Physics:Bloch–Grüneisen temperature
For typical three-dimensional metals, the temperature-dependence of the electrical resistivity ρ(T) due to the scattering of electrons by acoustic phonons changes from a high-temperature regime in which ρ ∝ T to a low-temperature regime in which ρ ∝ T5 at a characteristic temperature known as the Debye temperature. For low density electron systems, however, the Fermi surface can be substantially smaller than the size of the Brillouin zone, and only a small fraction of acoustic phonons can scatter off electrons.[1] This results in a new characteristic temperature known as the Bloch–Grüneisen temperature that is lower than the Debye temperature. The Bloch–Grüneisen temperature is defined as 2ħvskF/kB, where ħ is the Planck constant, vs is the velocity of sound, ħkF is the Fermi momentum, and kB is the Boltzmann constant. When the temperature is lower than the Bloch–Grüneisen temperature, the most energetic thermal phonons have a typical momentum of kBT/vs which is smaller than ħkF, the momentum of the conducting electrons at the Fermi surface. This means that the electrons will only scatter in small angles when they absorb or emit a phonon. In contrast when the temperature is higher than the Bloch–Grüneisen temperature, there are thermal phonons of all momenta and in this case electrons will also experience large angle scattering events when they absorb or emit a phonon. In many cases, the Bloch–Grüneisen temperature is approximately equal to the Debye temperature (usually written [math]\displaystyle{ \Theta_{\rm D} }[/math]), which is used in modeling specific heat capacity.[2] However, in particular circumstances these temperatures can be quite different.[3]
The theory was initially put forward by Felix Bloch[4] and Eduard Grüneisen.[5] The Bloch–Grüneisen temperature has been observed experimentally in a two-dimensional electron gas[3] and in graphene.[6]
Mathematically, the Bloch–Grüneisen model produces a resistivity given by:[2]
[math]\displaystyle{ \rho(T)=A\left(\frac{T}{\Theta_{\rm R}}\right)^n \int_0^{\Theta_R/T}\frac{t^n}{(e^t-1)(1-e^{-t})}dt }[/math].
Under Bloch's original assumptions for simple metals, [math]\displaystyle{ n=5 }[/math].[4] For [math]\displaystyle{ \Theta_{\rm R} \gg T }[/math], this can be approximated as [math]\displaystyle{ \rho \sim T^5 }[/math] dependence. In contrast, the so called Bloch–Wilson limit, where [math]\displaystyle{ n=3 }[/math] works better for s-d inter-band scattering, such as with transition metals.[7] The second limit gives [math]\displaystyle{ \rho \sim T^3 }[/math] at low temperatures.[8] In practice, which model is more applicable depends on the particular material.[9]
References
- ↑ Fuhrer, Michael (2010-12-13). "Textbook physics from a cutting-edge material". Physics (American Physical Society (APS)) 3: 106. doi:10.1103/physics.3.106. ISSN 1943-2879. Bibcode: 2010PhyOJ...3..106F.
- ↑ 2.0 2.1 Cvijović, D. (2011). "The Bloch-Gruneisen function of arbitrary order and its series representations". Theoretical and Mathematical Physics (Springer Science and Business Media LLC) 166 (1): 37–42. doi:10.1007/s11232-011-0003-4. ISSN 0040-5779. Bibcode: 2011TMP...166...37C.
- ↑ 3.0 3.1 Stormer, H. L.; Pfeiffer, L. N.; Baldwin, K. W.; West, K. W. (1990-01-15). "Observation of a Bloch-Grüneisen regime in two-dimensional electron transport". Physical Review B (American Physical Society (APS)) 41 (2): 1278–1281. doi:10.1103/physrevb.41.1278. ISSN 0163-1829. PMID 9993840. Bibcode: 1990PhRvB..41.1278S.
- ↑ 4.0 4.1 Bloch, F. (1930). "Zum elektrischen Widerstandsgesetz bei tiefen Temperaturen" (in de). Zeitschrift für Physik (Springer Science and Business Media LLC) 59 (3–4): 208–214. doi:10.1007/bf01341426. ISSN 1434-6001. Bibcode: 1930ZPhy...59..208B.
- ↑ Grüneisen, E. (1933). "Die Abhängigkeit des elektrischen Widerstandes reiner Metalle von der Temperatur" (in de). Annalen der Physik (Wiley) 408 (5): 530–540. doi:10.1002/andp.19334080504. ISSN 0003-3804. Bibcode: 1933AnP...408..530G.
- ↑ Efetov, Dmitri K.; Kim, Philip (2010-12-13). "Controlling Electron-Phonon Interactions in Graphene at Ultrahigh Carrier Densities". Physical Review Letters 105 (25): 256805. doi:10.1103/physrevlett.105.256805. ISSN 0031-9007. PMID 21231611. Bibcode: 2010PhRvL.105y6805E.
- ↑ Wilson, Alan Herries; Fowler, Ralph Howard (1938-09-23). "The electrical conductivity of the transition metals.". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences (The Royal Society) 167 (931): 580–593. doi:10.1098/rspa.1938.0156. ISSN 1364-5021. Bibcode: 1938RSPSA.167..580W.
- ↑ Suri, Dhavala; Siva, Vantari; Joshi, Shalikram; Senapati, Kartik; Sahoo, P K; Varma, Shikha; Patel, R S (2017-11-13). "A study of electron and thermal transport in layered titanium disulphide single crystals". Journal of Physics: Condensed Matter (IOP Publishing) 29 (48): 485708. doi:10.1088/1361-648x/aa90c5. ISSN 0953-8984. PMID 28975897. Bibcode: 2017JPCM...29V5708S.
- ↑ Allison, C.Y.; Finch, C.B.; Foegelle, M.D.; Modine, F.A. (1988). "Low-temperature electrical resistivity of transition-metal carbides". Solid State Communications (Elsevier BV) 68 (4): 387–390. doi:10.1016/0038-1098(88)90300-6. ISSN 0038-1098. Bibcode: 1988SSCom..68..387A.
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