1

Study of resistivity in the normal state ofiron pnictides

G.A. Ummarino

Prof of Theoretical Physics of Matter

Institute of Physics and Materials Engineering

Applied Science and Technology Departement

Politecnico di Torino, Italy

Turin Politexnika Universiteti, Tashkent, Uzbekistan

National Research Nuclear University MEPhI, Moscow

Engineering Physics Institute, Russia

International Scientific Seminar "Basov Workshop-2017" December 14-15, 2017 FIAN - MEPhI

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1. Iron pnictides superconductors

3

Phase diagrams

4

Eliashberg equations (superconductive state)

G.A. Ummarino, PRB 83, 092508 (2011)

6

Resistivity theory for a multiband metal

P.B. Allen, Phys. Rev. B 17 (1978) 3725–3734

ρ(T)=ρ0+

7

8

2. Theory

ARPES and de Haas-van Alphen data suggest that for Co-doped Ba-122 the

transport is dominated by the electronic bands and that the hole bands are

characterized by a smaller mobility**Lei Fang, Huiqian Luo, Peng Cheng, Zhaosheng Wang, Ying Jia, Gang Mu, Bing Shen, Mazin II, Lei Shan,

Cong Ren, and Hai-Hu Wen, Phys. Rev. B 80, 140508(R) (2009).

*E. G. Maksimov, A. E. Karakozov, B. P. Gorshunov, A. S. Prokhorov, A. A. Voronkov, E. S. Zhukova, V. S.

Nozdrin, S. S. Zhukov, D. Wu, M. Dressel, S. Haindl, K. Iida, and B. Holzapfel, Phys. Rev. B 83, 140502(R)

(2011).

A saturation at high temperature in the normal state electrical resistivity has been

observed in many alloys since the 1960s. This behaviour can be explained within a

phenomenological model containing two kinds of carriers with different scattering

parameters, then two parallel conductivity channels have to be considered so that:

where ρe(T) is the resistivity of the first group of carriers (electron), characterized by a

strong temperature-dependent scattering because of its weak scattering on defects,

and ρsat is the contribution of the second group of carriers (hole) that gives a

temperature-independent contribution with strong scattering on defects.

The simplest model: two bands model.

Parameters:

1) the electron boson constant coupling λ1tr and λ2tr

2) the impurities content Γ2 and Γ1 (Γ2 and Γ1 are connected by the experimental value of ρ0)

3) the plasma energy ωp1 and ωp2

4) the energy Ω0 of the peak of the trasport electron-boson spectralfunction (we assume that the shape is the same in every band)

BUT

3) ωp1 and ωp2 are provided by band theory

4) Ω0 can be provided by neutron measurements

AND, when temperature saturation is present,

1) only the band 1 contributes i.e., in the first approximation, λ2tr=0

2) the impurities are mainly in the band 2 (hole band) i.e. Γ2>>Γ1

so, in this case, we have only two free parameters: λ1tr and Γ2.

2. Theory

G.A. Ummarino , Sara Galasso , D. Daghero , M. Tortello , R.S. Gonnelli , A. Sanna,

Physica C 492 (2013) 21–24

10

λtr=0.77<<λsup=2.00

0 50 100 150 200 2500

100

200

300

400

500

600

700

Exp. data

1-band fit

Quadratic fit

2-band fit

(

cm

)

Temperature (K)

Two-band fit

0 10 20 30 40 50 60 700

25

50

75

100

125

cm

)

Temperature (K)

11

A.A. Golubov, O.V. Dolgov, A.V. Boris, A. Charnukha, D. L. Sun, C.T. Lin, A. F. Shevchun,

“Normal state resistivity of Ba1−xKxFe2As2: evidence for multiband strong-coupling behavior”

JETP Letters Vol.94, N 4, pages 333-337, (2011).

λtr=0.45<<λsup=2.84

4. (Ba,K)Fe2As2 (single crystal)

Two-band fit

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10% single crystal λtr=0.14<<λsup=2.22 and Ω0=40 meV

5. Ba(Fe,Co)2As2 (single crystal)

The value of Ω0 is in agreement

with D.S. Inosov et al., Nature

Phys. 6, 178 (2010).

Experimental data by Lei Fang, Huiqian Luo, Peng Cheng, Zhaosheng Wang, Ying Jia, Gang Mu, Bing Shen,

I. I. Mazin, Lei Shan, Cong Ren, and Hai-Hu Wen, Phys. Rev. B 80, 140508(R) (2009).

0 50 100 150 200 250 300 3500

50

100

150

200

250

300

350

0 50 100 150 2000,00

0,05

0,10

0,15

0,20

Ba(Fe0.9

Co0.1

)2As

2

Tc=24.6 K

exp data

two-band fit

cm

)

Temperature (K)

0=40 meV

F

()

(meV)

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FeSe1-x

Soon-Gil Jung, Nam Hoon Lee, Eun-Mi Choi, Won Nam Kang, Sung-Ik Lee, Tae-Jong Hwang, Dong Ho

Kim, Physica C 470, 1977 (2010).

The substrate affects the film resistivity

6. Films: effect of the substrate

Experimental details on thin films of Ba(Fe1−xCox)2As2

We measured the resistivity on Ba(Fe1−xCox)2As2 (x =0.08, 0.10, 0.15) epitaxial thin films with a thickness of the

order of 50 nm were deposited on (001) CaF2 substrates by Pulsed Laser Deposition using a polycrystalline target

with high phase purity. The surface smoothness was confirmed by in situ reflection high energy electron diffraction

during the deposition; only streaky pattern were observed for all films, indicative of a smooth surface

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6. Ba(Fe,Co)2As2

Films and single crystals

6. Ba(Fe,Co)2As2 films: data interpretation

[20] E. G. Maksimov, A. E. Karakozov, B. P. Gorshunov, A. S.Prokhorov, A. A. Voronkov, E. S. Zhukova,

V. S. Nozdrin, S. S. Zhukov, D. Wu, M. Dressel, S. Haindl, K. Iida, and B. Holzapfel, Phys. Rev. B 83,

140502(R) (2011).

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0 50 100 150 200 250 300 3500

50

100

150

200

250

0 200 400 600 800 10000,00

0,05

0,10

0,15

0,20

Ba(Fe0.9

Co0.1

)2As

2

Tc=26.6 K

exp data

one-band fit

two-band fit

cm

)

Temperature (K)

0=180 meV

c=

0/10

F

()

(meV)

10% film λtr=0.26<<λsup=2.22 but Ω0=180 meV

6. Ba(Fe,Co)2As2

The value of Ω0 is NOT in agreement

with D.S. Inosov et al., Nature Phys. 6,

178 (2010).

films

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6. Ba(Fe,Co)2As2

At first glance, the curves look very different!

0 50 100 150 200 250 300 3500

50

100

150

200

250

300

350

Ba(Fe0.9

Co0.1

)2As

2

single crystal

film

cm

)

Temperature (K)

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10% film λtr=0.13<<λsup=2.22, Ω0=40 meV as in single crystals!

0 50 100 150 200 250 300 3500

50

100

150

200

250

0 40 80 120 160 2000,00

0,05

0,10

0,15

0,20

0,25

0,30

Ba(Fe0.9Co0.1)2As2

Tc=26.6 K

cm

)

Temperature (K)

0=40 meV

c=

0

F

()

(meV)

6. Ba(Fe,Co)2As2

Possible solution α2F(Ω)~Ω3 not in the range 0<Ω<9Ω0/100<Ω<Ω0/10

Now the value of Ω0tr is in agreement with

D.S. Inosov et al., Nature Phys. 6, 178 (2010).

films

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6. Ba(Fe,Co)2As2

Thin film and single crystal

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8% film, λtr=0.13<<λsup=2.22 Ω0=44 meV

6. Ba(Fe,Co)2As2

0<Ω<Ω00<Ω<Ω0/10

Now the value of Ω0tr is in agreement with

D.S. Inosov et al., Nature Phys. 6, 178 (2010).

Possible solution α2F(Ω)~Ω3 not in the range

films

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6. Ba(Fe,Co)2As2

15% film, λtr=0.14<<λsup=1.82 Ω0=26 meV

0<Ω<Ω0/10 0<Ω<8Ω0/10

Now the value of Ω0tr is in agreement

with D.S. Inosov et al., Nature Phys. 6,

178 (2010).

Possible solution α2F(Ω)~Ω3 not in the range

films

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6. Ba(Fe,Co)2As2

There is a “hardening” of the electron–boson spectral function, with a transfer of spectral weight

from energies smaller than Ω0 to energies higher than Ω0. Probably this effect is so striking

because the main mechanism is not phononic. In fact the strain produced by the substrate on

the film can create changes in the electronic structure and in the Fermi surface nesting that is

crucial for raising superconductivity mediated by antiferromagnetic spin fluctuactions.

The effect of the substrate can be assimilated to that of a uniaxial pressure and in particular the

substrate used here (CaF2) has the largest effect on the volume cell of the Co doped Ba-122

because of the mismatch of the dimension of the unit cells.

films

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λ tr,tot<<λsup,tot Ω0>>Ωsup0

7. Iron pinictides

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It seems that the substrate, in the films, heavy affects the shape of the transport

spectral function in a range of energy bigger (0<Ω<Ω0) than a normal metal

(0<Ω<Ω0/10)

We also underline the possibility that the very different properties of the

mediating boson in the normal and superconductive state (λsup>>λtr and Ω0>>Ωsup0)

can be a unifying principle at the root of superconductivity in the iron-based materials

(In the HTCS there is an similar situation*)

A more systematic analysis must be carried out on different doping of Co and on

different iron compounds

•E.G. Maksimov, M.L. Kulic, and O.V. Dolgov, “Bosonic Spectral Function and the Electron-Phonon

Interaction in HTSC Cuprates”, Advances in Condensed Matter Physics Volume 2010 (2010) ID

423725

G.A. Ummarino, “Temperature-dependent spin resonance energy in iron pnictides and multiband s_+

Eliashberg theory” , Physical Review B 83, 092508 (2011).

G.A. Ummarino, Sara Galasso, Paola Pecchio, D. Daghero,R.S. Gonnelli, F. Kurth, K. Iida, B.

Holzapfel,” Resistivity in Co-doped Ba-122: comparison of thin films and single crystals.” Phys. Status

Solidi B, 1–7 (2015)

8. Conclusions

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Addendum: Transport Properties and Boltzmann Equation

At thermodynamic equilibrium, the probability of occupation of an energy level E(k)

is given by the Fermi-Dirac distribution function:

where T is the temperature of the sample and μ is the chemical potential (the chemical

potential will be denoted by μ or by EF , indifferently); f0(k) is a shorthand notation

for f0(E(k)).

When external perturbations (electric fields, magnetic fields, temperature gradients)

are applied to the sample, the electron distribution is disturbed from the equilibrium

Fermi-Dirac function. In general the disturbed distribution function f(r,k,t), in addition

to k, depends also on the real space coordinate r, and on time t; the quantity

f (r, k, t)drdk/4π3 gives the number of electrons at time t in the volume element

drdk around the point (r,k) of the phase space.

According to the semiclassical dynamics of carriers in a given energy band, an

electron in the point (r, k) at time t evolves toward the point (r +vk dt, k +(F/hr) dt)

(hr=h/2π is the reduced Planck constant) at time t +dt , where

(1)

(2)

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and F denotes the external force acting on the carriers. During the motion, collision processes

(due for instance to lattice vibrations, impurities, boundaries) may cause a net rate of change

[∂f/∂t ]coll of the number of electrons in the phase space volume drdk.

Using Liouville theorem (volumes in phase space are preserved by the semiclassical equations

of motion) we must have for the distribution function:

expresses the detailed balance in each volume drdk of the number

of carriers, when moving in the phase space under the action of external fields and

in the presence of collision processes (as indicated in pictorial form in the figure below).

(3)

Schematic representation of the

conservation of the number

of electrons moving in the space phase r,k. The

region dΩ around r, k at time t evolves into a new

region dΩ’, whose volume is the same as dΩ

(Liouville theorem). The distribution function

f (r+dr, k+dk, t+dt) equals f (r,k,t) supplemented

by the net change

[∂f/∂t]coll dt = [∂f/∂t]in coll dt −[∂ f/∂t]out coll dt of the

number of electrons forced in and ejected out

because of collision processes.

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The left member of Eq. (3) can be expanded in Taylor series up to first order, and we obtain the

Boltzmann equation

where ∂f/∂r and ∂f/∂k stand for ∇r f and ∇k f , respectively.

A crucial aspect in the transport theory is just the collision term, which makes the Boltzmann

equation a formidable integro-differential equation. In general, the Boltzmann collision operator

takes the form

where Icoll is an integral operator acting on f(r,k,t); it describes the effect on the distribution

function of the scattering processes (produced for instance by impurities, vacancies, lattices

vibrations, physical surfaces, etc.). The change of the distribution function at a given k vector

equals the number of electrons forced into k by collisions minus the number of electrons ejected

from k by collisions. The expression of Icoll is of the type

where Pk←k denotes the probability per unit time that an electron in the k state is scattered into

the k state; the factor of the type f(r,k,t)[1 − f(r,k,t)] is due to the Pauli principle and takes into

account the occupancy of the initial states and the non-occupancy of the final states.

(4)

(5)

(6)

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A major simplification is obtained when the deviation of f from the thermal equilibrium

distribution f0 is small: it is customary to assume that the rate of change of f due to collisions is

proportional to the deviation itself, i.e.

where τ denotes the appropriate proportionality coefficient and is called relaxation time; in

general τ = τ(k,r,T) depends on the energy E(k) and on position, and is often considered a

semi-empirical parameter. In the relaxation time approximation, the Boltzmann equation

becomes the rather manageable partial differential equation

The role of the relaxation time can be further clarified by considering the case of sudden

removal of external forces (F = 0) at t = 0 in a uniform system (∂f/∂r) = 0; from

Eq. (8) one obtains

and the distribution function moves to its equilibrium value f0 exponentially with time

constant τ. The non-equilibrium distribution function f , evaluated with the Boltzmann

equation, permits the investigation of a number of transport phenomena due to intraband

electronic processes.

(7)

(8)

(9)

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Transport coefficients can be inferred from the general expression of charge current density

and the energy current density (or energy flux) given by

where the factor 2/(2π)3 takes into account spin degeneracy and density of allowed points in k

space per unit volume (the k-dependence of the group velocity, band energy and distribution

function is left implicit, for simplicity of notations). Transport properties, dissipation of

irreversible heat, generation or subtraction of reversible heat, can be discussed using the above

equations, and the principles of the thermodynamics. The Boltzmann transport equation

describes a variety of transport phenomena in bulk crystalline solids. Whenever the scattering

processes are reasonably weak and the applied fields are reasonably low, it makes sense to

consider the joint effects of fields and collisions in driving the system out of equilibrium,

within the perturbative approach and mean-field spirit, implicit in the Boltzmann equation.

The description of transport phenomena by means of a distribution function is doomed to fail

for small or ultrasmall electronic devices, whose spatial scales (micron or submicron

dimensions) may become comparable or shorter than the semiclassical mean free path, or the

quantum wavelength of the particles responsible of transport; for these systems quantum

transport techniques must be applied.

(10)

(11)

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Static Conductivity with the Boltzmann Equation

We consider the static conductivity of a metal by using the Boltzmann approach. For a

homogeneous material in a uniform and steady electric field E, the distribution function f depends

only on k and the Boltzmann becomes

For low electric fields, we can assume that f − f0 is linear in the field strength, and

we can thus put f ≈ f0 in the first member of Eq. (12); we obtain

In ordinary situations eτ|E| <<kF thus Eq. (13), at the lowest order in eτ|E|/hrkF, can be written as

f(k) ≈ f0(k + eτE/hr). The nonequilibrium distribution function f(k) and the equilibrium

distribution function f0(k) are schematically indicated in the figure below:

(12)

(13)

Schematic representation of the equilibrium distribution function

f0(k) and of the nonequilibrium distribution function

f (k) ≈ f0(k+eτhr−1E) in the presence of a static electric field E.

The effect of the field is to shift the whole Fermi sphere by

-eτhr−1E in the k space. Forsake of clarity of the figure, the value

of -eτhr−1E has been magnified and taken of the sameorder as kF .

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It is convenient to write f = f0 + f1, and recast Eq. (13) in the form

where v = (1/hr)∂E(k)/∂k is the semiclassical expression of the velocity. Inserting f1 in the

expression (10) for the current density ( f0 gives zero contribution), one obtains

with α, β = x, y, z; the above relation defines the conductivity tensor linking the Cartesian

components of the current density to the Cartesian components of the electric field. For

simplicity we assume that the material is isotropic so that J and E are parallel,and the

proportionality constant is the static conductivity σ0; indicating by e = E/|E| the unit vector in

the direction of the electric field, and projecting J along e, we obtain

It is well known that the Fermi distribution function f0(E) changes sharply from unity to zero

within a small interval (≈kBT ) around the Fermi level, and that (−∂f0/∂E) is significantly

different from zero in this same interval. Thus the conductivity σ0 is basically determined from

the features of the conduction bands in the thermal shell kBT around the Fermi energy EF;

(13)

(14)

(15)

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this conclusion holds not only for the conductivity but also for any other transport coefficient in

metals. To estimate the conductivity given by expression (15), one can replace (e·v)2 with v2/3,

and approximate the derivative of the Fermi function with a δ-function; it follows

The integration can be performed exploiting the identity

The expression of the conductivity thus reduces to an integration over the Fermi surface

In the particular case of a parabolic conduction band with effective mass m∗, Eq. (18) gives

(16)

(17)

(18)

(19) (vF =hrkF/m∗ and k3F= 3π2n).

Notice that the rather naive result (19), in which all the electrons seem to take part to the transport, is valid only for parabolic

bands; in reality, the generally valid Eq. (18) shows that in any case (parabolic or nonparabolic bands) only the electrons at

(or near) the Fermi surface can change their state under perturbations and are relevant in the transport phenomena.

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There are two formulae for the electrical properties of metals: one of them is Matthiessen's

rule for the resistivity ρ:

The other formula expresses the conductivity σ as

This relation is often called the Drude formula. Its essential ingredient is the electron

scattering time τ. Matthiessen's rule says that the total resistivity is the sum of contributions

from the scattering of conduction electrons by the thermal vibrations and the scattering

by static lattice defects. In the second equation e and m are the electron charge and mass, n is

the electron number density and τ is an average relaxation time for the conduction electrons

carrying the current.

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The relation for the conductivity is deceptively simple. It might give the impression that the

electron number density n plays a central role. This can be true for doped semiconductors

where n varies by many orders of magnitude, but not for metals. The major problem is

hidden in the electron relaxation time τ.

An electron with velocity vk carries an electric current -evk. In equilibrium, for each electron

state characterised by the wave vector k, there is also a state -k, with velocity - vk, so that the

total electric current vanishes. In the presence of an electric field E, the Fermi-Dirac function

is changed from its equilibrium value f0k to the value fk. If the field is weak, we can assume

that fk-f0k is linear in E and write

where εk=E(k)-EF. The quantity τ(k) has the dimension of time. We shall call it a relaxation

time, although it usually does not fulfill the requirements of such a quantity (namely that the

electron states decay exponentially in t/τ as the field E is turned off at time t = 0). In the

following we may also write τ(εk), τ(ε), τ(εk,k) or τ(ε,k) instead of τ(k), to stress that τ is

energy dependent, or both energy dependent and anisotropic. The actual calculation of τ(k) is

difficult, but we can learn much just by assuming that it is a known quantity.

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36

37

Resistivity expressed in the Eliashberg transport coupling

function

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in free-electron-like metals.

The simplest choice is

α2trF(ω)=bω4θ(ωmax-ω)

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40

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θ(ωmax- ω)

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Einstein model

representative energy

43

The temperature dependence of resistivity is non affected by small amounts of alloys elements

ρ(T)=ρ0+hr/(ε0(hrωp)2τep)

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Above: calculated transport spectral function αtr2F(Ω) of Al, Au, Na, and Nb (solid lines) in comparison with the theoretical

phonon density of states (PDOS) (dashed lines). Below: calculated electrical resistivity (solid lines) as a function of temperature

of Al, Au, Na, and Nb in comparison with various experimental data (diamonds).

R. Bauer, A. Schmid, P. Pavone, and D. Strauch, PRB 57, 11 276, 1998.

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In the following we present and discuss in detail the more general formulation of the temperature dependency

of resistivity (P.B. Allen, PRB 17, 3725, 1978)

ALLEN THEORY, P.B. Allen, Physical Review B, 17, 3725 (1978)

49

Φx is a particular FSH proportional to x component

of speed and σ0 and σ1 are polynomials proportional

to 1 and ε, respectively.

50

51

52

If we put in the equation (1) and n=0 or 1

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54

55

56

57

ρ(T)=ρ0+hr/(ε0(hrωp)2τep)

58

NEXT STEP: HIGHER ORDER TERMS (>1)

TOO COMPLEX!

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Books:

Harald Ibach, Hans Luth

An Introduction to Principles of Materials Science,

2009, Springer-Verlag Berlin Heidelberg

Goran Grimwall,

THERMOPHYSICAL PROPERTIES OF MATERIALS

1999 Elsevier Science B.V.

Giuseppe Grosso, Giuseppe Pastori Parravicini,

SOLID STATE PHYSICS

2014 Elsevier Science B.V