Characteristics of the Eliashberg formalism on the example of high-pressure superconducting state in phosphor

0.4 MB·

by  A. M. Duda

Characteristics of the Eliashberg formalism on the example of high-pressure
superconducting state in phosphor
A. M. Duda(1),∗ R. Szcz¸e´sniak(1,2),† M. A. Sowin´ska(1), and I. A. Domagalska(1)
1 Institute of Physics, Cze¸stochowa University of Technology,
Ave. Armii Krajowej 19, 42-200 Cze¸stochowa, Poland and
2 Institute of Physics, Jan D lugosz University in Cze¸stochowa,
Ave. Armii Krajowej 13/15, 42-200 Cze¸stochowa, Poland
6 (Dated: January 25, 2016)
1
The work describes the properties of the high-pressure superconducting state in phosphor:
0
p 20,30,40,70 GPa. The calculations were performed in the framework of the Eliashberg
2 ∈ { }
formalism, which is the natural generalization of the BCS theory. The exceptional attention was

n paidtotheaccuratepresentationoftheusedanalysisscheme. Withrespecttothesuperconducting
a state in phosphor it was shown that: (i) the observed not-high values of the critical temperature
J ([TC]mp=a3x0GPa = 8.45 K) result not only from the low values of the electron - phonon coupling
1 constant, but also from the very strong depairing Coulomb interactions, (ii) the inconsiderable
2 strong - coupling and retardation effects force the dimensionless ratios R∆, RC, and RH - related
to the critical temperature, the order parameter, the specific heat and the thermodynamic critical

] field - to take thevalues close to the BCS predictions.
n
o
PACS:74.20.Fg, 74.25.Bt, 74.62.Fj
c
-
r
p
u Keywords: Eliashberg formalism, high-pressure supercon- interaction term. Asa result thefollowing can be obtained:
s ductivity,thermodynamic properties
.
t
a ′ 1 ε2k+|∆(T)|2
m 1=V tanh , (2)
- I. HAMILTONIAN AND FUNDAMENTAL Xk 2 ε2k+|∆(T)|2 q 2kBT
d EQUATIONS OF BCS MODEL AND q
n ELIASHBERG FORMALISM where kB is the Boltzmann constant. Let us notice that
o the equation (2) cannot be solved analytically. However,
c in the limit cases: T TC and T 0 K the relatively
[ The first microscopic theory of the superconducting state simple calculations allo→w us to obtain→the formulas for the
1 was formulated in 1957 by Bardeen, Cooper and Schrieffer critical temperature and the value of the order parameter:
v (theso-called ”BCS model”) [1],[2]. Intheframework of the kBTC = 1.13Ωmaxexp[ 1/λ], ∆(0) = 2Ωmaxexp[ 1/λ],
9 methodofthesecondquantizationtheBCS Hamiltoniancan where the electron-phono−n coupling constant (λ) in th−e BCS
1 be written with the following formula [3], [4]: model is given by: λ ρ(0)V (the quantity ρ(0) represents
≡
8 theelectrondensityofstatesontheFermisurface). TheBCS
.05 H =Xkσ εkc†kσckσ−V Xkk′′c†k↑c†−k↓c−k′↓ck′↑, (1) trahteioorsy, wprheidchictasrethdeeefixniesdtenbceeloowf:theuniversalthermodynamic
1
0 wherethefunctionεk representstheelectronbandenergy,V R 2∆(0) =3.53, (3)
6 istheeffectivepairingpotential,whichvalueisdeterminedby ∆ ≡ kBTC
1 thematrix elements of theelectron -phonon interaction, the
iv: ec†kleσctarnodncbkaσndrepenreesregnytatnhdectrheeatpiohnonaonndeannenrighyi.latTiohneospyemrabtoolrs RC ≡ CS(TCC)N−(TCCN) (TC) =1.43, (4)
X of theelectron statein themomentum representation (k)for
and
thespinσ , . Itshould benoted thatthesum denoted
ar bythesign∈′ o{u↑gh↓t}tobecalculatedonlyforthosevaluesofthe RH ≡ TCHC2N((0T)C) =0.168. (5)
momentums,forwhichthecondition: Ωmax <εk <Ωmax is C
−
fulfilled,whereΩmax representstheDebyeenergy. Inthecon- The symbols appearing in the formulas (4) and (5) denote
sidered case the effective pairing potential is positive, which respectively: (CS) the specific heat of the superconducting
allowstheformationofthesuperconductingcondensate. The state, (CN) the specific heat of the normal state, and (HC)
fundamental equation of the BCS theory for the order pa- thethermodynamiccritical field. Itshouldbenotedthatthe
′
rameter (∆ ≡ V k hc−k↓ck↑i) is derived directly from the predictions of the BCS theory quantitatively agree with the
Hamiltonian (1) using the mean field approximation to the experimentaldataonlywithinthescopeoftheweakelectron
P
- phonon coupling (λ 0.3).
≤
The Eliashberg formalism is the natural generalization of
theBCSmodel(explicitlycomplieswiththeelectron-phonon
∗Electronicaddress: [email protected] interaction). ThestartingpointofthetheoryistheHamilto-
†Electronicaddress: [email protected] nian, which models the linear coupling between the electron
2
and the phonon sub-system [5], [6]: should be calculated on the basis of the formula: K(z)
≡
2 ΩmaxdΩ Ω α2F(Ω). The symbol α2F(Ω) represents
H = Xkσ εkc†kσckσ+Xq ωqb†qbq (6) tthhRee0seole-ccatrlloendΩ2-E−plziha2sohnboenrginftuenractcitoionn,.whTihcheqfuunacnttiiotnatαiv2eFly(mΩ)odcealns
+ gk,k+qc†k+qσckσ b†−q+bq . bpeeridmeteenrtmoirnbedyreeitfehrerrinbgytoustihnegrtehsueltdsaotfatfhreomcaltchuelattuionnnseflroexm-
Xkσ (cid:16) (cid:17) the first principles. In the classical Eliashberg formalism the
Thesymbolωq representsthephononenergy,gk,k+q are the depairingelectron correlations wereincludedintheparamet-
matrix elements of the electron - phonon interaction. The ric manner: µ⋆(ωm) µ⋆θ(ωC ωm ), whereas µ⋆ is called
≡ −| |
operator b†q (bq) creates (annihilates) the phonon state with the Coulomb pseudopotential, the symbol θ is the Heaviside
the momentum q. Based on the operator (6) and using the function, and ωC denotes the cut-off frequency, which value
formalismofthethermodynamicGreenfunctionsoftheMat- is usually several times higher than the value of Ωmax. The
subara type it is possible to derive the Eliashberg equations Eliashbergequationscanbealsoderivedinthecase,inwhich
on theimaginary axis (i √ 1) [7], [8]: thefittingparameterµ⋆isnotpresent(mathematicallythisis
≡ − averytoughissue). Forthispurpose,boththeextendedHub-
∆nZn =πkBT M [K(iωn−iωm)−µ⋆(ωm)]∆m, (7) binartdheHapmapiletron[1ia0n] s[9h]oaunlddtbheeumseedth.oTdhoeftEhleiaasnhableyrsgisedqisucautsiosends
√ω2 +∆2
mX=−M m m ontheimaginary axisallow ustoprecisely calculate thecrit-
ical temperature and the free energy difference between the
and
superconducting and the normal state. They cannot, how-
M ever, be used to determine the exact physical values of the
K(iωn iωm)ωm
Zn =1+πkBT − Zm. (8) order parameter and the effective mass of the electron. For
m=−M √ωm2 +∆2m ωn thispurpose, theEliashberg equationsshouldbeanalytically
X
extendedinsuchawaythattheequationsinthemixedrepre-
The quantity ∆n ≡ ∆(iωn) denotes the order parameter, sentationcanbeobtained(determinedsimultaneouslyonthe
while Zn ≡Z(iωn) is the wave function renormalization fac- imaginary and real axis) [11]:
tor. The symbol ωn is the fermion Matsubara frequency:
ωn πkBT(2n 1). The values of the pairing kernel
≡ −
M
φ(ω+iδ) = πkBT [K(ω iωm) µ⋆(ωm)] φm (9)
− − ω2 Z2 +φ2
mX=−M m m m
+ iπ +∞dω′α2F ω′ f ω′ +fp ω′ ω φ ω−ω′+iδ
Z0 (cid:16) (cid:17)"h BE(cid:16) (cid:17) FD(cid:16) − (cid:17)i (ω ω′)2Z2(ω(cid:16) ω′+iδ) (cid:17)φ2(ω ω′+iδ)#
− − − −
q ′
+∞ ′ ′ ′ ′ φ ω+ω +iδ
+ iπ dω α2F ω f ω +f ω +ω ,
BE FD (cid:16) (cid:17)
Z0 (cid:16) (cid:17)"h (cid:16) (cid:17) (cid:16) (cid:17)i (ω+ω′)2Z2(ω+ω′+iδ) φ2(ω+ω′+iδ)#
−
q
M
i ωmZm
Z(ω+iδ) = 1+ πkBT K(ω iωm) (10)
ω − ω2 Z2 +φ2
m=−M m m m
X
+ iπ +∞dω′α2F ω′ f ωp′ +f ω′ ω ω−ω′ Z ω−ω′+iδ
ω Z0 (cid:16) (cid:17)"h BE(cid:16) (cid:17) FD(cid:16) − (cid:17)i (ω ω′)(cid:16)2Z2(ω (cid:17)ω′(cid:16)+iδ) φ2(ω(cid:17) ω′+iδ)#
− − − −
q ′ ′
iπ +∞ ′ ′ ′ ′ ω+ω Z ω+ω +iδ
+ dω α2F ω f ω +f ω +ω ,
ω Z0 (cid:16) (cid:17)"h BE(cid:16) (cid:17) FD(cid:16) (cid:17)i (ω+ω′)(cid:16)2Z2(ω+(cid:17)ω′(cid:16)+iδ) φ2(ω(cid:17)+ω′+iδ)#
−
q
wherethesymbolsfBE(ω)andfFD(ω)representrespectively tron - phonon interaction. All relevant issues have been dis-
the Bose - Einstein function and the Fermi - Dirac function. cussed on the example of the superconducting condensate in
Note that theorder parameter is defined as follows: ∆(ω) phosphor, which was subjected to the influence of the high
≡
φ(ω)/Z(ω). pressure.
In the remaining part of the work, we have described the
wayofusingtheEliashbergformalism todeterminethechar-
acteristics of the superconducting state induced by the elec-
3
Tab.I:Theselectedparametersdeterminingthepropertiesof
12
thehigh-pressure superconductingstate in phosphor.
10
Quantity Unit 20 GPa 30 GPa 40 GPa 70 GPa
K] 8
[C
T 6 TC K 6.39 8.45 8.05 5.4

4 m Ωmax meV 59.4 62.3 64.8 74
- 3
R
2 Cmca sc Cmmm sh µ⋆ 0.37 0.29 0.27 0.28
λ 0.795 0.771 0.739 0.676
0
0 20 40 60 80 100 120 140
ω meV 418.3 444.1 456.5 469.4
ln
p [GPa]
FIG. 1: The influence of the pressure on the value of the
critical temperature in phosphor [17].
withtherecentexperimentalresults[17]. Atthispoint,letus
mention the doctoral thesis of Nixon [26], which expects the
II. SUPERCONDUCTING PHASE IN
PHOSPHOR: THE STATE OF KNOWLEDGE growthofthecriticaltemperature(TC ∈h8.5,11iK)together
with the growing pressure (p 10,35 GPa). However, the
∈ h i
determined values of the Debye temperature are based only
Five structural phase transitions can be observed in phos- on the bulk modulus.
phorintherangeofthepressurefromnormalto262GPa. In
the normal conditions the black phosphor has the structure
Cmca,whichisstableuptothevalueofthepressureat5GPa
III. THERMODYNAMICS OF HIGH-PRESSURE
[12], [13]. The existence of the structure R3m proven above,
SUPERCONDUCTING STATE IN PHOSPHOR:
vanishes at the pressure of 11.1 GPa, going into the metallic
THE ELIASHBERG FORMALISM
phase (sc) [14]. In the range from 103 GPa to 137 GPa the
structureCmmmisstable[15]. Further,uptothepressureat
262 GPa, a simple structuresh has been observed. Whereas, Whenanalyzingthepropertiesofthesuperconductingstate
forthepressureatp>262GPa,thestabilityofthestructure in phosphor, we have taken into account the values of the
bcchasbeen noticed[15], [16]. Thesuperconductingstatein pressure equal to: 20 GPa, 30 GPa, 40 GPa, and 70 GPa
phosphorwasobservedforthefirsttimeaboutfiftyyearsago respectively, while the Eliashberg functions were determined
[18], [19]. It should be noted, however, that as of today the in [27]. It is worth noting that the required calculations for
exact dependenceof the critical temperature on thepressure the electron band structure, the phonon spectrum, and the
isnotfullyknownbecausethechangesofthevaluesofTC are electron-phononinteraction havebeenconductedinthefull
strongly correlated with theroad passed on the p-T diagram ab initio scheme. Unfortunately, the depairing electron cor-
[20],[21]. Theresultsfrom 1985suggestthatthevalueofthe relations have not been estimated in the same way. For that
critical temperature is equal to about 6 K (the structure sc) reason,thevaluesoftheCoulomb pseudopotentialhavebeen
andweaklydependsonthepressure[21]. Ontheotherhand, chosen on the basis of the newest experimental data related
the experimental data included in the work [22] indicate the tothecritical temperature [17] (see Table I). The Eliashberg
existenceoftwocharacteristicpeaksinthecourseofthefunc- equations on the imaginary axis have been solved for 1100
tionTC(p),whilethehighestvalueofthecriticaltemperature Matsubara frequencies (M = 1100). We have taken the ad-
is about 10 K (p=23 GPa). The latest results are quitedif- vantage of the numerical methods described and used in the
ferentandsuggesttheexistenceofthesinglemaximumofthe works: [28], [29], [30], [31], [32], and [33]. The functions ∆n
critical temperature (TC = 9.5 K) located in the vicinity of and Zn are stable for the temperature higher than T0 = 1.5
the pressure at 32 GPa, which is presented in Figure 1 [17]. K. It is assumed that the cut-off energy is equal to 5Ωmax,
Referring to the theoretical predictions, it should be noted where the exact values of the Debye energy are collected in
thatingeneraltheyareobtainedbytheuseofthesignificant Table I.
approximations. For example, in the paper [23] there is the In the first step we have calculated the physical values
evidenceofthemaximumTC located nearthesecondexperi- of the Coulomb pseudopotential corresponding to the given
mentalmaximumofthecriticaltemperatureindicated inthe pressure. For this purpose, the following condition has been
publication [22]. However, this theoretical work completely used: [∆n=1(µ⋆)]T=TC =0,where∆n=1 representsthemax-
left outtheeffectofthepressureonthephononspectrum. A imum value of the order parameter. Figure 2 presents the
similar approach in the work [24] caused on incorrect defini- dependence of the order parameter on the Coulomb pseu-
tion of the function TC(p). Relatively new results [25] sug- dopotentialat thecritical temperature. The obtainedresults
gest a very weak dependence of the critical temperature on provethat µ⋆ takes veryhigh values (see Table I) in relation
the pressure - which is in agreement with the experimental to the value generally taken into account in the calculations
data presented in [21]. However, this is in a sharp contrast (µ⋆ 0.1). It should be noted that this situation is often
∼
4

8 8
1/2
p=20 GPa 12 The Eliashberg equations Hc/( (0)) [meV]
p=30 GPa The experimental data 2
6 p=40 GPa 9 TThhee MAlcleMni-lDlaynn feosrm fourlmaula F/ (0) [meV]
p=70 GPa T [K]C6 4 4
V]
e 4 3
m

[1 0 0 0
n= 20 40 60 80
p [GPa]
2 20 GPa 30 GPa

2 4 6 8 10 2 4 6 8 10
8 8
0
0.0 0.1 0.2 0.3 0.4
*
4 4
FIG. 2: The maximum values of the order parameter as a

function of the Coulomb pseudopotential. The exact values
of the critical temperature have been collected in Table I.
High values of µ⋆ mean that the critical temperature cannot 0 0
be estimated on the basis of the McMillan or Allen - Dynes
expression [37], [38] (see thefigures insert). 40 GPa 70 GPa
2 4 6 8 10 2 4 6 8 10
T [K] T [K]
observed in the analysis of the high-pressure superconduct-
ing state. For example, for lithium the physical value of the FIG.3: Thefreeenergydifferencebetweenthesuperconduct-
Coulomb pseudopotential is equal to: [µ⋆] = 0.36 ingandthenormalstateasafunctionofthetemperature(the
p=29.7GPa
[34]. The anomalously high values of the Coulomb pseu- lower panels). The thermodynamic critical field - the upper
dopotentialcan beexplainedwhenanalyzing theinfluenceof panels.
the retardation effects on the value of the non-renormalized
Coulomb pseudopotential (µ Uρ(0)) in the second order
ofµ,whereU d3r1d3r2 Φ≡i(r1) 2VC(r1 r2) Φi(r2) 2,
≡ | | − | |
Φi(r) is the Wannier function. In the considered case the
retardation effecRtsRlead to the reduction: µ µ⋆, but not
→
as large [35] as it was predicted by the classical Morel and S
Anderson theory, which was limited to the linear order with 20 20 GPa C/ kB (0) [meV]
N
respect to µ [36]. 10 C/ kB (0) [meV]
In the Eliashberg formalism the free energy difference be-
tween the superconducting and the normal state should be 0

calculated on the basis of theformula [39]: 2 3 4 5 6 7 8 9 10
20 30 GPa
M
∆F = −2πkBTρ(0) [ ωn2 +(∆n)2−|ωn|] (11) 10
nX=1q 0
× [Zn(S)−Zn(N) ωn2|+ωn(|∆n)2]. 20 420 GPa3 4 5 6 7 8 9 10
q 10
In the next step, the thermodynamic critical field (HC =
√ 8π∆F) has been calculated, as well as the difference in 0
th−especificheatofthesuperconductingandthenormalstate: 2 3 4 5 6 7 8 9 10
∆C = CS −CN = −Tdd2T∆2F, where: CN = γT. The Som- 20 70 GPa
merfeld parameter is equal to: γ 2π2k2ρ(0)(1+λ). The
electron - phonon coupling consta≡nt 3shoulBd be calculated on 10
the basis of the formula: λ 2 ΩmaxdΩα2F(Ω) (Table I). 0
≡ 0 Ω
The lower panels in Figure 3 show the full form of the func- 2 3 4 5 6 7 8 9 10
R
tion ∆F(T). It can be very clearly seen that the free en- T [K]
ergy difference takes negative values in the whole range of
the temperature, up to TC, which is the evidence of the FIG. 4: The specific heat of the superconducting state and
thermodynamic stability of the superconducting phase. It thenormal state as a function of thetemperature.
shouldbenotedthatinthecaseofphosphorthelowest value
of the free energy difference has been obtained for 30 GPa
5
culated in the last step. The obtained results prove that the
10 10
Re [ ] T=1.5 K T=4.5 K valuesdonotsignificantly differfrom thevaluespredictedby
8 Im [ ] 8 the BCS theory. The biggest derogations of several percent
6 6 have been found for the pressure at 30 GPa. Let us note
V]
me4 4 that the result above is related to the insignificant strong -
[2 2 coupling and retardation effects, which are characterized by

0 0
1.5 1.80
20 GPa
-2 -2
0 10 20 30 40 50 60 0 10 20 30 40 50 60 30 GPa
108 T=6.25 K 108 T=8.0 K V]1.0 4700 GGPPaa 1.75
me me
meV]46 46 [n=1 * m/e
[2 2 0.5 1.70 20 GPa
0 0 30 GPa
40 GPa
-2 -2 70 GPa
0 10 20 30 40 50 60 0 10 20 30 40 50 60 0.0 1.65
[meV] [meV] 2 4 6 8 10 2 4 6 8 10
T [K] T [K]
FIG. 5: The order parameter on the real axis for the se-
lected valuesofthetemperature(p=30 GPa). Additionally, FIG.6: Thedependenceoftheorderparameterandtheelec-
theshape oftherescaled Eliashberg function (6α2F(ω))was tron effective mass on thetemperature.
plotted.
the ratio: r kBTC/ωln. Within the of BCS, the Eliash-
(∆F(T0)/ρ(0) = 1.4 meV2), while the highest has been berg equation≡s predict: r 0. For p = 30 GPa, it was
obtained for 70 GP−a (∆F(T0)/ρ(0) = −0.47 meV2). From obtained: r = 0.02. The qu→antity ωln is called the logarith-
the physical side the obtained result is related to the values micfrequency,andshouldbedeterminedwiththehelpofthe
of the electron - phonon coupling constant and the Coulomb formula: ω exp 2 ΩmaxdΩα2F(Ω)ln(Ω) . The valuesof
pseudopotential. The course of the function ∆F(T) directly ln ≡ λ 0 Ω
determines the thermodynamic critical field and the specific thelogarithmic freqhuenRcyhave been collectediin Table I.
heat. TheresultsarepresentedintheupperpanelsofFigure
3 and Figure 4, respectively.
Thephysicalvalueoftheorderparametershouldbecalcu-
IV. SUMMARY
lated by using the equation: ∆(T)= Re[∆(ω=∆(T),T)],
while the form of the order parameter on the real axis has
been determined by solving the Eliashberg equations in the The presented work discusses the Eliashberg formalism,
mixed representation. The exemplary courses have been col- whichisusedforthequantitativedescriptionofthethermody-
lectedin Figure5. Whenanalyzingthebehavioroftheorder namic properties of the superconducting condensate induced
parameter on the real axis, the attention should be paid to by the electron - phonon interaction. The detailed consid-
thefactthatforlowfrequenciesthenon-zerovaluesaretaken erations have been illustrated on the example of the super-
onlybytherealpartofthefunction∆(ω). Fromthephysical conducting state in phosphor underthe influenceof the high
point of view, this provesthe existence of the infinitely long- pressure. With respect to the considered system it has been
lived Cooper pairs. At the higher frequencies the non-zero foundthatthecritical temperatureis relatively low, whichis
is also the imaginary part of the order parameter, which, of connected with not very high valuesof theelectron - phonon
course,determinesthefinitelifetimeoftheelectronpairs. At coupling constant and the significant depairing Coulomb in-
this point it is worth mentioning that the particularly large teractions. Additionally, it has been shown that the strong -
changes in the values of the order parameter are very closely coupling and retardation effects do not cause the significant
correlatedwiththedistinctivegroupofpeaksoccurringinthe derogationsofthevaluesofthedimensionlessratiosR∆,RC,
course of theEliashberg function. and RH from thevalues predicted by theBCS theory.
Then,theelectroneffectivemass(m⋆)hasbeencalculated Letuspointourattentiontoward thefact thattheEliash-
e
on thebasis of theformula: m⋆e =Re [Z(T)]ω=0 me, where berg formalism discussed in the presented work is the main
the symbol me denotes the electron band mass. The values tool used for the description of the high - temperature su-
(cid:2) (cid:3)
of the order parameter and the electron effective mass have perconducting state induced by the electron - phonon. In
been plotted in Figure 6. particular, it can be used for the quantitativeanalysis of the
The dimensionless ratios R∆, RC, and RH have been cal- superconductingstate in H2S and H3S [40], [41], [42], [43].
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