The Clean and Dirty Limits

The purity of a superconductor is characterized by the ratio $l/ \xi_{\circ}$. $\xi_{\circ}$ is the coherence length of the pure material, and is given by Eq. (2.4). l is the electron mean free path, defined as:

$$l = \tau v_{F}$$ (9)

where $\tau$ is the time interval between collisions of conduction electrons with impurities in the sample. The magnitude of $\tau$ is determined in the normal state. A sample is clean if $l/ \xi_{\circ}\gg 1$, and dirty if $l/ \xi_{\circ}\ll 1$ [20]. The actual coherence length when impurities are considered, is dependent upon the mean free path l. Intuitively, one can define an effective coherence length $\xi (l)$ such that [13,17]:

$$\frac{1}{\xi (l)} = \frac{1}{\xi_{\circ}} + \frac{1}{l}$$ (10)

According to Eq. (2.10), the coherence length $\xi (l)$ becomes shorter with decreasing l, so that in the dirty limit:

$$\xi (l) = l,\ \ \ (l \ll \xi_{\circ})$$ (11)

and in the clean limit:

$$\xi (l) = \xi_{\circ},\ \ \ (l \gg \xi_{\circ})$$ (12)

Eq. (2.11) and Eq. (2.12) are valid only for T=0K. In the clean limit ( $l \gg \xi_{\circ}$) at T=0K, the magnetic penetration depth is the London penetration depth given in Eq. (2.1). The actual coherence length $\xi (T)$ and the observed penetration depth $\lambda (T)$as determined by microscopic theory are given in the clean limit ( $l \gg \xi_{\circ}$) as [21]:

$$\xi (T) = 0.74 \xi_{\circ}\left( \frac{T_{c}}{T_{c} - T} \right)^{\frac{1}{2}}$$ (13)

$$\lambda (T) = 0.71 \lambda_{L}\left( \frac{T_{c}}{T_{c} - T} \right)^{\frac{1}{2}}$$ (14)

and in the dirty limit ( $l \ll \xi_{\circ}$):

$$\xi (T) = 0.85 (\xi_{\circ}l)^{\frac{1}{2}} \left( \frac{T_{c}}{T_{c} - T} \right)^{\frac{1}{2}}$$ (15)

$$\lambda (T) = 0.62 \lambda_{L}\left( \frac{\xi_{\circ}}{l} \r . . . . . . c{1}{2}} \left( \frac{T_{c}}{T_{c} - T} \right)^{\frac{1}{2}}$$ (16)

where Equations (2.13) through (2.16) are valid only in the neighborhood of T_c, such that $(T_{c} - T)/T_{c} \ll 1$ [22]. The important result is that according to Eqs.(2.15) and (2.16), as l decreases (i.e. the superconductor becomes more impure), $\lambda (T)$ increases, while $\xi (T)$ decreases. Thus $\lambda (T) \gg \xi (T)$ at all temperatures in an impure material. The high-temperature superconductors have short coherence lengths on the order of $\xi \sim 12$ to 15Å. Since the electron mean free path l is typically $\sim 150$Å in these materials, then they are well within the clean limit [23].