Charge Kondo anomalies in PbTe doped with Tl impurities

0.17 MB·

by  T. A. Costi

Charge Kondo anomalies in PbTe doped with Tl impurities
T. A. Costi1 and V. Zlati´c1,2,3
1Peter Gru¨nberg Institut and Institute for Advanced Simulation,
Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany
2Institute of Physics, HR-10001 Zagreb, Croatia
3J. Stefan Institute, SI-1000 Ljubljana, Slovenia
(Dated: January 19, 2012)
WeinvestigatethepropertiesofPbTedopedwithasmallconcentrationxofTlimpuritiesacting
asacceptors and described byAnderson impurities with negativeonsite correlation energy. Weuse
2 the numerical renormalization group method to show that the resulting charge Kondo effect natu-
1 rally accounts for the unusual low temperature and doping dependence of normal state properties,
0 including the self-compensation effect in the carrier density and the nonmagnetic Kondo anomaly
2 intheresistivity. Thesearefoundtobeingoodqualitativeagreementwithexperiment. Ourresults
for theTl s-electron spectral function providea new interpretation of point contact data.
n
a
PACSnumbers: 71.27.+a,72.10.Fk,72.15.Qm
J

8
1 Introduction.— PbTe is a narrow gap semiconductor tures: whileforx<xc,theresidualresistivity,ρ0,isvery

with a band gap of 190meV at zero temperature [1]. smalland almostconstantas a function of x, for x>x ,
] c
el Upon doping with Tl impurities [1, 2] a number of strik- ρ0 increases linearly with x [9]. For x<xc, the resistiv-
- ing anomalies appear in the low temperature properties. ity, ρ(T), exhibits a positive slope at low temperature,
r These anomalies,described below, arecurrently not well while for x > x , the slope is negative and a Kondo-like
t c
s understood. They are believed to be due to the spe- contribution, ρ (T), is observed for T . 10K [6, 9].
. imp
t cial nature of Tl impurities in PbTe, which have been The origin of this anomaly is not due to magnetic impu-
a
m proposed to act as negative U centers, where U < 0 is rities, since the susceptibility is diamagnetic [6].
anattractiveonsite Coulombenergy. Incompounds, the
- In this Letter, we show that the unusual concentra-
d outer6sshellofTlcanbeempty,singlyoccupiedordou-
tion and low temperature dependence of a number of
n
bly occupied. A negative Coulomb correlation results in
o normal state properties of Pb1−xTlxTe can be naturally
thenonmagneticemptyanddoublyoccupiedstatesbeing
c explained within a picture of dilute Tl impurities acting
[ lower in energy than the magnetic singly occupied state.
asnegativeU centersinthePbTehost. Suchadiluteim-

Hybridizationofthes-stateswiththevalencebandstates
2 purity description is suggested by the observed concen-
of PbTe can then result, via dynamic valence fluctua-
v tration dependence of many properties, as summarized
4 tions, in a (nonmagnetic) charge Kondo (CK) analogue above, e.g., the linear dependence on x of the residual
2 [3, 4] of the conventional spin Kondo effect [5]. This
resistivity for x > x . We model the Tl impurities by
8 nonmagneticCK effecthasbeenarguedto explainmany c
the negative U Anderson model [14] and solve this for
1
. of the observed properties of Pb1−xTlxTe [6–10] which thestaticanddynamicpropertiesbyusingthenumerical
9 wouldthenconstitutethefirstphysicalrealizationofthis
renormalization group (NRG) method [15]. Our results,
0 effect.
1 withcomparisonstoexperimentaldata,andpreviousex-
:1 A remarkable feature of Pb1−xTlxTe is that the low ppreorivmideentsatrloanngdetvhiedoernecteicathlawtotrhkeonCKPbe1ff−excTtl[x3T]eis[6r,e7a,li1z6e]d,
v temperaturepropertiesdependsensitivelyontheTlcon-
i in this system.
X centration, with a qualitatively different behavior below
and above a critical concentration x ≃ 0.3 % Tl. For The idea that doping PbTe with Tl induces resonant
r c
a example, while Pb1−xTlxTe remains metalliclike down statesinthe valencebandofPbTewasconjecturedearly
to the lowest temperatures for x < x , for x > x it on(seeRef.1). ThesestatesaremainlyofTls-character
c c
becomes superconducting with a transition temperature and lie close to the top of the valence band, as re-
T (x) increasing linearly with x and reaching 1.5 K at cently shown by density functional theory calculations
c
x=1.5 % Tl [6, 11]. This is surprisingly high given the [17]. Resonant states alone, however, and generaliza-
low carrier density of less than 1020 holes/cm3. In ad- tions ofthis to animpurity bandofresonantstates,can-
dition, measurements of the Hall number p = 1/R e not explain properties such as the superconductivity of
H H
[8, 12] indicate that the number of holes grows linearly Pb1−xTlxTe. Forthis,acouplingofTlionstothelattice
with x for x < x , whereas for x > x the number of [18], or a static mixed valence model [19] have been pro-
c c
holesremainsalmostconstant: thesystemexhibits“self- posed. In the latter, Tl impurities, known to be valence
compensation” and the chemical potential is pinned to skippers in compounds, are assumed to dissociate into
a value µ = µ∗ ≃ 220meV [12, 13]. Transport mea- energetically close Tl1+ (6s2p1) and Tl3+ (6s0p3) ions,
surementsalsoshowanomalousbehavioratlowtempera- whiletheTl2+ (6s1p2)configurationlieshigherinenergy
2
[20]. In the highly polarizable PbTe host, this can re- terimpurity interactions and solve H for a collection of
sult in negative on-site U and provides a mechanism for singleindependentnegative-U centersbyusingthe NRG
superconductivity and a qualitative explanation for the [15]. ForeachxandeachT wesatisfytheaboveequation
observed self-compensation, chemical potential pinning byself-consistentlydeterminingthechemicalpotentialµ
e
and the diamagnetic behavior of Pb1−xTlxTe [19]. For (or equivalently the hole chemical potential µ, which we
a more quantitative explanation of the observed anoma- henceforth use).
lies, and in order to explain the Kondo anomalies in the The electricalresistivity of electrons scattering from a
resistivity, a more realistic model is needed, which in- dilute concentration x of Tl impurities is obtained from
cludes dynamic fluctuations between the Tl1+ and Tl3+ the usual expression, ρimp(T) = 1/e2L11, where L11 is
valence states. This motivates our use of the negative the static limit of the current-current correlation func-
U Anderson model [14], as formulated for Pb1−xTlxTe tion. Intheabsenceofnonresonantscatteringthevertex
in Ref. 7 and discussed as a model for this system in correctionsvanishandtherelevanttransportintegralL11
Refs.6–10,16,21. Forcompleteness,wementionalsothe can be written as [23],
valencebandmodelforPbTe,inwhichthemaineffectof
Tl doping is assumed to be a rigid shift of the chemical ∞ ∂f(ω)
L11 =σ0 dω − N(ω)τ(ω,T), (1)
potentialintothevalenceband. Thismodelmayberele- Z (cid:18) ∂ω (cid:19)
−∞
vant for the transport properties of PbTe doped with Tl
impurities [22] at high temperature (T > 300K), where where σ0 = hvk2Fi is the velocity factor vk2 averaged over
the charge Kondo effect is suppressed. It fails, however, the Fermi surface, f(ω) = 1/[1 + exp(ω/k T)] is the
B
to describe the low temperature anomalies that we are Fermi function, τ(ω,T) is the conduction-electrontrans-
addressing in this Letter, e.g. the Kondo upturn in the porttime[23]τ(ω,T)−1 =2nTlV02A(ω,T)andA(ω,T)=
resistivity at T <10K for x>xc. −π1ImG(ω+i0+) is the spectral function of s-electrons.
Model and calculations.— We consider n Tl impuri- The number of Tl impurities nTl per cm3 is related to x
ties in a PbTe crystal with N Pb sites described by the in percent by nTl =1.48×1020x [24].
Hamiltonian H =H +H +H . The first term, Choice of model parameters.— We use µ∗ ≈225meV,
band imp hyb
Hbabnadndof=PPbTkeσ,(wǫkhe−reµµee)ci†ksσtchkeσ(,edleecstcrroinb)eschtehmeivcaallepnocteenp-- c[1lo2s,e1t3o].thOetvhaelruepaorbatmaienteedrs,frosumchtuansne∆li0ng=ex∆p(eµri∗m),enrtes-
tial andc†kσ creates anelectronnwith energy ǫnk. The sec- aqrueireladrgteolyfixuntkhneowhynb.ridOizuartiionntefrupnrcettiaotnion∆(oωf)t,haendtunU-
ondterm,Himp =(ǫ0−µe) i=1σnˆisσ+U i=1nis↑nis↓, neling spectra (see discussion of Fig. 2 below), suggests
describes the Tl impuritiesP, where n =Ps† s is the
isσ iσ iσ U ≈ −30meV. The measured Kondo-like resistivity for
number operator for a Tl s-electron at site i with spin σ
x≥xc at low temperatures requires that |U|≫∆0. We
aTnhdeelansetrgteyrǫm0,,aHnhdybU=isthnie=1(negkaσtViv0e()c†kcσorsriσel+athio.cn.)e,nmerogdy-. ttaemkepUer/a∆tu0re=T−K11≈w1i.t2h3K∆0b=elo2w.7tmheeVhitgoheysitelTdca≈K1o.n5dKo
elsthehybridizationofPTls-sPtateswiththevalenceband at x = 1.5% ≫ x , where T is defined via the im-
c K
p-states and V0 is the matrix element for the s-p inter- purity resistivity ρ (T = T ) = 0.5ρ (T = 0) for
imp K imp
action. Its strengthis characterizedbythe hybridization
x ≫ x . The overall qualitative aspects of our results
function ∆(ω)=πV02 kδ(ω−ǫkσ)=πV02N(ω), where remainecd the same for values of U in the range 10meV
we retain the full enerPgy dependence of the p-band den- ≤|U|≤220meV and |U|/∆0 ≫1.
sity of states N(ω)= kδ(ω−ǫk) of PbTe [22]. Qualitative considerations.— It is instructive to first
The chemical poPtential µ determines n = makesomequalitativeremarks,startingfromtheatomic
e e
N1 kσhc†kσckσi and ns = n1 ni=1 σhnisσi, the aver- limit V0 = 0 [7]. For x = 0 the chemical potential lies
agePnumber of p and s electroPns perPsite. We denote by in the gap between the valence and conduction bands.
x = n/N the concentration of Tl impurities. Since Tl For finite but very small x each Tl impurity accepts one
acts as an acceptor, the ground state corresponds to the electron, i.e. ns ≈ 2 and n0 = x(ns−1) ≈ x grows lin-
Tl1+ (n = 2) configuration and the Tl3+ (n = 0) con- early with x. At the same time, the chemical potential
s s
figurationissplit-offfromthegroundstatebytheenergy shifts downwards into the valence band µ < E , where
v
3+ 1+
δ =E(Tl )−E(Tl )>0. AconcentrationxofTlim- E denotesthetopofthevalenceband. Thisimpliesthat
v
purities accommodates x(ns−1) electrons (per Tl site), the splitting δ(µ)=−(2(ǫ0−µ)+U)betweendonorand
where the number of accepted electrons in the 6s level acceptor configurations decreases. Eventually, at a criti-
of Tl is measured relative to the neutral Tl2+ (s1) con- cal concentration x=x∗, the chemical potential reaches
figuration having ns = 1. These electrons are removed µ = µ∗ = ε0 +U/2 where δ(µ) = 2(µ −µ∗) = 0 and
from the valence band leaving behind n0 =1−ne holes. the system is in a (static) mixed valence state where the
Thus,chargeneutralityimpliesn0 =x(ns−1) [7],which Tl1+ andTl3+ configurationsaredegenerate. Inthis sit-
for a given x and temperature T has to be satisfied by uation n = 1, and further doping cannot increase the
s
adjusting the chemicalpotential µe. Here,we neglectin- holecarrierdensitybeyondthevaluen0(µ∗),i.e. onehas
3
one hole/Tl (a) (b)
TT == 11..88 KK ((tMhaetosruys)hita et. al.) 100a.u.] a.u.]0.2 n0 [1019/cm3]5 =.74
T = 77 K (theory) 0) [ 0) [ 6.74
1 T = 77 K (Murakami et al.) u.] 50 A(E,T= N(E,T= 0.1 666...999367
a.
3m] =0) [ -2 0 2 -0100 0 100
20n [10/c00.5 300 µ* = 225 meV A(E,T100 E-EF [meV] |U|/2 E-EF [mexV ][%]0 =.4
eV]200 50 00..56
m 0.8
µ [100 TT= =1 .787 K K (a) 1.3
0 0
0 0.5 -30 -20 -10 0 10 20 30
00 0.5 1 x [%] 1.5 E-EF [meV]
x [%]
FIG. 2: (Color online) Tl spectral function A(E,T =0) ver-
FIG.1: (Color online)Holecarrierdensityn0 versusTldop- susE−EF forarangeofTldopingsx. Inset(a): regionnear
ingxin %forT =1.8KandT =77 K.Filledcircles: exper- E =EF showing the CK resonance. Inset (b): PbTe valence
imental data at T = 77 K [12], filled squares: experimental band density of states N(E) versus E−EF for x as in the
data at T = 1.8K [8], dot-dashed line: expected n0 for one main panel [22] (legend: hole densities n0 for each x).
hole per Tl. Inset (a): hole chemical potential µ versus x at
T =1.8K and T =77K [and henceδ(µ)=2(µ−µ∗))].
tuate between Tl1+ and Tl3+ so the average valence is
Tl2+,whichcorrespondstonoadditionalelectronsbeing
self-compensation with a pinning of the chemical poten- accepted or donated. The CK state may be destroyed
tial to µ∗ [7]. by lifting the degeneracy of the pseudospin states, i.e.
ForfiniteV0,quantumfluctuationsbetweenthedegen- in the language of the spin Kondo effect, by applying a
erate states Tl1+ and Tl3+ at µ=µ∗ become important “magneticfield”. A“magneticfield”intheCKeffectcor-
and lead to a CK effect. This significantly affects all respondsto dopingorshifting µ. This hasbeenachieved
staticanddynamicpropertiesandneedstobetakeninto by counterdoping with In ions [10], which act as donors.
account in describing the experiments. It is also impor- The Kondo anomalies, e.g. in the resistivity, were ob-
tant for µ > µ∗, since a finite charge splitting δ(µ) > 0 served to vanish, providing further support to the CK
in the negative-U Anderson model is similar to a Zee- picture. The self-compensationeffect canbe seen also in
man splitting in the conventionalspin Kondo effect [25]. the pinning of the hole chemical potential µ for x > x∗,
The latter is known to drastically influence all proper- shown in Fig. 1a. For x < x∗, µ grows nonlinearly with
ties [5]. Thus, for the whole range of concentrations x, x and rapidly approaches the value µ∗ = 225meV for
one expects fluctuations to playanimportantrolein the x>x∗, both at T =1.8K and at T =77K.
properties of Pb1−xTlxTe. Figure 2 shows the Tl s-electron spectral function
Numerical results.— Figure 1 shows n0(x) versus x at A(E,T = 0) at zero temperature and different dopings
T =1.8KandatT =77Kandacomparisonwithexper- x. For small doping, x<x∗, the hole chemical potential
imental data on Hall number measurements [8, 12]. At µ lies above µ∗ within the shallow part of the valence
lowdopings,n0 islinearinxbothintheoryandinexper- band density of states N(E) (see Fig. 2b), consequently
iment, as expected for Tl impurities acting as acceptors the splitting δ(µ)=2(µ−µ∗), see Fig.1a, is large. Such
(dot-dashed curve in Fig. 1). However, the efficiency, a large splitting acts like a large Zeeman splitting in the
n0(x)/nTl(x), of Tl dopants in supplying holes at low x conventional positive U Anderson model and polarizes
is only around65 % in the data of Ref. 12 as opposed to the spectralfunction so that its weightlies mostly below
100%inourmodelcalculationsandinthedataofRef.8. the Fermi level E [4]. For x≥x∗, µ approachesµ∗ and
F
At higher dopings n0(x) saturates rapidly with increas- a CK effect develops. The spectral function develops a
ing x for T = 1.8K and more slowly at higher tempera- sharp asymmetric Kondo resonance close to, but below
tures. The theoretical crossoverfrom linear to saturated E (see Fig. 2a) and an upper Hubbard satellite peak
F
behavior occurs at x=x∗ ≈0.5%, largerthan the value appears above E . Early tunneling experiments for low
F
x ≈ 0.3% for the onset of superconductivity. The the- dopings x < 0.3% showed only one resonant level be-
c
oretical saturation density n0 ≈ 0.7×1020/cm3 is close low EF [13], whereas more recent tunneling experiments
to the experimental value [12]. The self-compensation for x > 0.6% show two “quasilocal” peaks [12], a nar-
effect for x ≫ x∗ ≈ 0.5% is a characteristic signature row one of width 6meV close to E and a broader one
F
of the CK state: on entering this state the Tl ions fluc- of width 12meV at a nearly constant energy 13-15 meV
4
1 Tl impurities acting as negative U centers. Our NRG
1.5 static calculations supportthe suggestionthat the CK effect is
0.8 ρ0 1 Tl roefalloizwedteimnpPerba1t−uxrTelaxnToem[6a,lie7s].ofTPhbey1−exxTpllxaTine,ainnculumdbinegr
2.00 (1.33 K) 0.5 (a) the qualitatively different behavior below and above the
0.6 critical concentration x∗, where x∗ ≈ 0.5% is close to
u 1.30 (1.57 K) 0
ρ /p 0.80 (2.84 K) 0 0.5 x [1%] 1.5 xc ≈0.3%fortheonsetofsuperconductivity. Atx=x∗,
m
ρi0.4 0.60 (8.74 K) two nonmagnetic valence states of Tl become almostde-
generateandthe ensuing pseudospinCKeffectresultsin
0.50
aKondoanomalyintheresistivityforx>x∗andaresid-
0.40
0.2 0.30 ual resistivity approximately linear in x. Our results for
thesequantitiesandthecarrierdensityn0(x)areingood
qualitative agreementwith experiments [6, 8, 9, 12]. For
0.01 0.1 1 10 the Tl s-electron spectral function, we predict that one
T [K]
peakshouldbe presentfarbelowE forx<x∗ andthat
F
a second temperature dependent Kondo resonance peak
FIG.3: (Coloronline)Impurityresistivityρimp(T)/ρu versus
develops close to, but below E , on increasing x above
temperatureT andarangeofTlconcentrationsx(inpercent, F
legend: column 1). For x>x∗, ρimp is well described by the x∗. This provides a new interpretation of measured tun-
usual spin Kondo resistivity [23] (shown as open squares for nelingspectra[12],whichcouldbe testedby temperature
x = 1.3%), but with effective Kondo scales Teff (legend: dependentstudiesoftunnelingorphotoemissionspectra.
K
column 2). ρu = 2nTl/(e2π~NF2σ0) is the residual resistivity Inthe future, itwouldbe interestingtoextendthis work
for unitary scatterers with NF =N(µ∗). The vertical arrow: to include the effects of disorder within an Anderson-
TK =1.23K. Inset(a): ρ0=xρimp(T =0)/ρu versusx(filled Hubbard model description.
circles). Dashed line: ρ0 for static unitary scatterers in place
of Tl.
We thank K. M. Seemann, D. J. Singh, H. Murakami,
P. Coleman, G. Kotliar, and I. R. Fisher for discussions
and D. J. Singh, H. Murakami and I. R. Fisher for data
below this. Interpreting these as the Kondo resonance
[8, 22]. V.Z. acknowledges support by Croatian MZOS
and the lower Hubbard satellite peaks in Fig. 2 yields
GrantNo.0035-0352843-2849,NSF GrantDMR-1006605
|U|/2≈15meV and hence the value U ≈−30meV used
and Forschungszentrum Ju¨lich. T. A. C. acknowledges
in our calculations. Since the CK resonance is temper-
supercomputer support from the John vonNeumann In-
ature dependent [4], the above interpretation could be
stitute for Computing (Ju¨lich).
tested with temperature dependent studies of tunneling
or high resolution photoemission spectra.
The temperature and doping dependence of the impu-
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