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The Mooij criterion and weak localization
Solid State Communications, Vol. 42, No. 7, pp. 521-523, 1982. Printed in Great Britain.
0038-1098/190521-03503.00/0 Pergamon Press Ltd.
THE MOOIJ CRITERION AND WEAK LOCALIZATION D.S. McLachlan* IBM Thomas J. Watson Research Center, Yorktown Heights, IVY 10598, U.S.A. (Received 20 October 1981 by G. B u m s )
Formulae are obtained which show that when the residual resistivity of a metal lies in the range, 100-200 ttf2-cm, where the temperature coefficient of resistivity ot becomes zero (Mooij Criterion), the electron mean free path, the weak field coherence length and the Fermi wavelength are approximately equal. Arguments are made to show that the concept of weak localization must break down at the resistivity where the a ~ O. In degenerate semiconductors these formulae predict the residual resistivity at which the Hall mobility is a maximum and where the negative magnetoresistivity, associated with weak localization, increases rapidly. MOOIJ [1 ] FIRST POINTED OUT the correlation between the resistivity values and the occurrence of a negative temperature coefficient of resistance (~t) in a large number of amorphous and highly disordered metal conductors, r, is found to be generally positive for metals with p < 100#I2-cm and usually negative for p > 200 ttI2-cm. The reason for this phenomenon has never been explained. In this paper two equations for the resistivity Pm, where a is zero, as a function of carrier concentration are presented. Although the equations are arrived at from different limits, they qualitatively agree with each other and combining the two equations shows that at Pm the electron mean free path (l), the weak field coherence length (~) and the Fermi wavelength (~Y) are approximately equal. The implications of this observation are discussed. In the latter part of the paper a brief review of some of the experimental evidence supporting the above equations and the occurrence of weak localization when p > P m is presented. Using arguments based on the uncertainty principle, Ioffe and Regel [2] have written down relationships that show where the usual theory of electrical conduction in semiconductors break down, as the mean free path (/) of the carriers approaches either the temperature dependent wavelength (kT) of the electrons, or the interatomic spacing (a) (in metals kF [the Fermi wavelength] replaces ~'r)- While the latter condition is widely quoted in the literature the former, especially in the case of metals, seems to have been largely overlooked. This is probably because for metals ),F ~" a. Therefore, using the uncertainty relationship and the criterion that Ak/k = 1, the Ioffe-Regel criterion also holds for )re = / , , i n or Ax (the size of the electron wavepacket).
Mott and others [3] using the Kubo-Greenwood formula calculate a conductivity for l ~ a of o = p - I = SFe2a/241r2h = e2/2ha,
where SF (the Fermi surface area) is taken as 4rtk 2 and k as 1fla. In another modification [3] of the calculation the 2 in equation (1) is replaced by 2.56. Gurvitch [4] interprets the p in equation (1) as Psat, the maximum resistivity that a metal with a positive temperature coefficient of resistance can have as T ~ oo. However, at T = 0, equation (1) can be looked on as the criterion for the onset of weak localization in the material. The conductivity Ornin at the metal insulator transition [3] is Omin = Ce2/hn~ 1/3 = Cle~(rs/ao)/h
(2)
where Cand C 1 ~ 1/20, n c is the critical electron concentration and (rs/ao) is the radius of the free electron sphere r s in units of the Bohr radius ao. Using the criterion that l ~- ?,Y rather than a, together with SF = 41rk~. = 41r(21r/Xe) 2, in equation (1)
~VCS Pm ~ 3h/e2kF ~ 33.9 x (rs/ao)lJI2-cm.
(3)
As for most metals (rm/ao) ranges for "" 2 to 3.5, equation (3)gives Pm in the range 68 to 119/JI2-cm. This range of resistivity values is close to the range (100-200/JI2-cm) where, according to the Mooij [1 ] criterion, metallic conductors will have a zero or. Recently McLachlan [5] proposed a phenomenological formula to account for both positive and negative ot's in metallic systems and showed that this formula quantitatively fits the experimental results for TiOo.so- 1.23. a metallic oxide series. This formula is,
p(r) = p(o)+ [(X-~)/X]p~,,(T),
* Permanent address: University of the Witwatersrand, Johannesburg, South Africa.
(1)
(4)
where ~ is a length of the order of the elastic mean free 521
522
THE MOOIJ CRITERION AND WEAK LOCALIZATION
path let, ~ the coherence length [6] for weak localization and Pin(T) the inelastic or temperature dependent resistivity. Using the formula [6] p(0) ~ ~ / e ~
(5)
and [7] let ~ 92(rs/ao)2p(O) -~ A,
(6)
one obtains for the Mooij [1] resistivity, where le~ ~ X = ~ and 0t = 0, Pra ~ 61.4(rs/ao)la~2-cm.
(7)
For (rs/ao) in the range 2-3.5 this also approximately gives the range of resistivities given in the Mooij criterion [ 1]. Combining equations (3) and (7) shows that, in the range of resistivities where a is zero, ~ -~ t ~ X .~ A x .
(8)
In approaching the Mooij resistivity from the low resistivity side it is easy to visualize why the usual theory of conductivity breaks down [2] when the size wavepacket becomes equal to the mean free path or alternatively the lifetime of the wavepacket becomes equal to the relaxation time. This breakdown would appear to be the limit of positive a's with increasing p(0). ~, the coherence length, in the weak localization limit may be visualized as the smallest radius that a section of the material must have in order to exhibit a resistivity characteristic of the bulk material. When the region of radius ~ contains only a single wavepacket (~ ~ Ax), or a single mean free path (~ ~ l), it is obvious that the weak localization picture must break down. (For a discussion of the role the dimension of the coherence length ~ plays in phase transitions and other phenomena see [8].) References [1], [9], [10] and [5] have suggested that negative ct's are associated with the weak localization of the electrons. The above arguments which show how at the Mooij resistivity the usual conductivity and weak localization pictures simultaneously break down partially justifies this suggestion. An examination of equations (1), (2) and (3) show that if the above interpretation is correct, electrons in conductors can have an effective mean free paths of less than the electron wave length (or even the interatomic spacing) for resistivities between the onset of weak localization [equations (1) and (3)] and the metal insulator transition [equation (2)]. Ioffe and Regel [2] give several examples where, in semiconductors, the mobility is such that l < ;~ (or even a). Gurvitch [4] interprets Psat to be where l ~ a. However, due to thermal excitations, there can be no localization at Psat and therefore I cannot be less than a at high
Vol. 42, No. 7
temperatures. If I ~ a at Psat then systems, such [5] as TiOx, in which samples with large negative and positive a's have approximately the same Psat, the effective mean path at T = 0 in the negative a materials must be less than a. A major proof of the validity of equations (3) and (7) must rest on their prediction of the Mooij resisitivity as a function of electron concentration or rs/ao. Unfortunately a search of the literature showed that insufficient Hall effect measurements have been made on non-magnetic high residual resistivity materials to test equations (3) and (7). The width of the Mooij resistivity region (~ 100 ~-~ 200/a~2-cm) does however show that if there is a simple electron concentration dependence it must be rs/ao, i.e. (n -l/a) or (rs/ao) 3/2, i.e. n -1/2, as higher power dependencies give too wide a range of resistivities. If, in equation (1), kF were to be given by the free electron approximation but I were to remain equal to a, the electron concentration dependence would be n-2/a. This in the concentration dependence of Gurvitch's [4] Psat" Alternatively one can turn to degenerate semiconductors where the electron concentration differs by several orders of magnitude. Unfortunately in degenerate semiconductors there is no Mooij criterion and the situation is complicated by the fact that the electron concentration may vary with temperature. However, there is a maximum in the Hall monility as a function of electron concentration in phosphorus doped silicon at 4.2 and 77 K [ 11, 12]. This is not surprising as for a very degenerate semiconductor with a large number of ionized donors centers the mobility is low. As the number of donors decreases the Hall mobility increases at first but must pass through a maxima as it is zero again at the metal insulator transition or mobility edge [3]. This region of decreasing mobility with decreasing carrier (and donor) concentration may be thought of as one of weak localization. An examination of the data given in [ 11 ] shows that at the 4.2 K mobility maximum, p ~ 1600gl2-cm and nH ~ 1.6 x 1019 c m -3 while equations (3) and (7) give Pra ~ 1590 and 2870tzI2-cm respectively. The agreement is good considering the electron concentration has changed by three orders of magnitude. The results given by Tufte and Stelzer [12] for heavily doped n-type silicon are similar in that a flat maxima at n ~ 1019 c m -3 is observed in a plot of the Hall mobility against carrier concentration at both 4.2 and 77 K. Unfortunately the corresponding resistivities are not given. In [ 13] Rosenbaum et al. make an empirical evaluation of the region where precursive behavior, to the metal insulator, becomes important. Their results show that, for phosphorus doped silicon, deviation from the expected zero T conductivity sets in at
Vol. 42, No.7
THE MOOIJ CRITERION AND WEAK LOCALIZATION
n ~ 1.3 x 1019 cm 2, where the resistivity is approximately, 2400/aI2-cm and equations (3) and (7) give 1720 and 3100 ta~2-cm respectively. Again the agreement is excellent, especially if the numerical factors needed to bring equations (3) and (7) into more exact agreement with the Mooij criterion are inserted. Recently Kawabata [14] has shown that in the weak localization limit samples should exhibit negative magnetoresistivity and observed that this is indeed found in several semiconducting systems. Tufte and Stelzer [12] have shown that the magnitude of the negative magnetoresistance has a sharp peak at or near the metal insulator transition. At 4.2 K their inverse maximum at 20 kG would appear to lie at 7.7 x 10 is carriers while the negative magnetoresistivity for 3.3 x 1019 carriers is about half this value. Below 7.7 x 10 Is carriers a positive magnetoresistivity due to electron-electron interactions may mark an increasing negative magnetoresistance as the metal insulator transition is approached [ 15]. The situation regarding the magnetoresistivity of disordered and amorphous metals is less clear though a negative magnetoresistivity is sometimes observed (see for instance Cochrane and Strom-Olsen [16]). More experimental work on the Hall coefficient and magnetoresistance of high resistivity metallic systems is obviously needed to confirm (or otherwise) equation (3) and the ideas about weak localization expressed in this paper.
Acknowledgements - The author would like to acknowledge useful discussions with Y. Imry and P. Kes and thank M. Gurvitch for a preprint of his paper prior to publication. The remarks, suggestions and questions of M. Beasley, G. Deutscher and T. Geballe also played a role in formulating this paper.
523
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
J.H. Mooij, Phys. Status Solidi (a) 17, 521 (1973). A.F. Ioffe, Can. J. Phys. 34, 1393 (1956); A.F. Ioffe & A.R. Regel, Prog. Semicond. 4,237 (1960). N.F. Mott, Metal-Insulator Transitions. Taylor and Francis, New York (1974). M. Gurvitch, Phys. Rev. B, to be published December 1 (1981). D.S. McLachlan, Phys. Rev. (to be published). E. Abrahams, P.W. Anderson, D.C. Licciardello & T.V. Ramakrishnan, Phys. Rev. Lett. 42,673 (1979). N.W. Ashcroft & M.D. Mermin, Solid State Physics, p. 52. Holt, Rinehart and Winston, New York (1976). Y. Imry & D. Bergamn, Phys. Rev. A3, 1416, (1971); Y. Imry, G. Deutscher, D. Bergman & S. Alexander, Phys. Rev. A7,744 (1973). M. Jonson & S.M. Girvin, Phys. Rev. Lett. 43, 1447 (1979). Y. Imry, Phys. Rev. Lett. 44, 469 (1980). C. Yamanouchi, K. Mizuguchl & W. Sasaki, J. Phys. Soc. Japan 22,859 (1967). O.N. Tufte & E.L. Stelzer, Phys. Rev. 139, A265 (1965). T.F. Rosenbaum, K. Andres, G.A. Thomas & R.N. Bhatt, Phys. Rev. Lett. 45, 1723 (1980). A. Kawabata, Solid State Commun. 34, 431 (1980); J. Phys. Soc. Japan 49A, 375 (1980). T.F. Rosenbaum, R.F. Mllligan, G.A. Thomas, P.A. Lee, T.V. Ramakrishnan, R.N. Bhatt, K. DeConde, H. Hess & T. Perry (to be published). R.W. Cochrane & J.O. Strom-Olsen,J. Phys. F: MetalPhys. 7, 1799 (1977).
0038-1098/190521-03503.00/0 Pergamon Press Ltd.
THE MOOIJ CRITERION AND WEAK LOCALIZATION D.S. McLachlan* IBM Thomas J. Watson Research Center, Yorktown Heights, IVY 10598, U.S.A. (Received 20 October 1981 by G. B u m s )
Formulae are obtained which show that when the residual resistivity of a metal lies in the range, 100-200 ttf2-cm, where the temperature coefficient of resistivity ot becomes zero (Mooij Criterion), the electron mean free path, the weak field coherence length and the Fermi wavelength are approximately equal. Arguments are made to show that the concept of weak localization must break down at the resistivity where the a ~ O. In degenerate semiconductors these formulae predict the residual resistivity at which the Hall mobility is a maximum and where the negative magnetoresistivity, associated with weak localization, increases rapidly. MOOIJ [1 ] FIRST POINTED OUT the correlation between the resistivity values and the occurrence of a negative temperature coefficient of resistance (~t) in a large number of amorphous and highly disordered metal conductors, r, is found to be generally positive for metals with p < 100#I2-cm and usually negative for p > 200 ttI2-cm. The reason for this phenomenon has never been explained. In this paper two equations for the resistivity Pm, where a is zero, as a function of carrier concentration are presented. Although the equations are arrived at from different limits, they qualitatively agree with each other and combining the two equations shows that at Pm the electron mean free path (l), the weak field coherence length (~) and the Fermi wavelength (~Y) are approximately equal. The implications of this observation are discussed. In the latter part of the paper a brief review of some of the experimental evidence supporting the above equations and the occurrence of weak localization when p > P m is presented. Using arguments based on the uncertainty principle, Ioffe and Regel [2] have written down relationships that show where the usual theory of electrical conduction in semiconductors break down, as the mean free path (/) of the carriers approaches either the temperature dependent wavelength (kT) of the electrons, or the interatomic spacing (a) (in metals kF [the Fermi wavelength] replaces ~'r)- While the latter condition is widely quoted in the literature the former, especially in the case of metals, seems to have been largely overlooked. This is probably because for metals ),F ~" a. Therefore, using the uncertainty relationship and the criterion that Ak/k = 1, the Ioffe-Regel criterion also holds for )re = / , , i n or Ax (the size of the electron wavepacket).
Mott and others [3] using the Kubo-Greenwood formula calculate a conductivity for l ~ a of o = p - I = SFe2a/241r2h = e2/2ha,
where SF (the Fermi surface area) is taken as 4rtk 2 and k as 1fla. In another modification [3] of the calculation the 2 in equation (1) is replaced by 2.56. Gurvitch [4] interprets the p in equation (1) as Psat, the maximum resistivity that a metal with a positive temperature coefficient of resistance can have as T ~ oo. However, at T = 0, equation (1) can be looked on as the criterion for the onset of weak localization in the material. The conductivity Ornin at the metal insulator transition [3] is Omin = Ce2/hn~ 1/3 = Cle~(rs/ao)/h
(2)
where Cand C 1 ~ 1/20, n c is the critical electron concentration and (rs/ao) is the radius of the free electron sphere r s in units of the Bohr radius ao. Using the criterion that l ~- ?,Y rather than a, together with SF = 41rk~. = 41r(21r/Xe) 2, in equation (1)
~VCS Pm ~ 3h/e2kF ~ 33.9 x (rs/ao)lJI2-cm.
(3)
As for most metals (rm/ao) ranges for "" 2 to 3.5, equation (3)gives Pm in the range 68 to 119/JI2-cm. This range of resistivity values is close to the range (100-200/JI2-cm) where, according to the Mooij [1 ] criterion, metallic conductors will have a zero or. Recently McLachlan [5] proposed a phenomenological formula to account for both positive and negative ot's in metallic systems and showed that this formula quantitatively fits the experimental results for TiOo.so- 1.23. a metallic oxide series. This formula is,
p(r) = p(o)+ [(X-~)/X]p~,,(T),
* Permanent address: University of the Witwatersrand, Johannesburg, South Africa.
(1)
(4)
where ~ is a length of the order of the elastic mean free 521
522
THE MOOIJ CRITERION AND WEAK LOCALIZATION
path let, ~ the coherence length [6] for weak localization and Pin(T) the inelastic or temperature dependent resistivity. Using the formula [6] p(0) ~ ~ / e ~
(5)
and [7] let ~ 92(rs/ao)2p(O) -~ A,
(6)
one obtains for the Mooij [1] resistivity, where le~ ~ X = ~ and 0t = 0, Pra ~ 61.4(rs/ao)la~2-cm.
(7)
For (rs/ao) in the range 2-3.5 this also approximately gives the range of resistivities given in the Mooij criterion [ 1]. Combining equations (3) and (7) shows that, in the range of resistivities where a is zero, ~ -~ t ~ X .~ A x .
(8)
In approaching the Mooij resistivity from the low resistivity side it is easy to visualize why the usual theory of conductivity breaks down [2] when the size wavepacket becomes equal to the mean free path or alternatively the lifetime of the wavepacket becomes equal to the relaxation time. This breakdown would appear to be the limit of positive a's with increasing p(0). ~, the coherence length, in the weak localization limit may be visualized as the smallest radius that a section of the material must have in order to exhibit a resistivity characteristic of the bulk material. When the region of radius ~ contains only a single wavepacket (~ ~ Ax), or a single mean free path (~ ~ l), it is obvious that the weak localization picture must break down. (For a discussion of the role the dimension of the coherence length ~ plays in phase transitions and other phenomena see [8].) References [1], [9], [10] and [5] have suggested that negative ct's are associated with the weak localization of the electrons. The above arguments which show how at the Mooij resistivity the usual conductivity and weak localization pictures simultaneously break down partially justifies this suggestion. An examination of equations (1), (2) and (3) show that if the above interpretation is correct, electrons in conductors can have an effective mean free paths of less than the electron wave length (or even the interatomic spacing) for resistivities between the onset of weak localization [equations (1) and (3)] and the metal insulator transition [equation (2)]. Ioffe and Regel [2] give several examples where, in semiconductors, the mobility is such that l < ;~ (or even a). Gurvitch [4] interprets Psat to be where l ~ a. However, due to thermal excitations, there can be no localization at Psat and therefore I cannot be less than a at high
Vol. 42, No. 7
temperatures. If I ~ a at Psat then systems, such [5] as TiOx, in which samples with large negative and positive a's have approximately the same Psat, the effective mean path at T = 0 in the negative a materials must be less than a. A major proof of the validity of equations (3) and (7) must rest on their prediction of the Mooij resisitivity as a function of electron concentration or rs/ao. Unfortunately a search of the literature showed that insufficient Hall effect measurements have been made on non-magnetic high residual resistivity materials to test equations (3) and (7). The width of the Mooij resistivity region (~ 100 ~-~ 200/a~2-cm) does however show that if there is a simple electron concentration dependence it must be rs/ao, i.e. (n -l/a) or (rs/ao) 3/2, i.e. n -1/2, as higher power dependencies give too wide a range of resistivities. If, in equation (1), kF were to be given by the free electron approximation but I were to remain equal to a, the electron concentration dependence would be n-2/a. This in the concentration dependence of Gurvitch's [4] Psat" Alternatively one can turn to degenerate semiconductors where the electron concentration differs by several orders of magnitude. Unfortunately in degenerate semiconductors there is no Mooij criterion and the situation is complicated by the fact that the electron concentration may vary with temperature. However, there is a maximum in the Hall monility as a function of electron concentration in phosphorus doped silicon at 4.2 and 77 K [ 11, 12]. This is not surprising as for a very degenerate semiconductor with a large number of ionized donors centers the mobility is low. As the number of donors decreases the Hall mobility increases at first but must pass through a maxima as it is zero again at the metal insulator transition or mobility edge [3]. This region of decreasing mobility with decreasing carrier (and donor) concentration may be thought of as one of weak localization. An examination of the data given in [ 11 ] shows that at the 4.2 K mobility maximum, p ~ 1600gl2-cm and nH ~ 1.6 x 1019 c m -3 while equations (3) and (7) give Pra ~ 1590 and 2870tzI2-cm respectively. The agreement is good considering the electron concentration has changed by three orders of magnitude. The results given by Tufte and Stelzer [12] for heavily doped n-type silicon are similar in that a flat maxima at n ~ 1019 c m -3 is observed in a plot of the Hall mobility against carrier concentration at both 4.2 and 77 K. Unfortunately the corresponding resistivities are not given. In [ 13] Rosenbaum et al. make an empirical evaluation of the region where precursive behavior, to the metal insulator, becomes important. Their results show that, for phosphorus doped silicon, deviation from the expected zero T conductivity sets in at
Vol. 42, No.7
THE MOOIJ CRITERION AND WEAK LOCALIZATION
n ~ 1.3 x 1019 cm 2, where the resistivity is approximately, 2400/aI2-cm and equations (3) and (7) give 1720 and 3100 ta~2-cm respectively. Again the agreement is excellent, especially if the numerical factors needed to bring equations (3) and (7) into more exact agreement with the Mooij criterion are inserted. Recently Kawabata [14] has shown that in the weak localization limit samples should exhibit negative magnetoresistivity and observed that this is indeed found in several semiconducting systems. Tufte and Stelzer [12] have shown that the magnitude of the negative magnetoresistance has a sharp peak at or near the metal insulator transition. At 4.2 K their inverse maximum at 20 kG would appear to lie at 7.7 x 10 is carriers while the negative magnetoresistivity for 3.3 x 1019 carriers is about half this value. Below 7.7 x 10 Is carriers a positive magnetoresistivity due to electron-electron interactions may mark an increasing negative magnetoresistance as the metal insulator transition is approached [ 15]. The situation regarding the magnetoresistivity of disordered and amorphous metals is less clear though a negative magnetoresistivity is sometimes observed (see for instance Cochrane and Strom-Olsen [16]). More experimental work on the Hall coefficient and magnetoresistance of high resistivity metallic systems is obviously needed to confirm (or otherwise) equation (3) and the ideas about weak localization expressed in this paper.
Acknowledgements - The author would like to acknowledge useful discussions with Y. Imry and P. Kes and thank M. Gurvitch for a preprint of his paper prior to publication. The remarks, suggestions and questions of M. Beasley, G. Deutscher and T. Geballe also played a role in formulating this paper.
523
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
J.H. Mooij, Phys. Status Solidi (a) 17, 521 (1973). A.F. Ioffe, Can. J. Phys. 34, 1393 (1956); A.F. Ioffe & A.R. Regel, Prog. Semicond. 4,237 (1960). N.F. Mott, Metal-Insulator Transitions. Taylor and Francis, New York (1974). M. Gurvitch, Phys. Rev. B, to be published December 1 (1981). D.S. McLachlan, Phys. Rev. (to be published). E. Abrahams, P.W. Anderson, D.C. Licciardello & T.V. Ramakrishnan, Phys. Rev. Lett. 42,673 (1979). N.W. Ashcroft & M.D. Mermin, Solid State Physics, p. 52. Holt, Rinehart and Winston, New York (1976). Y. Imry & D. Bergamn, Phys. Rev. A3, 1416, (1971); Y. Imry, G. Deutscher, D. Bergman & S. Alexander, Phys. Rev. A7,744 (1973). M. Jonson & S.M. Girvin, Phys. Rev. Lett. 43, 1447 (1979). Y. Imry, Phys. Rev. Lett. 44, 469 (1980). C. Yamanouchi, K. Mizuguchl & W. Sasaki, J. Phys. Soc. Japan 22,859 (1967). O.N. Tufte & E.L. Stelzer, Phys. Rev. 139, A265 (1965). T.F. Rosenbaum, K. Andres, G.A. Thomas & R.N. Bhatt, Phys. Rev. Lett. 45, 1723 (1980). A. Kawabata, Solid State Commun. 34, 431 (1980); J. Phys. Soc. Japan 49A, 375 (1980). T.F. Rosenbaum, R.F. Mllligan, G.A. Thomas, P.A. Lee, T.V. Ramakrishnan, R.N. Bhatt, K. DeConde, H. Hess & T. Perry (to be published). R.W. Cochrane & J.O. Strom-Olsen,J. Phys. F: MetalPhys. 7, 1799 (1977).