Comment on "Unified Formalism of Andreev Reflection at a Ferromagnet/Superconductor Interface" by T.Y. Chen, et al

0.08 MB·English

by  M. Eschrig

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Comment on “Unified Formalism of Andreev the F/S interface) and conserved [10] 2D k , then, af-
k
Reflection at a Ferromagnet/Superconductor In- ter the usual quantization of k , one finds that the total
k
terface” by T. Y. Chen, Z. Tesanovic, and C. L. number of states available for transport in the x direc-
Chien tion (i.e., the number of the conductivity channels) for
A recent paper of Chen et al. [1], claims to have de- a given spin direction is proportional to the total area
rived an allegedly, previously unavailable “unified” An- of the Fermi surface projection on the interface plane,
dreev Reflection (AR) formalism for an arbitrary spin givenby the Fermi surface averageof the Fermi velocity,
polarization P that recovers earlier results as its special n=hN(EF)vFxi[3]. Aftersummationoverallelectronic
3 limits. In this Comment we show that, contrary to this states the total current turns out to be a linear combi-
1 claim, there are numerous works correctly solving the nation (with the weights defined by P), of the solutions
0 problem formulated in Ref. [1] for an arbitrary P [2–7], of the 1D BTK model with P = 0 and P = 1. This was
2 while Ref. [1] fails to correctly incorporate P 6=0 effects notpostulated,butderived,innumerouspapers(see the

n and violate basic physical principles. discussion in Ref. 7).
a In the P = 0 limit the approach of Ref. [1] is identi- In contrast to the “universal” P in the Ref. [1], inde-
J
cal to the one dimensional (1D) BTK model [8] for non- pendent of electronic mass, Fermi wavevectors, or any
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magnetic metals, while for P 6= 0 the only difference is band structure at all, the real transport spin polariza-
1
postulating an AR wavefunction that has an additional tion for AR spectroscopy depends on the overall Fermi

] evanescent wave contribution parameterized by a decay surface properties. Moreover, there is no unique spin
n 0 polarization (even for a uniform bulk metal), as it de-
o constant α, ψAR = a exp[(α + i)q+x], with α =
(cid:26)1(cid:27) pends on an experimental probe. In fact, the definition
c
- 2 P/(1−P). However, this wavefunction violates the usedinRef.[1](neglectingtheFermivelocities)doesnot
r
p chparge conservation: the corresponding charge current correspond to the AR experiments, but rather to spin-
u density is jAR(x)∝Im[ψA∗R∇ψAR]∝q+|a|2exp[2αq+x]. polarized photoemission.
s In the F-region, for any 0 < α < ∞, i.e., 0 < P < 1, To summarize, Ref. [1] has misinterpreted or ignored
.
t the divergence of the total current is finite and thus the previous works where the posed problem has been cor-
a
m total charge is not conserved (in particular, for 0 < x . rectly solved for an arbitrary P, and attempted to sup-
- 1/2αq+), signaling that ψAR is unphysical. Even for the planttheexistingsolutionwithanincorrectformula,pos-
d P =1half-metal(HM)state,wheretheaboveexpression tulatinganunphysicalwavefunctionforthereflectedelec-
n correctlygivesjAR =0,itisstillincorrect. ForaHMthe tron,whichdoesnotconservechargeandhasanincorrect
o
AR electron decays with a finite (not infinitely small, as HM limit. They proceeded by calculating the current
c
[ postulated in Ref.[1]) penetration depth, which depends due to ψAR at x = 0, even though the actual current is
on the electronic structure details, primarily the size of measuredfar awayfromthe interface (where they would
1
the gap in the nonmetallic spin channel [9]. have gotten zero Andreev contribution for any P and
v
1 TherationaleforinventinganewψAR [1]wasbasedon grave disagreement with the experiment). Furthermore,
1 the premise that: (a) the result in Ref. [3] was derived they have arbitrarilydecided that the penetration depth
5 only in the extreme case of a fully-polarized HM with foranelectroninsidethebandgapinthetransport-spin-
3 0 minority channel is uniquely defined by the spin polar-
. ψ˜AR = a exp(κx) and a nonmagnetic metal with
1 (cid:26)1(cid:27) ization(these two quantities are in fact unrelated). As a
0 0 result, they arrived at a formula that for their own ex-
3 ψ˜AR =a(cid:26)1(cid:27)exp(ikx), while postulating that for an in- perimentaldataprovidesafitthatisessentiallyidentical
1
termediate P the current will be a linear combination of tothatobtainedbyusingRef.[3](the differenceisbelow
:
v these cases, (b) no prior work had treated a 0 < P < 1 potential errors introduced by the BTK approximations
i
X case,and(c)onecanmeaningfullydefineP ofanindivid- of a step-shaped pairing potential and spin-independent
ual electron. Regarding (a-b), the derivation in Ref. [3], δ-shaped barrier [11]).
r
a aswellasinotherworks[4–7],isrigorousforanarbitrary Finally, we note that the inclusion of inelastic scat-
P. The lastpoint, (c)is particularlymisleading. Infact, tering using a phenomenological finite Γ 6= 0, another
one cannot define a BTK model in 1D with an arbitrary claimed novelty of Ref. [1] has already been previously
P becauseanygivenelectroninametalAndreev-reflects employed for AR with arbitrary P [6, 12].
either into a propagating, or into an evanescent wave. It may be that the formulas contrived in Ref. [1] fit
Finite P simply means that some current-carrying elec- a particular experiment. However, there is a maxim at-
trons belong to the former group and the others to the tributedto Niels Bohr,thatthere exists aninfinite num-
latter; one can only define transport spin polarizationin berofincorrecttheoriesthatcorrectlydescribethe finite
a multielectron system, where the numbers of electron number of experiments.
states at the Fermi surface (conductivity channels) for M. Eschrig1, A. A. Golubov2, I. I. Mazin3, B.
the two spin directions differ. If the electron wavevec- Nadgorny4, Y. Tanaka5, O. T. Valls6, Igor Zˇuti´c7
tor k is decomposed into a non-conserved kx (normal to 1 Department of Physics, Royal Holloway, University of London,
2
Egham, Surrey TW20 0EX, UK, 2Faculty of Science and Tech- Buhrman, Phys. Rev.B 75, 094417 (2007).
nology and MESA+ Institute of Nanotechnology, University of [7] R. Grein, T. Lofwander, G. Metalidis, and M. Eschrig,
Twente, The Netherlands, 3Naval Research Laboratory, Washing- Phys. Rev.B 81, 094508 (2010).
ton, DC 20375, USA, 4Department of Physics and Astronomy, [8] G. E. Blonder and M. Tinkham and T. M. Klapwijk,
Phys. Rev.B 25, 4515 (1982).
Wayne State University, Detroit, MI 48201, USA, 5Department
[9] Ref. [3], aswell as someothers, eventuallyusesthelimit
of Applied Physics, Nagoya University, Aichi 464-8603, Japan, of the infinitely small penetration depth, because the fi-
6School ofPhysics andAstronomy, UniversityofMinnesota, Min- nal result depends on this parameter very little, but it
neapolis, MN 55455, USA, 7Department of Physics, University at is important to use in the derivation a physically mean-
Buffalo,SUNY,Buffalo,NY14260, USA ingful wavefunction that allows for a finite penetration
depth.ThisinconsistencyinRef. [1]isapartofabigger
problemofusingthesamekF forbothspinchannels,de-
spitethefactthatasP →1oneofthetwok± gradually
F
vanishes [2, 4, 5, 7].
[10] For a ballistic flat interface. The generalization onto a
[1] T. Y. Chen, Z. Tesanovic, and C. L. Chien, Phys. Rev.
Lett.109, 146602 (2012). diffusive case, where only the total number of the con-
ductivity channels for each spin direction, but not the
[2] M. J. M. de Jong and C. W. J. Beenakker, Phys. Rev.
Lett.74, 1657 (1995). individual kk are conserved, is straightforward [3].
[11] For other parameters [see theirFig. 2(c)] their approach
[3] I. I. Mazin, A. A. Golubov, and B. Nadgorny, J. Appl.
Phys. 89, 7576 (2001); G. T. Woods, R. J. Soulen, Jr., leads to unphysical kinks at zero bias and unphysical
notches near thegap voltage.
I. I. Mazin, B. Nadgorny, M. S. Osofsky, J. Sanders, H.
Srikanth,W.F.Egelhoff,andR.Datla,Phys.Rev.B70, [12] OtherauthorsobservedthatinsteadofaddingaΓ,atan
extra computational cost, one can simply artificially in-
054416 (2004).
crease the temperature in the original expressions, since
[4] S.Kashiwaya,Y.Tanaka,N.Yoshida,andM.R.Beasley,
Phys.Rev.B 60, 3572 (1999). the effect of the two types of broadening on the con-
[5] I. Zˇuti´c and O. T. Valls, Phys. Rev. B 61, 1555 (2000), ductance curves is essentially the same, see, e.g.,Y. Bu-
see Eq. (2.8); I. Zˇuti´c and S. Das Sarma, Phys. Rev. B goslavsky,Y.Miyoshi,S.K.Clowes,W.R.Branford,M.
60, R16322 (1999). Lake, I. Brown, A. D. Caplin, and L. F. Cohen, Phys.
Rev. B 71, 104523 (2005)
[6] P. Chalsani, S. K. Upadhyay, O. Ozatay, and R. A.