Energy-resolved inelastic electron scattering off a magnetic impurity

Energy-resolved inelastic electron scattering off a magnetic impurity

Markus Garst,¹ Peter Wölfle,²László Borda,³ Jan von Delft,⁴ and Leonid Glazman¹

1William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA

2Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, 76128 Karlsruhe, Germany

3Budapest University of Technology and Economics, Institute of Physics and Research Group “Theory of Condensed Matter” of the Hungarian Academy of Sciences, H-1521 Budapest, Hungary

4Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center for NanoScience, Ludwig-Maximilians-Universität München, 80333 München, Germany

共Received 15 July 2005; published 23 November 2005兲

We study inelastic scattering of energetic electrons off a Kondo impurity. If the energyEof the incoming electron共measured from the Fermi level兲exceeds significantly the Kondo temperatureT_K, then the differential inelastic cross section ␴共E,␻兲, i.e., the cross section characterizing scattering of an electron with a given energy transfer ␻, is well defined. We show that ␴共E,␻兲 factorizes into two parts. The E dependence of

␴共E,␻兲is logarithmically weak and is due to the Kondo renormalization of the effective coupling. We are able to relate the␻dependence to the spin-spin correlation function of the magnetic impurity. Using this relation, we demonstrate that in the absence of the magnetic field, the dynamics of the impurity spin causes the electron scattering to be inelastic at any temperature. At temperaturesTlow compared to the Kondo temperatureT_K, the cross section is strongly asymmetric in␻ and has a well-pronounced maximum atប␻⬃T_K. At TⰇT_K, the dependence␴vs␻has a maximum at␻= 0; the width of the maximum exceedsT_K/បand is determined by the Korringa relaxation time of the magnetic impurity. Quenching of the spin dynamics by an applied magnetic field results in a finite elastic component of the electron scattering cross section. The differential scattering cross section may be extracted from the measurements of relaxation of hot electrons injected in conductors containing localized spins.

DOI:10.1103/PhysRevB.72.205125 PACS number共s兲: 72.15.Qm, 75.20.Hr, 73.23.⫺b

I. INTRODUCTION

Scattering of an electron off a magnetic impurity embed- ded in a conductor is known to be anomalously strong.¹The origin of the anomaly is rooted in the degeneracy of the localized spin states. This degeneracy, being removed by a weak exchange interaction with the itinerant electrons in a metal, gives rise to the strong scattering of electrons with low energy—the Kondo effect. Perturbation theory in the exchange interaction constantJis singular. The second-order contribution in J to the scattering amplitude diverges logarithmically²if the electron energyE共measured from the Fermi level兲 and temperature T are approaching zero. It is important to notice that the logarithmically divergent contri- bution to the amplitude corresponds to an elastic process.

Indeed, this contribution comes from the change of state of one electron: states of all other itinerant electrons are the same in the beginning and end of the scattering process.

Therefore, the energies of the electron before and after the scattering is unchanged.

The divergence noticed by Kondo is not unique to the second order of the perturbation theory. Its higher orders共n

⬎2兲 also contain divergent terms of the typeJⁿlnⁿ⁻¹共D/␧兲, where␧= max共E,T兲, andDis some ultraviolet energy cutoff, whose value depends on the specific model: DⰇ␧. These leading logarithmic terms may be summed up by diagram- matic method³ or by means of the “Poor man’s scaling”⁴ renormalization group共RG兲, yielding for the scattering am- plitude

A_{k,␴,S→k⬘,␴⬘,S⬘}= 1

ln共␧/TK兲s_␴,␴⬘·S_s,s⬘, 共1兲

wheres_␴,␴⬘andS_s,s⬘are the spin operators of the conduction electrons and the impurity, respectively. The so-called Kondo temperature is given in terms of the cutoff D and the ex- change interactionJas TK=De^{−1/共J␯兲}, where ␯is the density of states. Like the lowest-order perturbation theory result, the leading-logarithmic approximation Eq. 共1兲 corresponds to purely elastic electron scattering.

The leading-logarithmic approximation is adequate at

␧ⰇTK, but it fails at low temperatures. A convenient phe- nomenological description of the low-energy behavior of a single-channel Kondo model is given by Nozières’ effective Fermi liquid theory. In this theory, a scattering problem can be formulated, too. It is clearly seen,⁵however, that the scat- tering is not purely elastic at␧ⰆTK. AtT= 0, for example, the inelastic contribution to the electron scattering cross sec- tion scales as共E/T_K兲²and becomes comparable to the elastic part atE⬃TK.

The Kondo effect is a crossover phenomenon, rather than a phase transition. The measurable characteristics, such as the contribution to the susceptibility or resistivity due to magnetic impurities depend smoothly on temperature. Simi- larly, the electron scattering off a magnetic impurity, which is deeply inelastic at␧⬃TK, must have some inelastic compo- nent at any energyE. In this paper, we investigate in detail the inelastic scattering of a high-energy electron off a mag- netic impurity.

1098-0121/2005/72共20兲/205125共14兲/$23.00 205125-1 ©2005 The American Physical Society

A study of the energy-resolved, differential cross section,

␴共E,␻兲, is interesting in its own right, but it can, in prin- ciple, also be measured, e.g, in a modification of the experi- ments of Pothieret al.⁶Further motivation to study␴共E,␻兲 beyond perturbation theory comes from the recent theoretical work of Zarándet al.⁷ In Ref. 7, the energy dependence of the total scattering cross section, ␴tot共E兲=兰d␻␴共E,␻兲, was addressed. With the help of the optical theorem, the total cross section␴tot共E兲was compared with the elastic part of it.

The conclusion reached in Ref. 7 regarding the energy do- mainEⰇTK is striking: atT= 0, the scattering is deeply in- elastic: the elastic part turns out to be negligibly small. This seemingly contradicts the leading-logarithmic result for the scattering amplitude Eq.共1兲. The physical explanation of this phenomenon, however, remained unclear in Ref. 7 and mo- tivates us to revisit the problem of inelastic scattering. The dependence of the differential cross section ␴共E,␻兲 on ␻^, which we consider in this paper, clarifies the issue, as we are able to determine the distribution of energy losses in the inelastic electron scattering off a magnetic impurity.

The separation of the electron scattering cross section in the Kondo effect into elastic and inelastic parts atEⰇTKwas not addressed for decades, as it does not affect the routinely measured quantity, the resistivity. The anomalously fast elec- tron energy relaxation in some mesoscopic metallic wires,⁶ which was discovered in the last decade, prompted a search for relaxation mechanisms driven by impurities with internal degrees of freedom. A viable mechanism of energy relax- ation was suggested in Ref. 8 and was associated with the electron-electron scattering mediated by exchange interac- tion of electrons with magnetic impurities. The removal of degeneracy of the localized spin states by the exchange in- teraction results in an anomaly of the electron-electron scat- tering cross section at small energy transfers;⁸ the collision of two electrons with energiesE,E

⬘

^ⰇTKleads to a redistri- bution of the energies between the two particles, E,E

⬘

^→E

−␻^,E

⬘

⁺␻, and has cross section K共␻^,E,E

⬘

^兲⬀J4/␻^{2 in the} lowest-order perturbation theory.¹⁰The 1 /␻²dependence of K allowed the experimental observations⁶ to be explained qualitatively. Later experiments¹² performed in a magnetic field sufficient for the Zeeman splitting of impurity energy levels did confirm the origin⁸ of the inelastic electron- electron scattering, and indicated the irrelevance of more ex- otic mechanisms, which assumed a generic non-Fermi liquid behavior introduced by impurities.¹³

The existence of energy exchange between electrons me- diated by their interaction with a magnetic impurity indicates the inelastic nature of the electron scattering off a magnetic impurity. Indeed, using the Fermi Golden rule, we find

␴共E,␻兲⬀ J⁴

␻²

冕

_−⬁^{⬁ dE}

^⬘

^f共E

^⬘

^{兲关1 −f共E}

^⬘

^{+␻兲兴⬀J}␻^{4 共2兲} at ␻ⰇT. So already in the simplest perturbation theory, it becomes clear that there is an inelastic contribution to scat- tering. As long asE,␻ⰇT, temperature does not affect the inelastic cross section in this order. It is not clear, however, what the relation is between the inelastic cross section

␴共E,␻兲 and the leading-logarithmic result Eq. 共1兲: On one

hand,␴共E,␻兲⬀J⁴is parametrically smaller than the scatter- ing cross section following from Eq.共1兲. On the other hand, the total inelastic cross section obtained from Eq. 共2兲,

␴tot共E兲=兰d␻␴共E,␻兲, diverges at␻→0, indicating the inap- plicability of the lowest-order perturbation results at small energy transfers.

The lowest-order perturbation theory forK共␻^,E,E

⬘

^{兲 can} be controllably improved in two respects. First, at E,E

⬘

ⰇT_Kand兩␻兩ⰆE,E

⬘

the four constantsJ, entering as a prod- uct in the perturbative result, may be replaced⁸by the prop- erly renormalized⁴ quantities.¹⁴ Second, the divergence at

␻→0 is cutoff due to the dynamics of localized spin. An adequate theory may be developed for high temperatures, TⰇT_K, where the cutoff occurs due to the Korringa relaxation.⁸ These improvements allow one to see that

␴tot共E兲 is finite, but they are insufficient to investigate the details of the␴共E,␻兲dependence.

In this paper, we concentrate on the differential cross sec- tion,␴共E,␻兲, of inelastic scattering of a highly excited elec- tron with energy EⰇT_K. Despite the many-body nature of the Kondo effect, this quantity is well defined at␻ⰆE. We show in Sec. II that in the limitEⰇTK, the differential cross section is related to the dissipative part of the impurity spin susceptibility, ␹

⬙

. From the low-frequency and high- frequency asymptotes of ␹

⬙

, we extract in Secs. III and IV the behavior of the differential cross section␴共E,␻兲 in the absence and presence of a Zeeman energy, respectively. The analytical asymptotes thus obtained are complemented by results of the numerical renormalization group¹⁷ 共NRG兲, which allows us to access also the intermediate range of frequencies and magnetic fields. The connection between the result of Ref. 7 and the leading-logarithmic approximation for the scattering amplitude共1兲describing only elastic scat- tering will be explained in detail. Finally, in Sec. V, we dis- cuss possible hot-electron experiments in metallic mesos- copic wires and in a semiconductor quantum-dot setup in order to measure the differential scattering cross section

␴共E,␻兲.

II. RELATION BETWEEN INELASTIC SCATTERING CROSS SECTION AND SUSCEPTIBILITY

The relation between the scattering cross section of a “for- eign” spin-carrying particle and the spin-spin correlation function of a magnetic medium is well known from the theory of neutron scattering.¹⁸Here, we derive a similar re- lation for scattering off a magnetic impurity of a high-energy electron belonging itself to the Fermi liquid hosting the mag- netic impurity.

The exchange interaction between the impurity spin and spins of electrons forming the Fermi liquid,

H_int=J

兺

k,k⬘

S·s_␣␣⬘c_k^†_␣c_k⬘␣⬘ 共3兲

gives rise to the Kondo effect. Here, J is the constant of exchange interaction between the impurity spin and itinerant electrons with energies⑀k␴共measured from the Fermi level兲 confined to some energy band, 兩⑀k␴兩⬍D. Here, s_␣␣⬘ is ¹₂

times the vector of Pauli matrices. The Kondo problem al- lows for a logarithmic renormalization: the low-energy prop- erties of the system described by the Hamiltonian共3兲coin- cide with those for a Hamiltonian defined in a narrower band, say兩⑀k␴兩⬍E, upon the proper renormalization of the exchange constant,

J共E兲= J共D兲

1 −␯J共D兲ln共D/E兲, J共D兲=J, 共4兲 where ␯ is the density of states of itinerant electrons. The perturbative renormalization Eq. 共4兲 is valid as long as the running energy 关E in the case of Eq. 共4兲兴 significantly ex- ceeds the Kondo temperatureTK. An important property of the logarithmic renormalization is that only exponentially wide energy intervals 共␧1,␧2兲, such that ␯J兩ln共␧1/␧2兲兩⬃1 contribute significantly to the renormalization. That allows us to “skip” some relatively narrow strip of energies, say, 共E−⌬E, E+⌬E兲, with ⌬EⰆE, in the renormalization pro- cess, yielding a Hamiltonian

H_int=J共D˜兲

兺

兩⑀k␣兩,兩⑀k⬘^␣⬘^兩⬍D˜

S·s_␣␣⬘c_k^†_␣ck⬘^␣⬘

+J共E兲

兺

E−⌬E⬍⑀k␣,⑀k⬘^␣⬘^⬍E+⌬E

S·s_␣␣⬘c_k^†_␣ck⬘^␣⬘

+J共E兲

兺

E−⌬E⬍⑀_k␣⬍E+⌬E,兩⑀_k⬘^␣⬘^兩⬍D˜; E−⌬E⬍⑀k⬘^␣⬘^{⬍E+⌬E,兩⑀k␣兩⬍D˜}

S·s_␣␣⬘c_k^†_␣ck⬘^␣⬘, 共5兲

withD˜艋E−⌬E. The renormalized exchange constants here may be expressed in terms of the Kondo temperature,

␯J共␧兲= 1 / ln共␧/TK兲. There is no need to distinguish between J共E−⌬E兲,J共E兲, or J共E+⌬E兲 as long asEⰇTK.

If the scattering of an electron with initial energyEleaves it in the energy domain 共E−⌬E, E+⌬E兲, then the corre- sponding cross section, within the lowest-order perturbation theory inJ共E兲, can be evaluated with the help of the Hamil- tonian 共5兲. The first line of Eq. 共5兲 plays the role of the Hamiltonian of a magnetic medium in the neutron, scattering problem, and the second line describes the interaction of the energetic particle共we deal with an electron rather than with a neutron, though兲with the medium. The remaining part of the Hamiltonian does not contribute to the scattering cross sec- tion in the lowest-order calculation.

Consider such a scattering of an energetic electron with energyE and spin␴ in the initial andE−␻^and␴

⬘

^{, respec-} tively, in the final state with ␻ⰆE such that E−␻苸共E

−⌬E,E+⌬E兲. The state of the remaining system before and after scattering may be characterized by the wave functions

⌿iand⌿f, respectively. The initial and final state of the total system is then given by the product states for the initial state, 兩i典=兩E,␴典丢兩⌿i典, 共6兲 for the final state,

兩f典=兩E−␻^,␴

⬘

典丢兩⌿f典.

The differential cross section of inelastic scattering

␴_␴⬘␴共E,␻兲is determined by the probabilityP_␴⬘␴共E,␻兲d␻^of

scattering of an electron with initial energyEand spin␴^into a state within interval of energies共E−␻^,E−␻^−d␻兲and spin

␴

⬘

^,

P_␴⬘␴共E,␻兲d␻^=vF␴␴⬘^␴共E,␻兲d␻^, 共7兲 where v_F denotes the Fermi velocity. By energy conserva- tion, ␻⁼␰f−␰i, where energies ␰i,␰f are associated respec- tively with the functions ⌿i and ⌿f involving the states in the domain 兩⑀k␴兩⬍D˜. In the absence of a magnetic field, energyEis the orbital energy in the initial state, and␻^{is the} change in the orbital energy resulting from scattering. In the presence of Zeeman splitting, the initial energy and the en- ergy transfer include the orbital and Zeeman parts, e.g., E

=⑀k+␴^ge␮BB/ 2.

The standard application of the lowest-order perturbation theory in the interaction of the energetic electron with the remaining system yields for the scattering probability

wf←i=兩J共E兲s_␴␴⬘具⌿f兩S兩⌿i典兩²2␲␯␦共␰i−␰f+␻兲, where ␯ is the density of states for the energetic 共⑀⬃E兲 electron. After the summation over the final states and proper thermal averaging over the initial states, we are able to relate w_f←i with ␴_␴⬘␴共E,␻兲 and obtain the differential scattering cross section

␴␴⬘^␴共E,␻兲= ␯ 4v_FJ²共E兲

⫻关␦␴⬘^␴Szz共␻兲+s_␴⁺_⬘␴S+−共␻兲+s_␴⁻_⬘␴S−+共␻兲兴, 共8兲 where s_␴^±_⬘␴=s_␴^x_⬘␴±is_␴^y_⬘␴. As in the theory of neutron scattering,¹⁸the cross section involves a spin-spin correlation function. Here, it is the correlation function of the local mag- netic impurity spin,

Sab共␻兲=

冕

−⬁

⬁

dte^i␻t具S^a共t兲S^b共0兲典

=

兺

兵兩⌿i典,兩⌿f典其

e^−␤␰i

Z 具⌿i兩S^a兩⌿f典具⌿f兩S^b兩⌿i典2␲␦共␰i−␰f+␻兲. 共9兲 We thus reduced the scattering cross section to an expression where its dependence on the energy of the scattering hot electronEseparates from the dependence on the energy loss

␻. The dependence on the energy loss is determined by the dynamics of the impurity spin characterized by the correla- tion functionS. The spin correlator is related to the dissipa- tive part of the impurity susceptibility via the fluctuation- dissipation theorem,

共g␮B兲²Sab共␻兲= 2

1 −e^−␤␻␹ab

⬙

共␻兲. 共10兲 Here, ␮B is the Bohr magneton, and g is the impurity g factor. The behavior of␹

⬙

in various limits will be discussed in the following sections. The spin dynamics is thus included in a nonperturbative fashion. It will allow us to investigate

the behavior of the cross section at any energy transfer; at

␻ⰆT_K, we apply effective Fermi liquid theory, and the re- gion of intermediate energies, ␻⬃TK, is covered with the help of NRG calculations.

However, it is important to note that the total scattering cross section is fixed by the sum rule for the spin correlation function, Sab共␻兲. Consider the total cross section obtained after averaging over the initial electronic spin configurations

␴, summing over the final ones␴

⬘

, and integrating over the energy transfer␻^,

␴tot共E兲=1 2

兺

␴,␴⬘

冕

_−⬁^{⬁ d␻␴␴⬘␴共E,␻兲=38␲}␯^1vF

1 ln² E

TK

.

共11兲 We substituted the explicit form for the energy dependent exchange interaction, J共E兲= 1 /关␯ln共E/TK兲兴. The total scat- tering cross section will be used throughout the rest of the paper as a convenient basic unit of measurement for the dif- ferential cross section discussed below.

As we are mainly interested in the dependence of the scattering probability on the energy transfer␻, we will con- fine ourselves in the following to an analysis of the scattering cross section averaged over the initial electronic spin con- figurations␴and summed over the final ones␴

⬘

^,

␴共E,␻兲=␴tot共E兲 2

3␲

冋

^{Szz共␻兲+12关S+−共␻兲+S−+共␻兲兴}

册

^.

共12兲 Note that a Zeeman energy of electrons forming the Fermi sea was already incorporated in the definition of the energies E and␻. The generalization of our results to spin-resolved scattering is straightforward.

III. INELASTIC ELECTRON SCATTERING IN THE ABSENCE OF ZEEMAN SPLITTING

In the absence of a magnetic field, the expression for the scattering cross section共12兲simplifies considerably since the impurity spin correlator is diagonal, S共␻兲⬅Szz共␻兲

=¹₂S+−共␻兲,

␴共E,␻兲=␴tot共E兲2

␲^S共␻兲. 共13兲 Let us first establish the relation between Eq. 共13兲 and the well-known result of the leading-logarithmic approximation.^3,4For that, we need to substitute in Eq.共13兲 the function S共␻兲 evaluated in the zeroth order in the ex- change interaction J共D˜兲. In this order, S^共0兲共␻兲=共␲^{/ 2}兲␦共␻兲, which yields the well-known result^3,4for the cross section,

␴^共0兲共E,␻兲=␴tot共E兲␦共␻兲, 共14兲 i.e., scattering is elastic in the leading-logarithmic approxi- mation. The elasticity breaks down, however, if one accounts forJ共D˜兲⫽0. Indeed, the exchange interaction J共D˜兲leads to some dynamics of the impurity spin. The delta-function in

Eq.共14兲gets broadened, and spectral weight is transfered to finite energies ␻⫽0. The shape of the broadened peak is related to the character of the spin dynamics, which is dif- ferent in the limits of high,TⰇTK, and low,TⰆTK, tempera- tures. We study the shape of the peak in these limits below.

However, note that the broadening does not affect thetotal cross section, which is fixed by the sum rule and remains the same as for the elastic scattering, Eq.共14兲, evaluated in the leading-logarithmic approximation.

A. Inelastic electron scattering atTšT_K

AtTⰇTK, the local spin exhibits relaxational dynamics.

The Bloch equations for the impurity spin in the absence of a magnetic field,

⳵

⳵^{t具Sa典= −} 1

␶K

具S^a典, 共15兲

imply the following form for the imaginary part of the susceptibility,¹⁹ ␹ab

⬙

共␻兲=␦ab␹

⬙

^{共␻兲with}

␹

⬙

共␻兲=␹0共T兲 ␻^/␶K

␻²⁺共1/␶K兲². 共16兲 It involves the static susceptibility, which is given by␹0共T兲

=共g␮B兲²/共4T兲. The decay time␶K in the Bloch equations is the Korringa relaxation time,²⁰ 1 /␶K=␲关␯J共T兲兴²T. Inserting the scale dependent exchange interactionJ共T兲, the Korringa relaxation rate reads explicitly,

1

␶K

= ␲^T ln² T

TK

. 共17兲

It is parametrically smaller thanT at temperaturesTⰇT_K. Expression共16兲 adequately accounts for the behavior of

␹

⬙

at low frequencies,␻ⱗT, but fails at higher frequencies.

For ␻ⰇT, the susceptibility can be evaluated within the lowest-order perturbation theory in the exchange constant,²¹ J共D˜兲,

共g␮B兲⁻²␹

⬙

共␻兲=␲ 4

1

␻^ln2兩␻兩 T_K

. 共18兲

The additional logarithmic frequency dependence arises from the logarithmic enhancement of the exchange interac- tion due to the perturbative RG, which is now cutoff at a bandwidthD˜⬃␻^.

The resulting differential cross section, ␴共E,␻兲, can be found with the help of Eq.共10兲. It is symmetric in␻^{at small} energy transfers. It shows a narrow peak at␻= 0 and falls off significantly within the region of energies兩␻兩ⱗT

␴共E,␻兲=␴tot共E兲␦_⌫共␻兲, 共19兲 where we introduced a “broadened delta function,” which is a Lorentzian with linewidth⌫= 1 /␶K,

␦⌫共␻兲= 1

␲

1/␶K

␻²⁺共1/␶K兲². 共20兲 As 1 /␶KⰆT, see Eq.共17兲, almost the full weight of the total cross section is accounted for by Eqs. 共19兲 and 共20兲, see Fig. 1.

At higher energy transfers, 兩␻兩ⲏT, the cross section is asymmetric in␻^,

␴共E,␻兲=␴tot共E兲 1 1 −e^−␻/T

1

␻^ln2共兩␻兩/TK兲. 共21兲 The probability for the scattered electron to acquire energy 共␻⬍0兲 is exponentially suppressed. Although the contribu- tion of Eq. 共21兲 to the total cross section is parametrically small,⬀1 / ln共T/T_K兲, it is worth noting that its decay with␻ is remarkably slow.

The slow decay of␴共E,␻兲 vs ␻ is related to the depen- dence on the transfered energy of the cross section for inelas- tic electron-electron scattering mediated by a magnetic impurity.⁸The probability for such an inelastic scattering be- tween two electrons with initial energiesE andE

⬘

^{and final} energies E−␻ ^{and E}

⬘

⁺␻ was calculated in Ref. 8; all of these four energies were assumed to be large compared to TK. According to Ref. 8 关see also Eq. 共9兲 of Ref. 9兴, the contributionK共␻^;E,E

⬘

兲of a single magnetic impurity to this probability in the limit of high energy,EⰇ兩␻兩, reads

K共␻^;E,E

⬘

兲=3␲ 8␯

1 ln²兩E兩

TK

冉

^ln兩TEK

^⬘

^兩 4

+ ln兩E

⬘

⁺␻兩 TK

冊

^2␻12.

共22兲 The differential cross section共21兲 can be obtained by inte- gratingK共␻^;E,E

⬘

^兲over the available phase space volume of one of the scattering electrons,

vF␴共E,␻兲=

冕

^dE

^⬘

^f共E

^⬘

^{兲关1 −f共E}

^⬘

^{+␻兲兴K共␻;E,E}

^⬘

^兲.

共23兲 The Fermi functions in Eq.共23兲confine the energyE

⬘

^{to an} interval −␻ⱗE

⬘

^ⱗ0. This includes a regime where the argu- ments of theE

⬘

-dependent logarithmic factors are not mean- ingful anymore and should be replaced by temperature or the Korringa relaxation rate. After such a cutoff, integration over E

⬘

is easily performed, yielding ln⁻²兩␻兩/TKwithin logarith- mic accuracy. This way, starting from the collision integral kernel of Ref. 8, one recovers Eq.共21兲.

B. Inelastic electron scattering atT™T_K

When the temperature is below the Kondo temperature, the picture differs drastically from the zeroth order result 共14兲. ForTⰆTK, the low-frequency behavior of the scatter- ing cross section is beyond perturbation theory. Nevertheless, the cross section for small energy transfers,兩␻兩ⰆTK, may be found with the help of the Shiba relation²²for the suscepti- bility,

共g␮B兲²␹

⬙

共␻兲= 2␲␻关␹0共T= 0兲兴². 共24兲 The zero-temperature static susceptibility ␹0共0兲 is used conventionally²³ to define the pre-exponential factor of the Kondo temperature, ␹0共0兲=关共g␮B兲²W兴/共4TK兲; here W

= 0.413. . . is Wilson’s number.共We present a convenient deri- vation of the Shiba relation in Appendix A.兲The corrections to the Shiba relation are of orderO共␻^T2/TK

2,␻^3/TK

2兲and are subleading. We, thus, obtain for the cross section at 兩␻兩, TⰆTK,

␴共E,␻兲=␴tot共E兲W² 2

1 1 −e^−␻/T

␻

T_K². 共25兲 The high-frequency limit, 兩␻兩ⰇTK, of the scattering cross section can still be obtained perturbatively and is given by Eq.共21兲.

Comparing the results of Eqs.共21兲 and共25兲, we see that for temperatures TⰆTK, the differential cross section,

␴共E,␻兲, peaks at energy transfers of the order of␻⬃TK. It then decreases linearly upon further decrease of ␻^{, until it} crosses over 共at兩␻兩ⱗT兲into the exponential tail for ␻⬍0, see inset of Fig. 2. At zero temperature, the factor containing exp共−␻^/T兲 in Eq.共25兲 becomes a step function which for- bids any energy gain from the Kondo system,

␴共E,␻兲=␴tot共E兲W² 2 ⌰共␻兲␻

T_K², 共26兲 here⌰共x兲= 1 ifx⬎0 and 0 ifx⬍0.

The region between the asymptotes given in Eqs.共21兲and 共25兲can be bridged by calculations performed with the NRG method. In this method, after the logarithmic discretization of the conduction band, one maps the Kondo Hamiltonian onto a semi-infinite chain with the impurity at the end. As a consequence of the logarithmic discretization, the hopping along the chain decreases exponentially, tn⬃⌳^−n/2, where FIG. 1. 共Color online兲 Differential cross section, ␴共E,␻兲, at

large temperatures, TⰇT_K, without Zeeman splitting, B= 0, as given by Eq.共19兲. The Lorentzian peaks have a width given by the Korringa relaxation rate 1 /␶K, Eq.共17兲.

⌳⬎1 is the discretization parameter andn is the site index.

共We have used⌳= 2 throughout the calculations presented in the paper.兲The separation of energy scales provided by the exponential decay of the hopping rate allows us to diagonal- ize the Hamiltonian iteratively and keep the eigenstates with the lowest energy as the most relevant ones. Since we know the energy eigenvalues and eigenstates, we are able to calcu- late the impurity spin correlation function directly关see Eq.

共9兲兴given that the Dirac delta function appearing in the Le- hman representation must be broadened when performing a numerical calculation.^24,25The result of the NRG calculation is shown in Fig. 2.

To summarize this section, we demonstrated that the dy- namics of the impurity spin leads to inelastic electron scat- tering at all temperatures. The main contribution to the total scattering cross section comes from␻⬃T_K or 兩␻兩ⱗ1 /␶K at TⰆTK and TⰇTK, respectively. The total scattering cross section is fixed by the sum rule for the impurity spin corre- lation function, see Eq.共11兲, and is thus determined by the effective exchange constant J共E兲 evaluated within the leading-logarithmic approximation.³

IV. ZEEMAN EFFECT IN THE ELECTRON SCATTERING We now address the case when the degeneracy of the impurity spin is lifted by a magnetic field. The Zeeman split- ting of the impurity spin is described by the Hamiltonian

H_Zeeman= −g␮BS^zB. 共27兲

In the presence of the Zeeman splitting, the scattering elec- tron has to pay Zeeman energy in order to transfer spin to the

Kondo system. The resonance structure for electron scatter- ing involving a spin flip will, therefore, differ from the one of non-spin-flip scattering. Evaluating the impurity spin cor- relator in zeroth order inJ共D˜兲, we obtain for the scattering cross section共12兲in the leading-logarithmic approximation,

␴^共0兲共E,␻兲=␴tot共E兲 2 1 +e^−␤␻

⫻1

3兵␦共␻兲+␦关␻⁻␻Z共B兲兴+␦关␻⁺␻Z共B兲兴其.

共28兲 The single delta function forB= 0, Eq.共14兲, is now split into three contributions. In addition to a delta function at zero frequency, which is due to non-spin-flip scattering, there are two Zeeman satellites at ␻^{= ±}␻Z共B兲. In the limit of low temperatures,TⰆB, the satellite at negative Zeeman energy corresponding to an energy gain of the scattering electron is exponentially small as it is clear from Eq.共28兲.

The Zeeman energy␻Z共B兲depends on the renormalizedg factor, which is different from its bare valueg appearing in the Zeeman Hamiltonian共27兲. When we derived the effective interaction Hamiltonian共5兲, we integrated out a finite band of electronic degrees of freedom which lead to a renormal- ization of the exchange interactionJ. The Zeeman term共27兲 is not invariant under this perturbative renormalization of the Kondo model. Similar to the exchange interaction J, the g factor is also renormalized when the band is reduced fromD to D˜. As explained in Appendix B, the scale-dependent g factor in the leading-logarithmic order is given by

g共D˜兲

g =

冉

^{1 −2 ln1D˜/T}K

冊

^{. 共29兲}

To find the observable value of thegfactor, one needs to set D˜= max兵T,g␮BB其. The position of the Zeeman resonances, to the leading-logarithmic order, is given by²⁶

␻Z共B兲=g

冉

^{1 −}2 ln共max兵T,g¹ ␮BB其/TK兲

冊

^{␮BB. 共30兲}

Beyond the leading-logarithmic approximation, the dy- namics of the local spin is characterized by a further redis- tribution of the spectral weight of the scattering cross section 共28兲. However, a striking feature of the presence of a mag- netic field is that a finite weight of the delta resonance at␻

= 0 will still survive after accounting for the coupling of the impurity spin to the low-energy degrees of freedom of the Fermi sea. In other words, at any ratio T/T_K, a part of the scattering becomes elastic if a magnetic fieldB⫽0 is turned on. This can be best understood by considering the longitu- dinal spin correlation function in time. ForB⫽0, this corre- lation function will not fully decay with time but rather satu- rate at a value given by the finite expectation value of the impurity spin,具S^z共t兲S^z共0兲典→具S^z典² fort→⬁. This finite satu- ration value leads to a finite weight of the delta function␦共␻兲 in its Fourier transform and in Eq. 共8兲. Let us decompose

␴共E,␻兲into the elastic and inelastic parts, FIG. 2.共Color online兲NRG result for␴共E,␻兲 共solid line兲on a

logarithmic scale atT= 0 without Zeeman splitting,B= 0. A maxi- mum at finite ␻=T_K develops and scattering with small energy transfer␻is suppressed. Whereas, the high-frequency tail is pertur- batively accessible, see Eq. 共21兲 共short-dashed line兲, the low- frequency tail, Eq. 共25兲 共long-dashed line兲, is a property of the strong coupling fixed point described by Nozières’ Fermi liquid theory.共W= 0.413. . .. is Wilson’s number.兲The inset shows the tem- perature correction according to Eq.共25兲; the contribution for nega- tive␻is exponentially small for temperatures 0⬍TⰆT_K.

␴共E,␻兲=␴el共E,␻兲+␴inel共E,␻兲. 共31兲 The elastic part will be determined by the magnetization of the impurity spin

␴el共E,␻兲=␴tot共E兲4

3具S^z典²␦共␻兲. 共32兲 Being a thermodynamic quantity,具S^z典has a well-studied field and temperature dependence.^23,27 In the scaling regime, f共t,b兲=具S^z典 is a function of t=T/TK and b=g␮BB/TK. The asymptote of f共t,b兲 at max共t,b兲Ⰷ1 is with logarithmic ac- curacy given by

f共t,b兲=1

2

冉

^{1 −}2 ln关max共t,b兲兴¹

冊

⫻tanh

冋

^2tb

^冉

^{1 −}2 ln关max共t,b兲兴¹

冊 册

^{. 共33兲}

Note that in the limitt= 0,bⰇ1, Eq.共33兲yields the ground- state value of具S^z典in the perturbative regime. In the opposite limit of a weak field, bⰆ1Ⰶt, spin polarization is small according to the Curie law,f⬃b/ 4t. In the developed Kondo regime, max共t,b兲Ⰶ1, the average spin is f共t,b兲=共W/ 4兲b.

The weight of the elastic scattering, Eq. 共32兲, evaluated with NRG is shown in Fig. 3. In the limit of small magnetic fields, this weight increases as B². The saturation of the weight to its large-field limit, 1 / 3, is remarkably slow due to the logarithmic correction to the magnetization,²⁷ see Eq.

共33兲.

The inelastic part of the scattering cross section,

␴inel共E,␻兲, accounts for the remaining spectral weight. Note, however, that the total scattering cross section, i.e., the total spectral weight, is independent of the magnetic field: its value being fixed by the sum rule for the impurity spin cor- relator.

A. Dissipative part of magnetic susceptibility

To analyze the inelastic scattering cross section in more detail for the two limiting casesTⰇTKandTⰆTK, we start from presenting the proper details regarding the frequency dependence of the dissipative parts of longitudinal and trans- versal impurity spin susceptibilities 共␹zz

⬙

^and ␹+−

⬙

^{, respec-} tively兲.

At TⰇT_K, one may treat the exchange interaction J共D˜兲 perturbatively at any fieldB. The effect ofBon␹

⬙

^{is negli-} gible as long as the Zeeman splitting,␻Z共B兲, is smaller than the Korringa relaxation rate, 1 /␶K, see Eq. 共17兲. At higher fields, the susceptibility becomes anisotropic,␹zz

⬙

⫽¹₂␹+−

⬙

^{, and} its frequency dependence acquires a well-resolved structure.

The dissipative part of the susceptibility can be found from the Bloch equations.¹⁸The transversal part takes the form

␹+−

⬙

共␻兲= 2␹T

␻^/T2

关␻⁻␻Z共B兲兴²+共1/T2兲², 共34兲 where the static transversal differential susceptibility can be expressed with the help of Eq. 共33兲 as ␹T=共g␮B兲f共t,b兲/B.

The longitudinal part reads

␹zz

⬙

共␻兲=␹L

D ␻^/T1

␻²⁺共1/T1兲², 共35兲 where␹L

Dis given by

␹L D=

共g␮B兲²

冉

^{1 −}ln共max兵T,g¹␮BB其/TK兲

冊

4Tcosh²␻Z共B兲 2T

. 共36兲

The factor ␹L

D can be understood as the contribution to the static suceptibility which originates from the response of the occupation factors of the two Zeeman levels to a varying magnetic field, ␹L

D=g_eff␮B⳵具n+−n₋典/⳵^{B; here g}eff, see Eq.

共29兲, is the appropriately renormalized g factor. Note that only in the limit␻Z共B兲ⱗT, when the renormalizedg factor 共29兲 is insensitive to the magnetic field, ␹L

D does coincide with the full static longitudinal differential susceptibility␹L

=g␮B⳵^f共t,b兲/⳵^B.

In the case of a moderately high field, 1 /␶KⰆ␻Z共B兲ⱗT, the relaxation times, T₁ andT₂, equal each other¹⁸ and are given by Eq.共17兲,T₁=T₂=␶K.

At even higher fields, ␻Z共B兲ⲏT, the peak structure in

␹+−

⬙

共␻兲 is still described by a Lorentzian form of Eq. 共34兲, but the corresponding relaxation time is determined now by the Zeeman splitting rather than by temperature,¹⁹

1 T₂=␲

4

␻Z共B兲 ln² ␻Z共B兲

T_K

. 共37兲

The frequency dependence of the longitudinal susceptibility, however, requires additional discussion.

Generally, the susceptibility, ␹ij共␻兲, describes the re- sponse of the magnetic impurity to a local magnetic field that oscillates with frequency␻. At low frequencies, the variation of the dissipative part of the longitudinal component␹zz

⬙

共␻兲 FIG. 3. 共Color online兲 Weight of the elastic scattering cross

section,␴el共E兲=兰d␻␴el共E,␻兲, see Eq. 共32兲, determined by NRG.

The weight increases asB²for small magnetic fields and saturates logarithmically slowly to the limiting value for large B, see Eq.

共43兲.

given by Eq.共35兲can be understood in the framework of the Debye mechanism²⁸ of relaxational losses: At ␻= 0, relax- ation caused by the exchange interaction between the local magnetic moment and itinerant electrons establishes equilib- rium Gibbs occupation factors for the two Zeeman-split lev- els. At finite, but small frequency␻, the Zeeman splitting, which is caused by the sum of a constant and a slowly vary- ing magnetic field, changes with time slowly, and the relax- ation acts to adjust the occupation factors to the instant val- ues of the Zeeman splitting. The adjustment occurs via the emission 共or absorption兲 of particle-hole pairs with energy

␧ph⬃␻Z共B兲by flips of the local spin. It is the time variation of the occupation factors of the Zeeman-split levels that leads to dissipation. In the limit␻→0, the leading term in

␹zz

⬙

共␻兲, according to Eq.共35兲, is

␹zz

⬙

兩共␻兲兩Debye=␹L

DT₁␻^. 共38兲

As was already mentioned, in the weak-field case, 1 /T₁ is given by Eq.共17兲. In the limit, ␻Z共B兲ⰇT, the time T₁ was found¹⁹to beT₁=T₂/ 2 withT₂of Eq.共37兲. The contribution 共38兲 to ␹zz

⬙

from the Debye relaxational losses is valid at arbitrary ratio,␻Z共B兲/T. Note, however, that despite the fact that Eq.共38兲describes dissipation at low frequency, the De- bye mechanism is associated with the emission of particle- hole pairs with a comparatively high energy␧ph⬃␻Z共B兲. In the limit,␻Z共B兲ⰇT, the Debye mechanism thus yields only an exponentially small contribution to dissipation,

␹zz

⬙

兩共␻兲兩Debye= 2

␲

共g␮B兲²␻ T

ln² ␻Z共B兲 TK

␻Z共B兲 exp

冋

^{− ␻ZT共B兲}

册

^.

共39兲 The exponential smallness of␹L

Dcomes from the small prob- ability of the thermal occupation of the highly excited state, corresponding to the upper of the two Zeeman-split levels.

Temporal variations in this exponentially small quantity leads to an exponentially small contribution to␹zz

⬙

共␻兲.

Under these conditions, a second contribution, originating from the low-energy part of the spectrum, 兩⑀兩ⱗmax关␻,T兴, becomes important. The processes contributing here do not involve real impurity spin-flip processes共which are exponen- tially suppressed兲, but only virtual transitions. The starting point is the observation that the impurity magnetization lo- cally polarizes the Fermi sea. If the Zeeman splitting of the impurity is slowly varied with a small frequency␻^{, the mag-} netic polarization of the Fermi sea will adjust itself to the instantaneous adiabatic value of the impurity magnetization.

Since the spectrum of the particle-hole pairs is continuous, this adjustment results in dissipation via the emission of pairs with small frequency ␧ph⬃␻, which is in contrast to the Debye mechanism, where the emitted particle-hole pairs carry a large energy of the order of Zeeman splitting. As shown in Appendix A, this contribution to the susceptibility can be obtained by applying Nozières’ Fermi liquid theory and is adequately accounted for by the generalized Shiba

relation, Eq.共A6兲. Evaluating d具S^z典/dBwith the help of Eq.

共33兲atTⰆg␮BB, we find for the dissipative part of the lon- gitudinal susceptibility,

␹zz

⬙

共␻兲=␲ 8

共g␮B兲²␻

␻Z 2共B兲

1 ln⁴␻Z共B兲

TK

, ␻ⱗ␻Z共B兲. 共40兲

Comparing Eq.共40兲with the result for the Debye mecha- nism, we see that the strong-field asymptote Eq.共39兲for the latter mechanism is important only in a narrow interval of temperatures ␻Z共B兲ⲏTⲏ␻Z共B兲/ 6, as for all practical pur- poses in lnln共␻Z/T_K兲⬇1. Dispensing with that interval, we will use for the dissipative part of the longitudinal suscepti- bility, Eq.共35兲withT₁=␶Kin the case of␻Z共B兲ⱗTand Eq.

共40兲in the case of␻Z共B兲ⰇT.

At low temperatures, TⰆTK, there is little effect of the magnetic field on␹

⬙

^共␻兲 for weak fields, g␮BBⰆTK. In the strong-field regime,␻Z共B兲ⰇTKⰇT, the main contribution to the transversal part of the dissipative susceptibility is given by Eq. 共34兲 with the relaxation time T₂ of Eq. 共37兲. The longitudinal part is described by Eq. 共40兲 at ␻Ⰶ␻Z共B兲.

Equation共34兲adequately describes the nonmonotonic behav- ior of␹+−

⬙

共␻兲, but fails at higher frequencies; similarly, the linear dependence in␹zz

⬙

共␻兲does not stretch beyond ±␻Z共B兲. In the limit, 兩␻兩Ⰷ␻Z共B兲, the magnetic field does not affect significantly the dissipation, and Eq.共18兲is applicable.

B. Elastic and inelastic components of electron scattering The coupling of the impurity spin to the low-energy de- grees of freedom of the Fermi seas will lead to a broadening and redistribution of the spectral weight of the three delta functions in Eq.共28兲.

1. High temperatures: TšT_K

At high temperature, TⰇTK, and weak magnetic field,

␻Z共B兲ⰆT, the spin polarization is weak, and the elastic com- ponent of the scattering is small. Using Eqs. 共32兲and共33兲, we find

␴el共E,␻兲=␴tot共E兲4

3

冋

^{1 −ln共T/T2 K兲}

册冋

^g␮4TBB

册

^{2␦共␻兲.}

共41兲 The major contribution to the scattering cross section comes from the inelastic processes. At fields satisfying the condition

␻Z共B兲␶KⰇ1, which still belongs to the domain of weak fields,␻Z共B兲ⰆT, the single maximum in the␻^dependence of the cross section, see Eq.共19兲, splits into three

␴inel共E,␻兲 ⬇␴tot共E兲1

3兵␦⌫共␻兲+␦⌫关␻⁻␻Z共B兲兴

+␦_⌫关␻⁺␻Z共B兲兴其. 共42兲

The broadened delta function was defined in Eq.共20兲with a relaxation rate ⌫, given by the inverse Korringa time, ⌫

= 1 /␶K. 共We neglected a small part of the spectral weight

which moved to the elastic component of the scattering cross section兲.

With the increase of the ratio␻Z共B兲/T, the intensity of the elastic scattering increases, and in the strong-field limit, we find

␴el共E,␻兲=␴tot共E兲1

3

冋

^{1 −ln共g␮1BB/TK兲}

册

^{␦共␻兲. 共43兲}

Simultaneously, the maximum of ␴inel共E,␻兲 at negative ␻ gets suppressed, and the structure at 兩␻兩Ⰶ␻Z共B兲 broadens and becomes asymmetric. In the limit ␻Z共B兲/TⰇ1, only a single maximum at positive␻ remains in the inelastic cross section,

␴inel共E,␻兲=␴tot共E兲 2 3␲

1 1 −e^−␻/T

1

␻Z共B兲

⫻ ␻^/T2

关␻⁻␻Z共B兲兴²+共1/T2兲². 共44兲 Here the relaxation timeT₂is defined by Eq.共37兲. This main contribution to the inelastic scattering is proportional to

␹₊₋

⬙

共␻兲and comes from the spin-flip processes. The compari- son of Eqs. 共39兲 and 共40兲 with Eq. 共34兲 shows that at

␻Z共B兲ⰇTK, the effect of the dissipative part of longitudinal susceptibility is small starting from␻Z共B兲/Tⲏ4. Under this condition, ␹zz

⬙

共␻兲 yields a contribution to ␴共E,␻兲, which is small compared to Eq.共44兲.

The high-frequency tail,兩␻兩Ⰷmax关T_K,g␮BB,T兴, is unaf- fected by the Zeeman splitting and is still given by Eq.共21兲.

2. Low temperatures: T™T_K

We turn now to the opposite limit of small temperature, TⰆTK. At weak magnetic field, g␮BBⱗTK, the low- frequency behavior of the scattering cross section is beyond perturbation theory. In this regime, the electron scatters from a fully developed, many-body Kondo singlet. Here we can use the Shiba relation, Eq.共25兲, to access the low-frequency tail of the cross section. In the presence of a magnetic field, there are additional corrections to the Shiba relation of order O关␻共g␮BB兲²/T_K²兴 which are subleading and are neglected in the following. We get for the low-frequency part兩␻兩ⰆTK,

␴共E,␻兲=␴tot共E兲W² 2

1

1 −e^−␻/T

冋

¹⁶

^冉

^g␮TKBB

^冊

^{2␦共␻兲+T␻K2}

册

^,

共45兲 where W is again Wilson’s number.²³ The scattering cross section decreases linearly with frequency. At␻ⱗT, the lin- ear decrease crosses over into an exponential tail which ex- tends to negative frequencies. In Fig. 4, NRG results at T

= 0 for the inelastic cross section at small magnetic fields are compared with the NRG data atB= 0. In finite field, the slope in the linear low-frequency regime is reduced. The difference in slope is of orderO共g␮BB/T_K兲², a correction alluded to but neglected in Eq.共45兲. This difference, however, accounts for the reduction of the inelastic scattering weight. The weight of order O共g␮BB/TK兲² is transfered from the inelastic to the elastic scattering contribution leading to a delta peak at

␻= 0, as sketched in the inset of Fig. 4. In contrast to the case of high temperatures 共TⰇTK,g␮BB兲, the elastic scattering contribution now does not sit on top of a large Lorentzian peak, but is rather located within the scattering pseudogap.

Although its weight is small, here it is easily distinguishable from the background. The crossover from the linear depen- dence on ␻ to the high-frequency behavior occurs at

␻⬃TK, where the inelastic scattering cross section has a maximum. The high-frequency tail is still given by the per- turbative expression共21兲.

When the magnetic field is increased above the Kondo temperature,g␮BBⰇTK, the elastic and inelastic components of the scattering cross section are given by Eqs. 共43兲 and 共44兲, respectively. The elastic peak at ␻= 0 now exhausts almost the full spectral weight of the longitudinal correlator, i.e., it accounts for approximately 1 / 3 of the total scattering cross section, see Fig. 3. The remaining 2 / 3 of the total spectral weight are to be found in the extended structure of the Zeeman satellite共44兲centered at␻⁼␻Z共B兲. The effect of Zeeman splitting on the cross section is confined to the re- gion of energies兩␻兩ⱗ␻Z共B兲. At兩␻兩Ⰷmax关T_K,g␮BB,T兴, the behavior of␴共E,␻兲is again given by Eq.共21兲.

In Fig. 5, the inelastic cross section is shown in the limit of large magnetic fields, g␮BBⰇTK, as given by Eq. 共44兲.

The inset compares the result with the NRG. The low- frequency and high-frequency asymptotes are reproduced in the numerical calculation fairly well. The deviation in the width of the Zeeman peak, however, demonstrates the limi- tation of the NRG method. Due to the logarithmic frequency resolution, the NRG tends to overbroaden any peak in the spectral function centered around anonzerofrequency.

V. POSSIBLE EXPERIMENTS

As we have shown above, the differential scattering cross section of the magnetic impurity shows a rich structure in FIG. 4. 共Color online兲NRG result for␴共E,␻兲atT= 0 for mag- netic fieldsg␮BB⬍T_K. The difference of the curves indicates the scattering weight which forB⬎0 is transfered from the inelastic to the elastic component leading to a delta-function peak at␻= 0, as sketched in the inset.