1485:
1372:
1328:
1295:
1262:
1245:
1230:
1152:
1098:
1033:
1009:
947:
840:
820:
563:
159:Independent electron approximation
87:the temperature dependence of the
61:and hence it is also known as the
14:
3228:Hall effect and magnetoresistance
2361:{\textstyle c_{V}^{\text{Drude}}}
1217:is the temperature and we define
4434:
4422:
4410:
4399:
4398:
1352:{\displaystyle E_{\rm {F}}(T=0)}
441:is the external electric field,
427:
401:
316:
305:
2881:is the Lorenz number, given by
338:
320:
277:classical velocity distribution
3749:10.1103/PhysRevLett.112.196603
3702:University of Nebraska-Lincoln
3265:of a periodic lattice of ions.
3186:
3175:
1941:
1926:
1701:
1695:
1686:
1680:
1631:
1625:
1460:
1454:
1390:
1378:
1346:
1334:
1051:
1039:
1021:
1015:
734:
728:
465:(number of electrons/volume),
1:
3997:Spontaneous symmetry breaking
3829:Elektronentheorie der Metalle
3343:. More exotic phenomena like
3308:Boltzmann transport equations
3268:Diversity in the conductivity
2099:{\displaystyle 3nk_{\rm {B}}}
993:) at higher temperatures as:
532:Properties of an electron gas
3372:Two-dimensional electron gas
3289:valence and conduction bands
2044:Corrections to Drude's model
1719:electron degeneracy pressure
1403:are practically equivalent.
434:{\displaystyle \mathbf {E} }
408:{\displaystyle \mathbf {J} }
94:the shape of the electronic
3326:empty lattice approximation
3209:Inaccuracies and extensions
2399:model and the more refined
2013:{\textstyle \mu _{\rm {B}}}
1359:and the chemical potential
1304:{\displaystyle k_{\rm {B}}}
956:{\displaystyle E_{\rm {F}}}
38:model for the behaviour of
4499:
4473:Electronic band structures
4177:Spin gapless semiconductor
4086:Nearly free electron model
3584:Ashcroft & Mermin 1976
3563:Ashcroft & Mermin 1976
3548:Ashcroft & Mermin 1976
3536:Ashcroft & Mermin 1976
3519:Ashcroft & Mermin 1976
3507:Ashcroft & Mermin 1976
3495:Ashcroft & Mermin 1976
3478:Ashcroft & Mermin 1976
3466:Ashcroft & Mermin 1976
3454:Ashcroft & Mermin 1976
3442:Ashcroft & Mermin 1976
3427:Ashcroft & Mermin 1976
3415:Ashcroft & Mermin 1976
3330:nearly free electron model
2052:
535:
264:
104:electrical conductivities;
18:
4394:
4126:Density functional theory
4101:electronic band structure
3968:
3093:classical heat capacity.
1441:of the system, we obtain
694:In three dimensions, the
192:Pauli exclusion principle
119:thermal electron emission
4468:Condensed matter physics
4296:Bogoliubov quasiparticle
4040:Quantum spin Hall effect
3932:BoseâEinstein condensate
3896:Condensed matter physics
3693:Tsymbal, Evgeny (2008).
3509:, pp. 47 (Eq. 2.81)
3377:BoseâEinstein statistics
2180:electron gas, given by:
2065:volumetric heat capacity
2055:Electronic specific heat
2038:electron magnetic moment
1860:BohrâVan Leeuwen theorem
511:electron electric charge
57:with quantum mechanical
3727:Physical Review Letters
2475:{\displaystyle \gamma }
2306:where the prefactor to
2028:contribution (known as
1868:magnetic susceptibility
1710:{\textstyle U(T)=u(T)V}
123:field electron emission
78:electrical conductivity
3695:"Electronic Transport"
3627:Zeitschrift fĂŒr Physik
3271:Not all materials are
3217:Temperature dependence
3195:
3132:
3086:
3041:
2875:
2852:
2744:
2696:
2617:
2551:
2496:
2476:
2452:
2382:
2362:
2330:
2329:{\displaystyle nk_{B}}
2296:
2166:
2100:
2014:
1981:
1951:
1840:
1711:
1664:
1641:
1543:
1498:
1431:
1397:
1353:
1305:
1272:
1211:
1188:
987:
986:{\displaystyle T>0}
957:
930:and, like the related
920:
885:
856:
703:
669:
668:{\displaystyle \hbar }
646:
503:
479:
455:
435:
409:
384:
325:
196:FermiâDirac statistics
182:
89:electron heat capacity
63:DrudeâSommerfeld model
59:FermiâDirac statistics
4172:Topological insulator
4106:Anderson localization
3273:electrical conductors
3196:
3133:
3087:
3042:
2876:
2853:
2745:
2697:
2618:
2552:
2497:
2477:
2453:
2383:
2363:
2331:
2297:
2167:
2101:
2053:Further information:
2043:
2030:Landau's diamagnetism
2015:
1982:
1980:{\textstyle \mu _{0}}
1952:
1841:
1712:
1665:
1642:
1544:
1499:
1432:
1398:
1354:
1306:
1273:
1212:
1189:
988:
958:
921:
886:
857:
693:
670:
647:
504:
480:
478:{\displaystyle \tau }
456:
436:
410:
385:
326:
183:
181:{\displaystyle \tau }
146:Ideas and assumptions
113:thermoelectric effect
4050:AharonovâBohm effect
3937:Fermionic condensate
3846:Ziman, J.M. (1972).
3315:exchange interaction
3142:
3104:
3054:
3027:free electron model.
2888:
2865:
2761:
2706:
2646:
2561:
2523:
2486:
2466:
2410:
2372:
2340:
2310:
2187:
2116:
2075:
1995:
1964:
1877:
1733:
1674:
1654:
1556:
1511:
1448:
1415:
1363:
1319:
1286:
1221:
1201:
1000:
971:
965:Sommerfeld expansion
938:
919:{\displaystyle \mu }
910:
884:{\textstyle E\geq 0}
869:
722:
659:
554:
493:
469:
445:
423:
397:
335:
301:
261:From the Drude model
204:Sommerfeld expansion
172:
82:thermal conductivity
4441:Physics WikiProject
4116:tight binding model
4096:Fermi liquid theory
4081:Free electron model
4030:Quantum Hall effect
4011:Electrons in solids
3801:Solid State Physics
3639:1928ZPhy...47....1S
3417:, Ch. 2 & Ch. 3
3341:Fermi liquid theory
3098:Seebeck coefficient
2752:WiedemannâFranz law
2357:
2133:
1989:vacuum permittivity
1922:
109:Seebeck coefficient
74:WiedemannâFranz law
50:, who combined the
32:free electron model
28:solid-state physics
4002:Critical phenomena
3821:Sommerfeld, Arnold
3647:10.1007/bf01391052
3623:Sommerfeld, Arnold
3362:Electronic entropy
3191:
3128:
3082:
3037:
3032:
3022:
2954:
2871:
2848:
2740:
2692:
2613:
2547:
2492:
2472:
2448:
2378:
2358:
2343:
2326:
2292:
2162:
2119:
2096:
2010:
1977:
1947:
1906:
1836:
1707:
1670:is the volume and
1660:
1637:
1539:
1494:
1427:
1393:
1349:
1313:Boltzmann constant
1301:
1268:
1207:
1184:
983:
953:
916:
905:chemical potential
881:
852:
704:
665:
642:
499:
475:
463:electronic density
451:
431:
405:
380:
321:
178:
36:quantum mechanical
4483:Arnold Sommerfeld
4450:
4449:
4336:Exciton-polariton
4221:
4220:
4193:Thermoelectricity
3812:978-0-03-083993-1
3782:978-0-471-49024-1
3345:superconductivity
3028:
3006:
2982:
2960:
2938:
2914:
2874:{\displaystyle L}
2837:
2772:
2611:
2495:{\displaystyle A}
2355:
2275:
2256:
2226:
2145:
2131:
1894:
1858:According to the
1854:Magnetic response
1816:
1800:
1771:
1620:
1591:
1522:
1474:
1280:Fermi temperature
1210:{\displaystyle T}
1158:
1134:
1104:
1080:
847:
846:
826:
807:
794:
774:
713:density of states
707:Density of states
696:density of states
636:
599:
522:magnetoresistance
502:{\displaystyle e}
454:{\displaystyle n}
375:
125:from bulk metals.
96:density of states
48:Arnold Sommerfeld
4490:
4438:
4437:
4426:
4414:
4413:
4402:
4401:
4341:Phonon polariton
4233:Amorphous magnet
4213:Electrostriction
4208:Flexoelectricity
4203:Ferroelectricity
4198:Piezoelectricity
4055:Josephson effect
4035:Spin Hall effect
4015:
3992:Phase transition
3974:
3957:Luttinger liquid
3904:States of matter
3889:
3882:
3875:
3866:
3861:
3842:
3816:
3804:
3795:Mermin, N. David
3786:
3753:
3752:
3742:
3718:
3712:
3711:
3709:
3708:
3699:
3690:
3684:
3683:
3681:
3680:
3665:
3659:
3658:
3619:
3603:
3597:
3587:
3586:, pp. 58â59
3581:
3566:
3560:
3551:
3545:
3539:
3533:
3522:
3516:
3510:
3504:
3498:
3497:, pp. 38â39
3492:
3481:
3480:, pp. 45â48
3475:
3469:
3468:, pp. 32â37
3463:
3457:
3451:
3445:
3444:, pp. 49â51
3439:
3430:
3424:
3418:
3412:
3244:
3200:
3198:
3197:
3192:
3190:
3189:
3178:
3172:
3167:
3166:
3165:
3164:
3137:
3135:
3134:
3129:
3091:
3089:
3088:
3083:
3081:
3080:
3046:
3044:
3043:
3038:
3036:
3033:
3029:
3026:
3017:
3016:
3011:
3007:
3002:
3001:
3000:
2990:
2983:
2978:
2977:
2968:
2961:
2958:
2949:
2948:
2943:
2939:
2934:
2933:
2932:
2922:
2915:
2907:
2880:
2878:
2877:
2872:
2857:
2855:
2854:
2849:
2838:
2836:
2835:
2834:
2818:
2814:
2813:
2801:
2800:
2791:
2790:
2789:
2778:
2773:
2765:
2749:
2747:
2746:
2741:
2739:
2738:
2734:
2721:
2720:
2701:
2699:
2698:
2693:
2688:
2680:
2679:
2664:
2663:
2622:
2620:
2619:
2614:
2612:
2610:
2609:
2600:
2595:
2594:
2593:
2580:
2575:
2574:
2573:
2556:
2554:
2553:
2548:
2543:
2542:
2541:
2517:diffusive motion
2501:
2499:
2498:
2493:
2481:
2479:
2478:
2473:
2457:
2455:
2454:
2449:
2447:
2446:
2422:
2421:
2393:DulongâPetit law
2387:
2385:
2384:
2379:
2367:
2365:
2364:
2359:
2356:
2353:
2351:
2335:
2333:
2332:
2327:
2325:
2324:
2301:
2299:
2298:
2293:
2291:
2290:
2289:
2276:
2274:
2273:
2272:
2259:
2257:
2252:
2251:
2242:
2237:
2236:
2231:
2227:
2225:
2217:
2209:
2199:
2198:
2171:
2169:
2168:
2163:
2161:
2160:
2159:
2146:
2138:
2132:
2129:
2127:
2105:
2103:
2102:
2097:
2095:
2094:
2093:
2069:DulongâPetit law
2019:
2017:
2016:
2011:
2009:
2008:
2007:
1986:
1984:
1983:
1978:
1976:
1975:
1956:
1954:
1953:
1948:
1940:
1939:
1938:
1921:
1916:
1915:
1905:
1904:
1895:
1887:
1845:
1843:
1842:
1837:
1832:
1831:
1830:
1817:
1809:
1801:
1793:
1788:
1787:
1776:
1772:
1770:
1762:
1754:
1716:
1714:
1713:
1708:
1669:
1667:
1666:
1661:
1646:
1644:
1643:
1638:
1621:
1613:
1608:
1607:
1596:
1592:
1590:
1582:
1574:
1548:
1546:
1545:
1540:
1535:
1534:
1533:
1523:
1515:
1503:
1501:
1500:
1495:
1490:
1489:
1488:
1475:
1467:
1436:
1434:
1433:
1430:{\textstyle T=0}
1428:
1402:
1400:
1399:
1394:
1377:
1376:
1375:
1358:
1356:
1355:
1350:
1333:
1332:
1331:
1310:
1308:
1307:
1302:
1300:
1299:
1298:
1277:
1275:
1274:
1269:
1267:
1266:
1265:
1255:
1250:
1249:
1248:
1235:
1234:
1233:
1216:
1214:
1213:
1208:
1193:
1191:
1190:
1185:
1180:
1176:
1169:
1168:
1163:
1159:
1157:
1156:
1155:
1142:
1135:
1130:
1129:
1120:
1115:
1114:
1109:
1105:
1103:
1102:
1101:
1088:
1081:
1076:
1075:
1066:
1038:
1037:
1036:
1014:
1013:
1012:
992:
990:
989:
984:
962:
960:
959:
954:
952:
951:
950:
934:, often denoted
925:
923:
922:
917:
890:
888:
887:
882:
861:
859:
858:
853:
848:
845:
844:
843:
830:
829:
827:
825:
824:
823:
810:
808:
800:
795:
790:
789:
777:
775:
773:
772:
771:
762:
761:
751:
750:
741:
674:
672:
671:
666:
651:
649:
648:
643:
638:
637:
629:
627:
623:
619:
618:
600:
598:
597:
596:
583:
582:
573:
568:
567:
566:
518:plasma frequency
508:
506:
505:
500:
484:
482:
481:
476:
460:
458:
457:
452:
440:
438:
437:
432:
430:
414:
412:
411:
406:
404:
389:
387:
386:
381:
376:
374:
373:
364:
360:
359:
346:
330:
328:
327:
322:
319:
308:
187:
185:
184:
179:
163:screening effect
4498:
4497:
4493:
4492:
4491:
4489:
4488:
4487:
4453:
4452:
4451:
4446:
4390:
4371:Granular matter
4366:Amorphous solid
4352:
4277:
4263:Antiferromagnet
4253:Superparamagnet
4226:Magnetic phases
4217:
4181:
4130:
4091:Bloch's theorem
4064:
4006:
3987:Order parameter
3980:Phase phenomena
3975:
3966:
3898:
3893:
3858:
3845:
3839:
3819:
3813:
3789:
3783:
3767:Kittel, Charles
3765:
3757:
3756:
3720:
3719:
3715:
3706:
3704:
3697:
3692:
3691:
3687:
3678:
3676:
3667:
3666:
3662:
3621:
3620:
3616:
3606:
3598:
3594:
3590:
3582:
3569:
3561:
3554:
3546:
3542:
3534:
3525:
3517:
3513:
3505:
3501:
3493:
3484:
3476:
3472:
3464:
3460:
3452:
3448:
3440:
3433:
3425:
3421:
3413:
3409:
3400:
3357:Bloch's theorem
3353:
3295:can be seen as
3238:
3232:
3211:
3155:
3140:
3139:
3102:
3101:
3069:
3052:
3051:
3031:
3030:
3023:
2991:
2985:
2984:
2969:
2963:
2962:
2955:
2923:
2917:
2916:
2897:
2886:
2885:
2863:
2862:
2826:
2819:
2805:
2792:
2780:
2779:
2759:
2758:
2722:
2712:
2704:
2703:
2671:
2655:
2644:
2643:
2637:
2601:
2584:
2564:
2559:
2558:
2532:
2521:
2520:
2509:
2484:
2483:
2464:
2463:
2438:
2413:
2408:
2407:
2370:
2369:
2338:
2337:
2316:
2308:
2307:
2280:
2263:
2243:
2218:
2210:
2204:
2203:
2190:
2185:
2184:
2150:
2114:
2113:
2084:
2073:
2072:
2057:
2051:
2046:
1998:
1993:
1992:
1967:
1962:
1961:
1929:
1896:
1875:
1874:
1856:
1821:
1763:
1755:
1749:
1748:
1731:
1730:
1672:
1671:
1652:
1651:
1583:
1575:
1569:
1568:
1554:
1553:
1524:
1509:
1508:
1479:
1446:
1445:
1413:
1412:
1409:
1366:
1361:
1360:
1322:
1317:
1316:
1289:
1284:
1283:
1256:
1239:
1224:
1219:
1218:
1199:
1198:
1146:
1137:
1136:
1121:
1092:
1083:
1082:
1067:
1058:
1054:
1027:
1003:
998:
997:
969:
968:
941:
936:
935:
908:
907:
901:
893:conduction band
867:
866:
834:
814:
781:
763:
753:
752:
742:
720:
719:
709:
677:Planck constant
675:is the reduced
657:
656:
610:
606:
602:
601:
588:
584:
574:
557:
552:
551:
540:
534:
491:
490:
467:
466:
443:
442:
421:
420:
417:current density
395:
394:
365:
351:
347:
333:
332:
299:
298:
269:
263:
247:
233:
223:Bloch's theorem
170:
169:
148:
40:charge carriers
24:
17:
12:
11:
5:
4496:
4494:
4486:
4485:
4480:
4475:
4470:
4465:
4463:Quantum models
4455:
4454:
4448:
4447:
4445:
4444:
4432:
4429:Physics Portal
4420:
4408:
4395:
4392:
4391:
4389:
4388:
4383:
4378:
4376:Liquid crystal
4373:
4368:
4362:
4360:
4354:
4353:
4351:
4350:
4345:
4344:
4343:
4338:
4328:
4323:
4318:
4313:
4308:
4303:
4298:
4293:
4287:
4285:
4283:Quasiparticles
4279:
4278:
4276:
4275:
4270:
4265:
4260:
4255:
4250:
4245:
4243:Superdiamagnet
4240:
4235:
4229:
4227:
4223:
4222:
4219:
4218:
4216:
4215:
4210:
4205:
4200:
4195:
4189:
4187:
4183:
4182:
4180:
4179:
4174:
4169:
4167:Superconductor
4164:
4159:
4154:
4149:
4147:Mott insulator
4144:
4138:
4136:
4132:
4131:
4129:
4128:
4123:
4118:
4113:
4108:
4103:
4098:
4093:
4088:
4083:
4078:
4072:
4070:
4066:
4065:
4063:
4062:
4057:
4052:
4047:
4042:
4037:
4032:
4027:
4021:
4019:
4012:
4008:
4007:
4005:
4004:
3999:
3994:
3989:
3983:
3981:
3977:
3976:
3969:
3967:
3965:
3964:
3959:
3954:
3949:
3944:
3939:
3934:
3929:
3924:
3919:
3914:
3908:
3906:
3900:
3899:
3894:
3892:
3891:
3884:
3877:
3869:
3863:
3862:
3856:
3843:
3838:978-3642950025
3837:
3817:
3811:
3791:Ashcroft, Neil
3787:
3781:
3762:
3761:
3755:
3754:
3733:(19): 196603.
3713:
3685:
3660:
3613:
3612:
3611:
3610:
3605:
3604:
3591:
3589:
3588:
3567:
3552:
3540:
3523:
3511:
3499:
3482:
3470:
3458:
3446:
3431:
3419:
3406:
3405:
3404:
3399:
3396:
3395:
3394:
3389:
3384:
3379:
3374:
3369:
3364:
3359:
3352:
3349:
3319:ferromagnetism
3301:
3300:
3297:quasiparticles
3281:semiconductors
3269:
3266:
3262:
3259:
3247:band structure
3236:
3229:
3226:
3218:
3210:
3207:
3188:
3184:
3181:
3177:
3171:
3163:
3158:
3153:
3150:
3147:
3127:
3124:
3121:
3118:
3115:
3112:
3109:
3079:
3076:
3072:
3068:
3065:
3062:
3059:
3048:
3047:
3035:
3024:
3021:
3015:
3010:
3005:
2999:
2994:
2988:
2981:
2976:
2972:
2965:
2964:
2956:
2953:
2947:
2942:
2937:
2931:
2926:
2920:
2913:
2910:
2904:
2903:
2900:
2896:
2893:
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2583:
2578:
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2567:
2546:
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2535:
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2513:mean free path
2508:
2507:Mean free path
2505:
2491:
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2460:
2459:
2445:
2441:
2437:
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2431:
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2397:Einstein solid
2381:{\textstyle T}
2377:
2350:
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2323:
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2006:
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1663:{\textstyle V}
1659:
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581:
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536:Main article:
533:
530:
498:
487:mean free time
474:
450:
429:
403:
391:
390:
379:
372:
368:
363:
358:
354:
350:
344:
341:
318:
314:
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265:Main article:
262:
259:
255:band structure
251:electron holes
245:
237:effective mass
231:
212:
211:
189:
177:
166:
156:
147:
144:
127:
126:
116:
105:
102:
99:
92:
85:
76:which relates
15:
13:
10:
9:
6:
4:
3:
2:
4495:
4484:
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4479:
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4184:
4178:
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4173:
4170:
4168:
4165:
4163:
4160:
4158:
4155:
4153:
4152:Semiconductor
4150:
4148:
4145:
4143:
4140:
4139:
4137:
4133:
4127:
4124:
4122:
4121:Hubbard model
4119:
4117:
4114:
4112:
4109:
4107:
4104:
4102:
4099:
4097:
4094:
4092:
4089:
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4053:
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4041:
4038:
4036:
4033:
4031:
4028:
4026:
4023:
4022:
4020:
4016:
4013:
4009:
4003:
4000:
3998:
3995:
3993:
3990:
3988:
3985:
3984:
3982:
3978:
3973:
3963:
3960:
3958:
3955:
3953:
3950:
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3930:
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3910:
3909:
3907:
3905:
3901:
3897:
3890:
3885:
3883:
3878:
3876:
3871:
3870:
3867:
3859:
3857:0-521-29733-8
3853:
3849:
3844:
3840:
3834:
3830:
3826:
3822:
3818:
3814:
3808:
3803:
3802:
3796:
3792:
3788:
3784:
3778:
3774:
3773:
3768:
3764:
3763:
3759:
3758:
3750:
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3741:
3736:
3732:
3728:
3724:
3717:
3714:
3703:
3696:
3689:
3686:
3675:
3671:
3664:
3661:
3656:
3652:
3648:
3644:
3640:
3636:
3633:(1â2): 1â32.
3632:
3629:(in German).
3628:
3624:
3618:
3615:
3608:
3607:
3601:
3596:
3593:
3585:
3580:
3578:
3576:
3574:
3572:
3568:
3564:
3559:
3557:
3553:
3549:
3544:
3541:
3538:, pp. 52
3537:
3532:
3530:
3528:
3524:
3520:
3515:
3512:
3508:
3503:
3500:
3496:
3491:
3489:
3487:
3483:
3479:
3474:
3471:
3467:
3462:
3459:
3455:
3450:
3447:
3443:
3438:
3436:
3432:
3429:, pp. 60
3428:
3423:
3420:
3416:
3411:
3408:
3402:
3401:
3397:
3393:
3390:
3388:
3385:
3383:
3382:Fermi surface
3380:
3378:
3375:
3373:
3370:
3368:
3367:Tight binding
3365:
3363:
3360:
3358:
3355:
3354:
3350:
3348:
3346:
3342:
3338:
3333:
3331:
3327:
3322:
3320:
3316:
3311:
3309:
3304:
3298:
3294:
3290:
3286:
3282:
3278:
3274:
3270:
3267:
3263:
3260:
3256:
3252:
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3242:
3235:
3230:
3227:
3224:
3219:
3216:
3215:
3214:
3208:
3206:
3204:
3182:
3179:
3169:
3156:
3151:
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3119:
3116:
3113:
3110:
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3066:
3063:
3060:
3057:
3019:
3013:
3008:
3003:
2992:
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2974:
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2797:
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2755:
2753:
2735:
2731:
2727:
2717:
2713:
2689:
2685:
2676:
2672:
2665:
2660:
2656:
2652:
2649:
2640:
2634:
2632:
2630:
2626:
2606:
2602:
2597:
2585:
2581:
2576:
2565:
2544:
2533:
2529:
2526:
2518:
2514:
2506:
2504:
2489:
2469:
2443:
2439:
2435:
2432:
2429:
2426:
2423:
2418:
2414:
2406:
2405:
2404:
2402:
2398:
2394:
2389:
2375:
2348:
2344:
2321:
2317:
2313:
2281:
2277:
2264:
2260:
2253:
2248:
2244:
2238:
2233:
2228:
2222:
2214:
2205:
2200:
2195:
2191:
2183:
2182:
2181:
2177:
2151:
2147:
2142:
2139:
2134:
2124:
2120:
2112:
2111:
2110:
2107:
2085:
2081:
2078:
2070:
2066:
2062:
2061:heat capacity
2056:
2049:Heat capacity
2048:
2041:
2039:
2035:
2031:
2027:
2023:
2022:Bohr magneton
1999:
1990:
1972:
1968:
1944:
1930:
1923:
1918:
1907:
1901:
1897:
1891:
1888:
1883:
1880:
1873:
1872:
1871:
1869:
1865:
1861:
1853:
1851:
1833:
1822:
1818:
1813:
1810:
1805:
1802:
1797:
1794:
1789:
1784:
1781:
1778:
1773:
1767:
1759:
1750:
1745:
1742:
1739:
1736:
1729:
1728:
1727:
1726:of the metal
1725:
1720:
1704:
1698:
1692:
1689:
1683:
1677:
1657:
1634:
1628:
1622:
1617:
1614:
1609:
1604:
1601:
1598:
1593:
1587:
1579:
1570:
1565:
1562:
1559:
1552:
1551:
1550:
1536:
1525:
1519:
1516:
1491:
1480:
1476:
1471:
1468:
1463:
1457:
1451:
1444:
1443:
1442:
1440:
1424:
1421:
1418:
1406:
1404:
1387:
1384:
1381:
1367:
1343:
1340:
1337:
1323:
1314:
1290:
1281:
1257:
1252:
1240:
1236:
1225:
1204:
1181:
1177:
1173:
1170:
1165:
1160:
1147:
1143:
1138:
1131:
1126:
1122:
1116:
1111:
1106:
1093:
1089:
1084:
1077:
1072:
1068:
1062:
1059:
1055:
1048:
1045:
1042:
1028:
1024:
1018:
1004:
996:
995:
994:
980:
977:
974:
966:
942:
933:
929:
913:
906:
898:
896:
894:
878:
875:
872:
849:
835:
831:
815:
811:
804:
801:
796:
791:
786:
782:
778:
768:
758:
754:
747:
743:
737:
731:
725:
718:
717:
716:
714:
706:
701:
697:
692:
688:
686:
685:electronvolts
682:
678:
639:
633:
630:
624:
620:
615:
611:
607:
603:
593:
589:
585:
579:
569:
558:
550:
549:
548:
546:
539:
531:
529:
527:
523:
519:
514:
512:
496:
488:
472:
464:
448:
418:
377:
370:
366:
361:
356:
352:
348:
342:
339:
312:
309:
297:
296:
295:
293:
289:
284:
282:
278:
274:
268:
260:
258:
256:
252:
248:
241:
238:
234:
228:
227:electron mass
224:
219:
217:
216:free particle
209:
205:
201:
197:
193:
190:
175:
167:
164:
160:
157:
153:
152:
151:
145:
143:
141:
137:
133:
124:
120:
117:
114:
110:
106:
103:
100:
97:
93:
90:
86:
83:
79:
75:
71:
70:
69:
66:
64:
60:
56:
53:
49:
45:
41:
37:
33:
29:
22:
4439:
4427:
4415:
4403:
4321:Pines' demon
4080:
4060:Kondo effect
3962:Time crystal
3847:
3828:
3800:
3770:
3730:
3726:
3716:
3705:. Retrieved
3701:
3688:
3677:. Retrieved
3674:HyperPhysics
3663:
3630:
3626:
3617:
3595:
3565:, p. 23
3543:
3521:, p. 49
3514:
3502:
3473:
3461:
3449:
3422:
3410:
3334:
3323:
3312:
3305:
3302:
3240:
3233:
3212:
3095:
3049:
2860:
2641:
2638:
2510:
2461:
2390:
2305:
2178:
2175:
2108:
2058:
1959:
1864:paramagnetic
1857:
1848:
1724:bulk modulus
1649:
1506:
1410:
1196:
932:Fermi energy
902:
864:
710:
698:of a gas of
681:Fermi energy
654:
545:Fermi energy
541:
515:
392:
287:
285:
270:
243:
239:
235:becoming an
229:
220:
213:
149:
140:noble metals
128:
67:
62:
31:
25:
4358:Soft matter
4258:Ferromagnet
4076:Drude model
4045:Berry phase
4025:Hall effect
3825:Bethe, Hans
3668:Nave, Rod.
3600:Kittel 1972
3456:, p. 7
3387:White dwarf
3261:Directional
3239:= â1/|
3225:scattering.
2401:Debye model
2026:diamagnetic
1439:phase space
928:Fermi level
899:Fermi level
526:Hall effect
273:Drude model
267:Drude model
208:Fermi level
155:collisions.
55:Drude model
4457:Categories
4273:Spin glass
4268:Metamagnet
4248:Paramagnet
4135:Conduction
4111:BCS theory
3952:Superfluid
3947:Supersolid
3707:2018-04-21
3679:2018-03-21
3609:References
3398:References
3337:Lev Landau
3285:Semimetals
3277:insulators
1870:given by
294:, that is
198:(see also
4331:Polariton
4238:Diamagnet
4186:Couplings
4162:Conductor
4157:Semimetal
4142:Insulator
4018:Phenomena
3942:Fermi gas
3740:1311.6805
3655:0044-3328
3403:Citations
3255:aluminium
3251:magnesium
3152:−
3123:∇
3117:−
3108:∇
3075:−
3067:×
2971:π
2816:⟩
2803:⟨
2770:σ
2767:κ
2724:⟩
2710:⟨
2682:⟩
2669:⟨
2666:τ
2650:κ
2625:Ängströms
2545:τ
2527:λ
2470:γ
2427:γ
2424:≈
2245:π
2220:∂
2212:∂
2071:of about
2067:given by
2000:μ
1969:μ
1908:μ
1898:μ
1881:χ
1785:μ
1765:∂
1757:∂
1743:−
1605:μ
1585:∂
1577:∂
1566:−
1174:⋯
1123:π
1117:−
1069:π
1063:−
914:μ
876:≥
765:ℏ
755:π
663:ℏ
612:π
576:ℏ
538:Fermi gas
473:τ
362:τ
340:σ
313:σ
292:Ohm's law
281:Fermi gas
200:Fermi gas
176:τ
52:classical
21:Fermi gas
4478:Electron
4405:Category
4386:Colloids
3827:(1933).
3797:(1976).
3769:(1972).
3351:See also
1991:and the
1866:and its
700:fermions
44:metallic
4417:Commons
4381:Polymer
4348:Polaron
4326:Plasmon
4306:Exciton
3760:General
3635:Bibcode
3602:, Ch. 6
3392:Jellium
3203:lithium
2629:defects
2557:(where
2020:is the
1987:is the
1278:as the
711:The 3D
520:, the
509:is the
485:is the
461:is the
415:is the
111:of the
4316:Phonon
4311:Magnon
4069:Theory
3927:Plasma
3917:Liquid
3854:
3835:
3809:
3779:
3653:
3243:|
3223:phonon
2861:where
2462:where
1960:where
1650:where
1197:where
963:. The
865:where
679:. The
655:where
393:where
136:alkali
30:, the
4291:Anyon
3912:Solid
3735:arXiv
3698:(PDF)
3293:holes
3258:does.
2959:Drude
2354:Drude
2130:Drude
331:with
42:in a
34:is a
4301:Hole
3852:ISBN
3833:ISBN
3807:ISBN
3777:ISBN
3651:ISSN
3313:The
3253:and
3064:2.44
2482:and
2034:spin
1385:>
978:>
903:The
489:and
290:for
138:and
132:ions
121:and
107:the
80:and
72:the
3922:Gas
3745:doi
3731:112
3643:doi
1311:is
547:as
26:In
4459::
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