Optical properties

Typical frequency-dependent optical conductivity of a heavy fermion compound. Blue line: T > T_coh. Red line: T < T_coh.

In order to obtain the optical properties of heavy fermion systems, these materials have been investigated by optical spectroscopy measurements.^[7] In these experiments the sample is irradiated by electromagnetic waves with tunable wavelength. Measuring the reflected or transmitted light reveals the characteristic energies of the sample.

Above the characteristic coherence temperature ${\displaystyle T_{\rm {coh}}}$, heavy fermion materials behave like normal metals; i.e. their optical response is described by the Drude model. Compared to a good metal however, heavy fermion compounds at high temperatures have a high scattering rate because of the large density of local magnetic moments (at least one f electron per unit cell), which cause (incoherent) Kondo scattering. Due to the high scattering rate, the conductivity for dc and at low frequencies is rather low. A conductivity roll-off (Drude roll-off) occurs at the frequency that corresponds to the relaxation rate.

Below ${\displaystyle T_{\rm {coh}}}$, the localized f electrons hybridize with the conduction electrons. This leads to the enhanced effective mass, and a hybridization gap develops. In contrast to Kondo insulators, the chemical potential of heavy fermion compounds lies within the conduction band. These changes lead to two important features in the optical response of heavy fermions.^[1]

The frequency-dependent conductivity of heavy-fermion materials can be expressed by ${\displaystyle \sigma (\omega )={\frac {ne^{2}}{m^{*}}}{\frac {\tau ^{*}}{1+\omega ^{2}\tau ^{*2}}}}$, containing the effective mass ${\displaystyle m^{*}}$ and the renormalized relaxation rate ${\displaystyle {\frac {1}{\tau ^{*}}}={\frac {m}{m^{*}}}{\frac {1}{\tau }}}$.^[8] Due to the large effective mass, the renormalized relaxation time is also enhanced, leading to a narrow Drude roll-off at very low frequencies compared to normal metals.^[8][9] The lowest such Drude relaxation rate observed in heavy fermions so far, in the low GHz range, was found in UPd₂Al₃.^[10]

The gap-like feature in the optical conductivity represents directly the hybridization gap, which opens due to the interaction of localized f electrons and conduction electrons. Since the conductivity does not vanish completely, the observed gap is actually a pseudogap.^[11] At even higher frequencies we can observe a local maximum in the optical conductivity due to normal interband excitations.^[1]

Specific heat for normal metals

At low temperature and for normal metals, the specific heat ${\displaystyle C_{P}}$ consists of the specific heat of the electrons ${\displaystyle C_{P,{\rm {el}}}}$ which depends linearly on temperature ${\displaystyle T}$ and of the specific heat of the crystal lattice vibrations (phonons) ${\displaystyle C_{P,{\rm {ph}}}}$ which depends cubically on temperature

${\displaystyle C_{P}=C_{P,{\rm {el}}}+C_{P,{\rm {ph}}}=\gamma T+\beta T^{3}\ }$

with proportionality constants ${\displaystyle \beta }$ and ${\displaystyle \gamma }$.

In the temperature range mentioned above, the electronic contribution is the major part of the specific heat. In the free electron model — a simple model system that neglects electron interaction — or metals that could be described by it, the electronic specific heat is given by

${\displaystyle C_{P,{\rm {el}}}=\gamma T={\frac {\pi ^{2}}{2}}{\frac {k_{\rm {B}}}{\epsilon _{\rm {F}}}}nk_{\rm {B}}T}$

with Boltzmann constant ${\displaystyle k_{\rm {B}}}$, the electron density ${\displaystyle n}$ and the Fermi energy ${\displaystyle \epsilon _{\rm {F}}}$ (the highest single particle energy of occupied electronic states). The proportionality constant ${\displaystyle \gamma }$ is called the Sommerfeld coefficient.

Relation between heat capacity and "thermal effective mass"

For electrons with a quadratic dispersion relation (as for the free-electron gas), the Fermi energy ε_F is inversely proportional to the particle's mass m:

${\displaystyle \epsilon _{\rm {F}}={\frac {\hbar ^{2}k_{\rm {F}}^{2}}{2m}}}$

where ${\displaystyle k_{\rm {F}}}$ stands for the Fermi wave number that depends on the electron density and is the absolute value of the wave number of the highest occupied electron state. Thus, because the Sommerfeld parameter ${\displaystyle \gamma }$ is inversely proportional to ${\displaystyle \epsilon _{\rm {F}}}$, ${\displaystyle \gamma }$ is proportional to the particle's mass and for high values of ${\displaystyle \gamma }$, the metal behaves as a Fermi gas in which the conduction electrons have a high thermal effective mass.

Example: UBe₁₃ at low temperatures

Experimental results for the specific heat of the heavy fermion compound UBe₁₃ show a peak at a temperature around 0.75 K that goes down to zero with a high slope if the temperature approaches 0 K. Due to this peak, the ${\displaystyle \gamma }$ factor is much higher than the free electron model in this temperature range. In contrast, above 6 K, the specific heat for this heavy fermion compound approaches the value expected from free-electron theory.