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One-phonon processes in phonon-assisted tunnelling jumps at low temperatures
Volume 82A, number 2
PHYSICS LETTERS
9 March 1981
ONE-PHONON PROCESSES IN PHONON-ASSISTED TUNNELLING JUMPS
AT LOW TEMPERATURES H. TEICHLER Institut fi~r Metallphysik, Universit~t G6ttingen, G~ttingen, Germany and
A. SEEGER Institut ffir Physik, Max-Planck-lnstitutfiir Metallforschung and Institut j~'r Theoretische und Angewandte Phystk, Universitiit Stuttgart, Stuttgart, Germany Received 25 November 1980
The conditions for light-interstitialdiffusion by "incoherent tunnelling" via one-phononprocessesare investigated.If elementary diffusionjumps are involvedwhich changethe crystallographicorientations of the defects, one-phononprocesses may give rise to a linear temperature dependenceof the diffusion coefficient.
Phonon-assisted tunnelling jumps of defects (in the following called "incoherent tunnelling") play an important r61e in several areas of solid-state physics, among them small-polaron hopping (where the theoretical ideas were originally introduced [ 1,2]), reorienration processes of dipolar defects [3-9], and lowtemperature diffusion of light interstitials in crystals (so-called quantum diffusion [10,11]). An important question is that of the temperature variation of the transition rate at low temperatures. The temperature law depends critically on whether the lowest nonvanishing contribution comes from one-phonon or from two-phonon processes. Experiments on dipolar reorientation are often carried out in an applied field which removes the energetic degeneracy of the configurations between which the transitions take place. It has been shown by both experiment and theory [3,7] that even if this degeneracy is restored (i.e., if the applied field is reduced to zero) the so-called one-phonon processes give the lowest non-vanishing contribution to the transition rate. This means that in this case the limiting law at low temperatures for incoherent tunnelling is a T I law.
The situation is quite different in the case of quantum diffusion in metals. Both Flynn and Stoneham [10] and Kagan and Klinger [11] claim that the lowest non-vanishing contribution comes from two-phonon processes, with the consequence that the limiting lowtemperature law for incoherent tunnelling is predicted to be T7. In the present note we investigate the origin of the difference between these results and those summarized in the preceding paragraph. We shall see that the Flynn-Stoneham re.~ult [I0] is conditional upon assumptions which in practice are often violated. In such cases one-phonon processes do contribute to the rate of diffusion by incoherent tunnelling and lead to low-temperature diffusion coefficients which are proportional to T 1. As most papers in the field, we restrict ourselves to dilute systems (i.e., we consider only one defect at any one time) and make the assumptions of (i) adiabatic dec0upling of defect and host-material dynamics, (ii) linear coupling approach for the defect-lattice interaction, (iii) Condon approximation for the transfer integrals,
0 031-9163/81/0{300-0000/$ 02.50 © North-Holland Publishing Company
91
Volume 82A, number 2
PHYSICS LETTERS
(iv) harmonic approximation for the lattice vibrations, and start with the hamiltonian (for further details see ref. [10])
i
l,n
+
ai aignUn
(z)
+
wi/= lim (a/at)(n/(t)nl(O))/(ni(O)) , n k = a~ak , t--, ®
dr e - i ~ r
X Iexp (~q Pq COS(COqr)/sirth(hCOq/2kT) ) - I1 • In eq. (3) the following abbreviations have been used:
,oxp
,
~ 2 ----e / - ei is the energy difference between the final and initial states of the defect. (M is the atomic mass of the host material, N is the number of translation cells in the Born-yon Karman volume.) Eq. (3) may be derived by an approach analogous to that used by Gosar and Pirc [7]. The term - 1 occurring in the square brackets in eq. (3) guarantees that the coherent transitions are properly eliminated. In the following we consider the configurational average of wi/me ,
92
eq)12 sin2(q • all2)
(6)
In eq. (6)
p/),
(7)
denote the even and odd linear combinations, respectively,of the double force tensors pi and P/of the defect in its initialand final state,whose (ta,v) cartesian components are given by (see, e.g.,ref. [12])
(k= i,:)
<8)
n
--oa
f da
1-'q = (2MN'h6Oq3)-1 [1(q" pe.
--
(2)
the rate of transitions of the defect from state i to state/, becomes to second order with respect to J
=
where p(I2) is the appropriate distribution function of the energy difference ~2. In investigating the contribution of one-phonon processes we may confine ourselves to the limit of long wavelengths and rewrite eq. (4) a s
+ I(q" p o . %)12 cos2(q, d/2)] .
Here a~c,a k denote creation and annihilation operators for the defect in the state k, gnk the force of the defect in sta~e k on the host atom at lattice point R n, u n the + displacement of this atom, bq, bq creation and annihilation operators for phonon mode q with frequency c~q and polarization e~, J the defect transfer integral. In this model the incoherent contribution to
Iv~.ne = ]J,/~}2 f
9 March 1981
fS)
Furthermore, d means the distance vector between the initial and Final site of the defect. The usual treatments of quantum diffusion are equivalent to the choice po = 0, i.e., only the first term in eq. (6) is retained. This corresponds to the assumption that in a diffusional jump the strain fields in the initial and in the final states are the same, imp!ying that the two equilibrium configuration s differ only by a translation. In theories of dipolar reorientation without diffusion the first term in eq. (6) vanishes because o f d = 0. The only contribution comes from the second term, which reflects the change of orientation of the defect. In general such a change in orientation also occurs during diffusion of interstitial atoms, as is illustrated by the nearest-neighbour jumps of interstitial atoms located on either the octahedral or the tetrahedral sites in bcc metals. Both types of sites possess tetragonal symmetry. Nearest-neighbour jumps involve a change in the tetragonality axes. This means that in a general treatment of diffusion we must retain both the pe and the po term. The terms corresponding to one-,two-, etc. phonon processes are obtained by expanding the exponential in the square brackets in eq. (3) in a power series. If the width of p(~2) is small compared to the I)ebye frequency w D, the one-phonon contribution to w ine becomes for a three-dimensional system with linear dispersion of the long-wavelength phonons and kT /ic~D:
Volume 82A, number 2
wnO=
PHYSICS LETTERS
oxp( (9)
oo
× / d[2 p([2) h~2 exp(-h[2/2kT) (CO+/i2122ce) 2kT sinh (h[2/2kT) - -
oo
C°=~V- I ~q [5(Wq+[2)+5(Wq-[2)] (10a) X Iq" pa
C ~ -~Vq~
"eq [2/(hWq) 4 , [5 (¢oq + [2) + 5(t% - [2)]
X (q. d/2) 2 [q" ps.
eq [2/(h~q)6 ,
(lOb)
where under the above assumptions C ° and C e are independent of I2 for [I2l "¢ ~D. Under the additional assumption that the width of p([2) is small compared to 2kT, the one-phonon processes give for the incoherent tunnelling rate: w~he=
perimentally found to be proportional to T. If the conditions (i)-(iv) are satisfied one may conclude that the diffusion jumps either must involve a change in the double-force tensor of the d___efectconfiguration or are made possible by a finite I22 produced by lattice imperfections. Whether the latter possibility obtains may be tested experimentally either by investigating the influence of the crystal perfection or by studying the temperature dependence ofw ine outside the range of validity of the T 1 law. Consider as an example a distribution p ([2) restricted to [2 I> 0 (this corresponds to thermally activated "up-hill" hopping), varying only slowly for 0 < I2 < ([22)1/2 and dropping rapi_~y to zero for I2 ([22)1/2. Here for kT ~-,~ (I22) 1/2 a transition takes place from the high-temperature T 1 law to a low-temperature relationship:
w~ne=~rr(lrkT)2J2 exp ( - q~ I'q)
(13)
X [C° +-~ (rtkT)2Ce]p(O)/h.
lrkTJ2 exp ( - ~ r'q)(C°+ ~i212---2ce), (ll) co
- f
9 March 1981
d[2p(~)[2
2 .
(12)
po contributes to wjne a term linear in T which is independent of any further details of p ([2), whereas pe yields a__contribution linear in T which is proportional to ~2 and which experimentally may be made small by using crystals of high purity and high perfection. The most important result from the preceding discussion is the prediction of a finite contribution of one-phonon processes to the transition rate of diffusional incoherent-tunnelling jumps (leading to a T 1 law at low temperatures) even if the energy levels are practically degenerate, i.e. p([2) "~ 5 (~2), inall those situations in which the diffusion jumps involve changes of the crystallographic orientation of the defect configuration. As already discussed, such a situation is realized in the diffusion of fight interstitial atoms in bcc metals. The predicted behaviour has indeed been recently observed for the diffusion of positive muons in ~-Fe [13]. Let us now reverse the problem and ask which conclusions can be drawn if the low-temperature diffusion coefficient associated with incoherent tunnelling is ex-
Since this transition depends on the degree of perfection of the crystal it should not be too difficult to distinguish it from the case that the jumps occur between physically distinct sites (e.g., octahedral and tetrahedral interstices) with e / - ei = Ae = constant (i :/:/'),. corresponding to p([2) = 5 ([2 - Ae). In this case wJne "- T 1 as valid for kT >~Ae will be replaced at low temperatures by an approximate Arrhenius law with activation energy equal to Ae. Positive-muon diffusion experiments are particularly well suited for identifying the origin of the onephonon contributions to w inc. The possibility of level shifts [2 due to the interaction between different diffusing particles is excluded and therefore a small width of p([2) may be realized by using crystals of high purity and high perfection. Once a T 1 dependence of w inc, independent of the lattice perfection, is established we may conclude that po is f'mite, provided assumptions (i)-(iv) are admissible. For transitions between equivalent sites in cubic crystals this means that the defect configuration cannot have cubic symmetry, e.g., that in a face-centred cubic metal the diffusing interstitials cannot be located at the centres of the octahedral or the tetrahedral interstices. The conclusions of the preceding paragraph are critically dependent on the validity of assumption (iii). 93
Volume 82A, number 2
PHYSICS LETTERS
If (iii) is violated, additional one-phonon processes may occur as has been discussed very recently [ 14] within a quantum theory of diffusion beyond the Condon approximation [15] which combines the quantum-diffusion approaches o f Sussmann [16] and of Flynn and Stoneham [10]. If one goes beyond the Condon approximation, a linear T dependence of w inc may result from a reduction o f the tunnelling barrier via one-phonon processes, a phenomenon not included in common small-polaron theories. According to ref. [14] this leads to a low-temperature T 1 law for w inc in all those systems in which the most effective transition channels are characterized by coincidence configurations that do not possess point-inversion symmetry.
References [1] J. Yamashita and T. Kuromwa, J. Phys. Chem. Solids 5 (1958) 34. [2] T. Holstein, Ann. Phys. (NY) 8 (1959) 343.
94
9 March 1981
[3] W. K~lzig, J. Phys. Chem. Solids 23 (1962) 479. [4] S. Kapphan and F. Liity, Solid State Commun. 8 (1970) 349. [5] J.A. Sussmann, Phys. Konden~ Mater. 2 (1964) 146. [6] R. Pirc, B. ~ek] and P. Gosar, J. Phys. Chem. Solids 27 (1966) 1219. [7] P. Gosar and R. Pirc, in: CoUoque Ampere XIV, ed. R. ' Blinc (North-Holland, Amsterdam, 1967) p. 636. [8] B.G. Dick and D. Strauch, Phys. Rev. B2 (1970) 2200. [9] H.B. Shore and L.M. Sander, Phys. Rev. B12 (1975) 1546. [I0] C.P. Flynn and A.M. Stoneham, Phys. Rev. B1 (1970) 3966. [11] Yu. Kagen and M.I. Klinger, J. Phys. C7 (1974) 2791. [12] G. Le~fried and N. Breuer, in: Point defects in metals I, Springer tracts in modern physics, Vol. 81 (Springer, Berlin, 1978). [13] E. Yagi et al., in: Proc. 2nd Intern. Topical Meeting on Muon spin rotation, to be published in J. Hyp. Int. [14] H. Teichler, in: Proc. 2nd Intern. Topical Meeting on Muon spin rotation, to be published in J. Hyp. Int. [15] H. Teichler, submitted to Phys. Star. Sol. Co). [16] J.A. Sussmann, Ann. de Phys. (Paris) 6 (1971) 135.
PHYSICS LETTERS
9 March 1981
ONE-PHONON PROCESSES IN PHONON-ASSISTED TUNNELLING JUMPS
AT LOW TEMPERATURES H. TEICHLER Institut fi~r Metallphysik, Universit~t G6ttingen, G~ttingen, Germany and
A. SEEGER Institut ffir Physik, Max-Planck-lnstitutfiir Metallforschung and Institut j~'r Theoretische und Angewandte Phystk, Universitiit Stuttgart, Stuttgart, Germany Received 25 November 1980
The conditions for light-interstitialdiffusion by "incoherent tunnelling" via one-phononprocessesare investigated.If elementary diffusionjumps are involvedwhich changethe crystallographicorientations of the defects, one-phononprocesses may give rise to a linear temperature dependenceof the diffusion coefficient.
Phonon-assisted tunnelling jumps of defects (in the following called "incoherent tunnelling") play an important r61e in several areas of solid-state physics, among them small-polaron hopping (where the theoretical ideas were originally introduced [ 1,2]), reorienration processes of dipolar defects [3-9], and lowtemperature diffusion of light interstitials in crystals (so-called quantum diffusion [10,11]). An important question is that of the temperature variation of the transition rate at low temperatures. The temperature law depends critically on whether the lowest nonvanishing contribution comes from one-phonon or from two-phonon processes. Experiments on dipolar reorientation are often carried out in an applied field which removes the energetic degeneracy of the configurations between which the transitions take place. It has been shown by both experiment and theory [3,7] that even if this degeneracy is restored (i.e., if the applied field is reduced to zero) the so-called one-phonon processes give the lowest non-vanishing contribution to the transition rate. This means that in this case the limiting law at low temperatures for incoherent tunnelling is a T I law.
The situation is quite different in the case of quantum diffusion in metals. Both Flynn and Stoneham [10] and Kagan and Klinger [11] claim that the lowest non-vanishing contribution comes from two-phonon processes, with the consequence that the limiting lowtemperature law for incoherent tunnelling is predicted to be T7. In the present note we investigate the origin of the difference between these results and those summarized in the preceding paragraph. We shall see that the Flynn-Stoneham re.~ult [I0] is conditional upon assumptions which in practice are often violated. In such cases one-phonon processes do contribute to the rate of diffusion by incoherent tunnelling and lead to low-temperature diffusion coefficients which are proportional to T 1. As most papers in the field, we restrict ourselves to dilute systems (i.e., we consider only one defect at any one time) and make the assumptions of (i) adiabatic dec0upling of defect and host-material dynamics, (ii) linear coupling approach for the defect-lattice interaction, (iii) Condon approximation for the transfer integrals,
0 031-9163/81/0{300-0000/$ 02.50 © North-Holland Publishing Company
91
Volume 82A, number 2
PHYSICS LETTERS
(iv) harmonic approximation for the lattice vibrations, and start with the hamiltonian (for further details see ref. [10])
i
l,n
+
ai aignUn
(z)
+
wi/= lim (a/at)(n/(t)nl(O))/(ni(O)) , n k = a~ak , t--, ®
dr e - i ~ r
X Iexp (~q Pq COS(COqr)/sirth(hCOq/2kT) ) - I1 • In eq. (3) the following abbreviations have been used:
,oxp
,
~ 2 ----e / - ei is the energy difference between the final and initial states of the defect. (M is the atomic mass of the host material, N is the number of translation cells in the Born-yon Karman volume.) Eq. (3) may be derived by an approach analogous to that used by Gosar and Pirc [7]. The term - 1 occurring in the square brackets in eq. (3) guarantees that the coherent transitions are properly eliminated. In the following we consider the configurational average of wi/me ,
92
eq)12 sin2(q • all2)
(6)
In eq. (6)
p/),
(7)
denote the even and odd linear combinations, respectively,of the double force tensors pi and P/of the defect in its initialand final state,whose (ta,v) cartesian components are given by (see, e.g.,ref. [12])
(k= i,:)
<8)
n
--oa
f da
1-'q = (2MN'h6Oq3)-1 [1(q" pe.
--
(2)
the rate of transitions of the defect from state i to state/, becomes to second order with respect to J
=
where p(I2) is the appropriate distribution function of the energy difference ~2. In investigating the contribution of one-phonon processes we may confine ourselves to the limit of long wavelengths and rewrite eq. (4) a s
+ I(q" p o . %)12 cos2(q, d/2)] .
Here a~c,a k denote creation and annihilation operators for the defect in the state k, gnk the force of the defect in sta~e k on the host atom at lattice point R n, u n the + displacement of this atom, bq, bq creation and annihilation operators for phonon mode q with frequency c~q and polarization e~, J the defect transfer integral. In this model the incoherent contribution to
Iv~.ne = ]J,/~}2 f
9 March 1981
fS)
Furthermore, d means the distance vector between the initial and Final site of the defect. The usual treatments of quantum diffusion are equivalent to the choice po = 0, i.e., only the first term in eq. (6) is retained. This corresponds to the assumption that in a diffusional jump the strain fields in the initial and in the final states are the same, imp!ying that the two equilibrium configuration s differ only by a translation. In theories of dipolar reorientation without diffusion the first term in eq. (6) vanishes because o f d = 0. The only contribution comes from the second term, which reflects the change of orientation of the defect. In general such a change in orientation also occurs during diffusion of interstitial atoms, as is illustrated by the nearest-neighbour jumps of interstitial atoms located on either the octahedral or the tetrahedral sites in bcc metals. Both types of sites possess tetragonal symmetry. Nearest-neighbour jumps involve a change in the tetragonality axes. This means that in a general treatment of diffusion we must retain both the pe and the po term. The terms corresponding to one-,two-, etc. phonon processes are obtained by expanding the exponential in the square brackets in eq. (3) in a power series. If the width of p(~2) is small compared to the I)ebye frequency w D, the one-phonon contribution to w ine becomes for a three-dimensional system with linear dispersion of the long-wavelength phonons and kT /ic~D:
Volume 82A, number 2
wnO=
PHYSICS LETTERS
oxp( (9)
oo
× / d[2 p([2) h~2 exp(-h[2/2kT) (CO+/i2122ce) 2kT sinh (h[2/2kT) - -
oo
C°=~V- I ~q [5(Wq+[2)+5(Wq-[2)] (10a) X Iq" pa
C ~ -~Vq~
"eq [2/(hWq) 4 , [5 (¢oq + [2) + 5(t% - [2)]
X (q. d/2) 2 [q" ps.
eq [2/(h~q)6 ,
(lOb)
where under the above assumptions C ° and C e are independent of I2 for [I2l "¢ ~D. Under the additional assumption that the width of p([2) is small compared to 2kT, the one-phonon processes give for the incoherent tunnelling rate: w~he=
perimentally found to be proportional to T. If the conditions (i)-(iv) are satisfied one may conclude that the diffusion jumps either must involve a change in the double-force tensor of the d___efectconfiguration or are made possible by a finite I22 produced by lattice imperfections. Whether the latter possibility obtains may be tested experimentally either by investigating the influence of the crystal perfection or by studying the temperature dependence ofw ine outside the range of validity of the T 1 law. Consider as an example a distribution p ([2) restricted to [2 I> 0 (this corresponds to thermally activated "up-hill" hopping), varying only slowly for 0 < I2 < ([22)1/2 and dropping rapi_~y to zero for I2 ([22)1/2. Here for kT ~-,~ (I22) 1/2 a transition takes place from the high-temperature T 1 law to a low-temperature relationship:
w~ne=~rr(lrkT)2J2 exp ( - q~ I'q)
(13)
X [C° +-~ (rtkT)2Ce]p(O)/h.
lrkTJ2 exp ( - ~ r'q)(C°+ ~i212---2ce), (ll) co
- f
9 March 1981
d[2p(~)[2
2 .
(12)
po contributes to wjne a term linear in T which is independent of any further details of p ([2), whereas pe yields a__contribution linear in T which is proportional to ~2 and which experimentally may be made small by using crystals of high purity and high perfection. The most important result from the preceding discussion is the prediction of a finite contribution of one-phonon processes to the transition rate of diffusional incoherent-tunnelling jumps (leading to a T 1 law at low temperatures) even if the energy levels are practically degenerate, i.e. p([2) "~ 5 (~2), inall those situations in which the diffusion jumps involve changes of the crystallographic orientation of the defect configuration. As already discussed, such a situation is realized in the diffusion of fight interstitial atoms in bcc metals. The predicted behaviour has indeed been recently observed for the diffusion of positive muons in ~-Fe [13]. Let us now reverse the problem and ask which conclusions can be drawn if the low-temperature diffusion coefficient associated with incoherent tunnelling is ex-
Since this transition depends on the degree of perfection of the crystal it should not be too difficult to distinguish it from the case that the jumps occur between physically distinct sites (e.g., octahedral and tetrahedral interstices) with e / - ei = Ae = constant (i :/:/'),. corresponding to p([2) = 5 ([2 - Ae). In this case wJne "- T 1 as valid for kT >~Ae will be replaced at low temperatures by an approximate Arrhenius law with activation energy equal to Ae. Positive-muon diffusion experiments are particularly well suited for identifying the origin of the onephonon contributions to w inc. The possibility of level shifts [2 due to the interaction between different diffusing particles is excluded and therefore a small width of p([2) may be realized by using crystals of high purity and high perfection. Once a T 1 dependence of w inc, independent of the lattice perfection, is established we may conclude that po is f'mite, provided assumptions (i)-(iv) are admissible. For transitions between equivalent sites in cubic crystals this means that the defect configuration cannot have cubic symmetry, e.g., that in a face-centred cubic metal the diffusing interstitials cannot be located at the centres of the octahedral or the tetrahedral interstices. The conclusions of the preceding paragraph are critically dependent on the validity of assumption (iii). 93
Volume 82A, number 2
PHYSICS LETTERS
If (iii) is violated, additional one-phonon processes may occur as has been discussed very recently [ 14] within a quantum theory of diffusion beyond the Condon approximation [15] which combines the quantum-diffusion approaches o f Sussmann [16] and of Flynn and Stoneham [10]. If one goes beyond the Condon approximation, a linear T dependence of w inc may result from a reduction o f the tunnelling barrier via one-phonon processes, a phenomenon not included in common small-polaron theories. According to ref. [14] this leads to a low-temperature T 1 law for w inc in all those systems in which the most effective transition channels are characterized by coincidence configurations that do not possess point-inversion symmetry.
References [1] J. Yamashita and T. Kuromwa, J. Phys. Chem. Solids 5 (1958) 34. [2] T. Holstein, Ann. Phys. (NY) 8 (1959) 343.
94
9 March 1981
[3] W. K~lzig, J. Phys. Chem. Solids 23 (1962) 479. [4] S. Kapphan and F. Liity, Solid State Commun. 8 (1970) 349. [5] J.A. Sussmann, Phys. Konden~ Mater. 2 (1964) 146. [6] R. Pirc, B. ~ek] and P. Gosar, J. Phys. Chem. Solids 27 (1966) 1219. [7] P. Gosar and R. Pirc, in: CoUoque Ampere XIV, ed. R. ' Blinc (North-Holland, Amsterdam, 1967) p. 636. [8] B.G. Dick and D. Strauch, Phys. Rev. B2 (1970) 2200. [9] H.B. Shore and L.M. Sander, Phys. Rev. B12 (1975) 1546. [I0] C.P. Flynn and A.M. Stoneham, Phys. Rev. B1 (1970) 3966. [11] Yu. Kagen and M.I. Klinger, J. Phys. C7 (1974) 2791. [12] G. Le~fried and N. Breuer, in: Point defects in metals I, Springer tracts in modern physics, Vol. 81 (Springer, Berlin, 1978). [13] E. Yagi et al., in: Proc. 2nd Intern. Topical Meeting on Muon spin rotation, to be published in J. Hyp. Int. [14] H. Teichler, in: Proc. 2nd Intern. Topical Meeting on Muon spin rotation, to be published in J. Hyp. Int. [15] H. Teichler, submitted to Phys. Star. Sol. Co). [16] J.A. Sussmann, Ann. de Phys. (Paris) 6 (1971) 135.