HAL Id: jpa-00247010
https://hal.archives-ouvertes.fr/jpa-00247010
Submitted on 1 Jan 1994
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Two dimensional weak localization beyond the diffusion approximation
A. Cassam-Chenai, B. Shapiro
To cite this version:
A. Cassam-Chenai, B. Shapiro. Two dimensional weak localization beyond the diffusion approxi- mation. Journal de Physique I, EDP Sciences, 1994, 4 (10), pp.1527-1537. �10.1051/jp1:1994103�.
�jpa-00247010�
J. Phys. I France 4 (1994) 1527-1537 OCTOBER1994, PAGE 1527
Classification Physics Abstracts
72,15G
Two dimensional weak localization beyond trie diffusion
approximation
A. Cassam-Chenai(~) and B. Shapiro(~,~)
(~) CNRS, LMM, 196 Av. Henry Ravera, BP107, 92225 Bagneux Cedex, France (~) Department of Physics, Technion-Israel Institute of Technology, 32000 Haifa, Israel
(Received 30 March 1994, received in final form 15 June 1994, accepted 30 June 1994)
Abstract, The standard theory of weak localization is based on a diffusion approximation.
It produces quantitatively accurate results only when trie phase breaking length L~ and trie
magnetic length LH are very large compared with trie mean free path f. In some practical cases
L~ and LH are only moderately large m comparison with f. For such cases a better quantita-
tive description of weak localization is obtained by gomg beyond trie diffusion approximation.
We present theoretical results for the weak localization correction which are not based on the diffusion approximation. We aise present some experimental data and compare them with trie theory.
1. Introduction.
The phenomenon of weak localization is due to quantum interference which accompanies the dilfusive propagation of electrons in a weakly disordered metal or sen~iconductor. Within the
quasiclassical picture the quantum correction to the diffusion coefficient D, or to the conduc- tivity a, is proportional to the retum probability for a diffusing partiale [1]. More rigorously [2-6], the weak locahzation correction is described with the help of a diagran~matic series for the Cooperon ladder r (Fig. l). Solid fines represent the averaged, retarded or advanced,
Green functions GR and GA, and dashed fines represent scattering on impurities. An expres- sion &/2grpir is assigned to a dashed fine, where pi and r are the density of states (per spin)
and the momentum relaxation time. For short range impurities (in the Bom approxin~ation)
or for a white-noise potential the series con be summed up, in the momentum representation.
The result, in two dimensions, is [2]: r(q) = u~PoIl Po)~~, where u~ e &/2grpir and p~j~~
~ j~ ~ ~~~~~-i/~ j~~
is the Fourier transform of u~ GR(ri rz) Î~. The elastic mean free path is desigi~ated by é.
Equation il) is vahd for q smaller than the Fermi wave number kF.
Fig. 1. The Cooperon ladder.
r~ rz
~
j'i'1
r~ rz
r ', ,' r' r r' r r'
,~', rz
r~ r
hi rz ) fi
~ ', _,'
(a) (b) (c)
Fig. 2. The two-impunty diagrams for the conductivity correction.
In the diffusion approximation r(q) is approximated by 2u~/q~é~ which leads to the well- known logarithmic correction for the conductivity:
~'~ 212& ~~ l
~ iÎ& ~~ ~~~
where rj and Lj = /~ are the phase breaking tinte and length, respectively. The accuracy of the diffusion approximation improves with the increase of the parameter L~lé. In many
practical cases, however, this parameter is not extremely large and a more accurate treatment, not based on the diffusion approximation, might lead to a better quantitative description of the quantum correction ôa. In section 2 we will see that such a treatment requires a careful
analysis of the twc-impurity diagrams for the conductivity. In particular, it will be shown that the contribution to ôa of the first diagram in figure 1 exactly cancels with the contribution from the two other two-impurity diagrams given in figure 2. Thus, as far as the conductivity
is concerned, the Cooperon ladder should start with the three-impurity diagram.
In the presence of a magnetic field H a new length, the magnetic length LH % (&cleH)~/~,
appears in the problem. It was pointed out by Kawabata
[GI that the standard treatment of the quantum correction ôa(H), based on the diffusion approximation, is quantitatively accurate
only for exceedingly large values of the ratio LHlé. In order to obtain better quantitative results for moderately large values of LHIi, Kawabata has derived an essentially exact expression for the Cooperon. However, his expression for the conductivity correction ôa(H) contains an
error. Although the error cancels out in the final expression for the magnetoconductivity
Aa(H) e ôa(H) ôa(0), that expression is somewhat inconvenient: it certains a difference of two terms, each of which depends on an upper cut-off No (H) and diverges when No - oc.
Following, with some modification, the approach of Kawabata
[GI we derive in section 3 an expression for ôa(H) with no upper cut-off:
~2f2 °° p3
~'l~~
" ~7r2&Lj
~
i -§~v
where
N°10 WEAK LOCALIZATION BEYOND DIFFUSION 1529
~~~ ~ ~~~ ~ ~lrj~'
This expression is quite convenient for comparison with experimental data, which are pre- sented and analyzed in section 4. The paper is closed by some conclusions ai~d remarks (Sect. 5).
2. Cancellation of tue two-impurity diagrams for tue conductivity.
The diffusion approximation relies on the fact that the main contribution to the Cooperon
comes from long dilfusive trajectories, up tu a scale of order Lj. Short trajectories, corre-
spondii~g tu the begii~ning of the series in figure 1, are of no importance in this diffusion approximation. For instance, adding to the series a single impurity diagram as some authors do [4-6] dues net affect the approximate expression 2u~/q~é~ for the Cooperon.
When the ratio Lj Ii land hence the number ofterms in the Cooperon ladder) is not too large,
there can be significant dilferences between the results based on the diffusion approximation and the results based on the exact treatment for the Cooperon. Moreover, the expression for the Cooperon does depend on whether one starts the series with one, two or three-impurity diagram. Clearly, the one-impurity diagram should not be included in the Cooperon ladder
smce it has already been counted in the diffusion ladder. Below we calculate the contribution to the conductivity of the two-impurity diagram in figure and show that it cancels with the
contribution from the two other diagrams.
In the second order perturbation theory for the four-point function land for a short-range scattering potential) only three diagrams cari make a non-zero contribution to the conductivity.
These diagrams are depicted in figure 2. These diagrams have been considered by Neal [7] many
years ago. His conclusion was that in two dimensions, unlike the three-dimensional case, the three diagrams do trot cancel and give a contribution of order (e~/&) In (kil) tu the conductivity [8]. We will prove that the diagrams in figure 2 do cancel also in two dimensions. We use the
coordinate representation, with the intention to extend the approach to finite magnetic fields (Sect. 3).
Let us start with the diagram (2a). The corresponding analytic expression is
ôafj =
/ / d~rid~r2Jo(r2,
ri, ~ù)ri(ri, r2, ~ù)Jp(r2, ri, ~ù)
,
(3)
gr
where S is the sarnple area, ri (ri, r2,~ù) = U~GR(ri, r2,~ù)GA(ri, r2, 0) is the first term of the
Coo$eron ladder m figure 1, and the vertex function Jo(r2, ri,~ù) is given by j~j~~,r~,çu)
=
~ / ~rGA(~2>~>Ù)) ~
~R(~'~~'~°~ ~~~
The retarded and advanced Green functions
~pn
GR,A(~É'~~ (~~ ~-
~ ~ ~~ ~~~Î~
~ ~~~~
~_ ~~~
jj ~~~j~~i/2 expiiik~ + [~~ ~i ~
'
where the approximate expression for GR,A is valid for R > kj~. Here we introduced a finite frequency ~ù in order to introduce later the loss of phase coherence. Using the identity for
~ùr < 1
/d~T~A(~2 r, 0)~R(~ ~l ld) " ~)(~A(~2 rl 0) ~R(r2 rl )>ld) (~)
and the approximate expression ikFÉOGR,A(R,~ù) for the derivative iI~GR,A(R, ~ù), we obtain
a simple expression for the vertex function
Jlr2, ri, ùJ)
= eié21lGRlr2 ri, ùJ) + GAlr2 ri, 0)) 17)
where1
= &kfr/m and é21 denotes a unit vector in the direction r2 ri. Substituting
the expression for ri and Jo into equation (3), we obtain the following expression for the
longitudinal component of the conductivity:
ôa(~)
=
-Îi~u~
/d~R[GR(R,~u) + GA(R,0)]~ GR(R,~u)GA(R,0) (8)
It is suliicient to keep only terms containing equal powers of GR aud GA (terms contaiu- ing nonequal powers rapidly oscillate with R and give only a negligible contribution to ôa).
Therefore, ôa(~)
= -£i~u~2/d~RG((R, 0)G((R,0) m
-$ In (kil) (9)
7r 7r
Here we have set ~u
= 0. This quantity is not necessary as long as we do not sum the series.
Consider now the sum ôa(~+~) of the two other diagrams in figure 2. The corresponding
expression dilfers from equation (3) in two respects: first, ri (r2, ri,0) should be replaced by
G((ri,r2, 0) + G((ri, r2, 0) and, second, due to the property Ja(r2, ri, 0) = -Ja(ri, r2, 0) the diagram has a dilferent sign. Thus, mstead of equation (8) we have
ôa(~+~)
= £i~u~ / d~R[GR(R,0)
+ GA(R, 0)]~[G((R,0) + G((R,0)]
7r
m £i~u~/d~R 2G((R)G((R)
,
(10)
7r
which cancels with the terni ôaa (Eq. (9)). This result implies that, when defining the
Cooperon ladder for the conductivity problem, one should discard not only the one-impurity but also the two-impurity diagram.
Using equation (7) for the vertex function, it becomes rather straightforward to compute the contribution of the n-th term m the Cooperon ladder to the conductivity. The n-th term of the Cooperon ladder is
~n(rl, rn, Ld)
# U~ / dr2.. dru-ll~O(rl r2>ld)I~O(r2 r3>ld).. ~O(rn-1 rn, Ld)
Ill)
where the kemel Po(r~ -rj, ~u) e U~GR(r~ -rj, ~u)GA(r~ -rj, 0). The corresponding conductivity diagram contains, m addition to rn, two vertex functions. These functions, as was shown above, mtroduce a factor [GR(rn ri, ~u + GA(rn ri, 0)]~ and additional integration over ri and rn.
The main contribution comes from the cross-term 2GRGA
~ Po(rn ri,~u). The additional factor Po(rn ri,~u) completes the chain of Po's which appears in equatioi~ Ill) and ei~ables
us to write the n-th conductivity diagram as
ôa~"~ ~2
= -~i~lYlÉ?) (12)
N°10 WEAK LOCALIZATION BEYOND DIFFUSION lS31
Summing ail the terms, starting with n = 3, gives
2 É3
(13)
à? " à~~~~
i -°Ao
The operator Po is diagonal in the q-representation [2, 9]. In this representation Tr is replaced by an integral over q. The phase breaking time rj is introduced, as usual, by the formai
replacement -i~u + 1/rj, so that
P0(~,Té)
" ll~ + ))~ + ~~é~l~~/~ 1~4)
4
and, via equation (13), we have à, =
~~
12/°° d2q Pllq, ri)
~h
° (2~)~ P01~,Té)
~2 ~~
2gr2h ~~ ~i ~ ~~ (là)
To leading order m the small parameter r/r~, equation (là) reduces to equation (2). However, for moderately small values of r/r~, equation (15) is expected to be more accurate thon equa-
tion (2) which was based on the diffusion approximation. Below we extend the treatment to the case of a finite magnetic field and demonstrate the advantages of going beyond the diffusion
approximation.
3. Finite magnetic fields.
In the presence of a magnetic field H, going beyond the diffusion approximation becomes necessary when one wants to study not only very weak fields but also fields of intermediate strength (LH smaller or of the saine order as 1). In any case, however, the condition ~ucr « 1
must be satisfied, ~ùc being the cyclotron frequency. Dur treatment in this section is close to that of I<awabata [6], and we will compare the results at the end of the section.
The Cooperon ladder is still defined by figure 1 but the averaged Green functions are now
given by
~fÎ(~2> rl, ld) # GR,A(r2, rl, ld)~XP(~) /r2 J~' d~) (l~) ri
where GR,A are the averaged Green functions at zero field and Air) is the vector potential.
The curvelinear integration is performed here along a straight fine going front ri to r2. The
conductivity diagrams remam the same as in the previous section but we can no longer use
the commutativity of the momentum operator with the Green function to simplify the current vertices. In equation (3), one should replace the vertex functions Ja (r2, ri, ~ù) and Jp(r2, ri, ~ù) respectively by
jAR(H)j~ ~ ~)_ ~ /~2~~(H)j~ ~~~jÎÎ
Ô
~~ j~~~~(H)j~~ ~~
OE 2, 1, £ A 2, , j~~ a R , 1>
a
~~~~~~(~21~l,ld) /
d~T~~~(~2,r>ld)(~ £ ~~fl(~))~~~(~> ~l, Ù)
p
11?)
The kernel Po(ri r2, ~ù) now becomes
P(ri,r2,~ù)
= u~Gf~(ri,r2,~ù)Gf~(ri,r2,0) (18) Equation (18) defines the operator É(~ù) in r-representation.
We can still obtain an expression similar to the one we obtained in the zero magnetic field
case. For this we use the followmg property given in appendix
3~~~~~(r2,rl>ld) 3~~~~~(r2>~l>ld)
" 3~~~~(~2,rl>ld) '3~~~~(r2>rl>ld)
where
~~~~(~2> ri, ld) ~ ~ ~ (
+ ~OE(~l)j / d~T~~~(~2,
~, Ù)~~~(~, ~l >ld)
la
(19)
This property, deduced from the isotropy, allows us ta commute the two current operators with the Green functions m the weak localization correction to the conductivity tensor.
Using the identity (6), which is valid also for the Green functions m the presence of H, one
can then simplify the expression of the new current vertices J'
3~(~2,rl)
" )~~é21 (GÙ~(~2,~l) + ~~~(~2,rl)) (2Ù)
1-e- the same as equation (7), with G(~) instead of G.
The cancellation of the diagrams in figure 2 is proven exactly m the same way as in the previous section. The replacement of J by J' also holds m the case of diagrams b) and c). In
fact, apart from small corrections, ôa(~) is not at ail affected by the magnetic field, since it contains only the combination Gf~(r2,
ri )Gf~(ri,r2) Gf~(r2,
ri)Gf~ (ri, r2) which is field mdependent (the same is true for ôa(~+~)). One can also, in complete analogy with section 2, relate ôa(") ii-e- the contribution of the n-th term in the Cooperon ladder to the conductivity)
to the n-th power of the operator É ôa(")(H)
=
-£i~ Tr É" (21)
The operator É is diagonal in the basis of the Landau functions for a partiale with a charge 2e in the magnetic field H. The projection of this operator onto the n-th Landau level is given by [6]
~ oe ~2
~~
l + z
~~~~~~~~~~~~ "
2 ' ~~~~
where LN is the Laguerre polynomial, s e il + z)(2~ùcr~eF)~~/~, and the parameter z e r/r~
accounts for the phase-breaking processes. The time rj is introduced in the saine way as for the zero field case. Writing equation (21) in the diagonal representation and summing over n, from n
= 3 to oc, we obtain the final expression for the conductivity correction
e~i~ ffpn ~~~~ f ~Î~
~~~~~ 123)
~~~~~
ji=O~=~
~ ~~~~~
~ ~
~~
N°10 WEAK LOCALIZATION BEYOND DIFFUSION 1533
The power 3, in the term P( of equation (23), appears because the first non-zero contribution to trie conductivity comes from the three-impurity diagram in the Cooperon ladder. If une had ignored the cancellation of the two-impunty diagrams and included into ôa(H) also the first diagram in figure 1, one would have obtained a term P( (instead of P() in equation (23).
Let us note that Kawabata [6] includes in the Cooperon ladder not only the twc-impurity diagram but also the one-impunty diagram, and ends up with a first power of FN in the
numerator of equation (23). The first power makes the correction ôa very large, of order of
the Boltzmann conductivity aB. This is a mistake which is due to the incorrect treatment of
the current vertices in the one-impurity diagram m reference [6] (this diagram should make
no contribution at ail to the conductivity). The mistake cancels out in the expression for the
magnetoconductance Aa(H) e ôa(H) ôa(0). Moreover, since the contribution to ôa(H)
of the two-impunty diagram m the Cooperon ladder is independent of H, this diagram also cancels out in the expression for Aa(H). Thus, although Kawabata's expression for ôa(H) (and ôa(0)) is incorrect, his final expression for the magnetoconductance Aa(H) is essentially correct, albeit somewhat inconvenient: it is written as a dif§erence of two large terms which depend on an upper cutoff and diverge when the cutoff is taken to infinity. Dur expression for
ôa(H) (Eq. (23)), does trot require any upper cutoff: for large N, FN decreases as lllR, so
that the term P( (in contrast to P(, or FN as in Ref. [6]) insures convergence of the series in
equation (23).
4. Experimental data and comparison with theory.
The experiment was performed on a superlattice GaAs Gai-~A[As with delta doping in the middle of each barrer. The width of the wells and the barriers is respectively a
= 190 À, b
=
140 À, for aluminum concentration of x
= 0.15. For such x the coupling between the wells
is negligible and the superlattice breaks up into independent two-dimensional systems. The initial number of wells was 30 but after depletion only 26 wells have been Ieft. The mobility of the electron gas, m each well, is /t
= 5700 cm~ V~~ s~~ and the Fermi energy
eF # 22 mev.
Because of the low mobility, the variation of Aa(H) with H occurs on large scale. The elastic
mean free path 1a 75 nm, so that the magnetic length LH becomes equal to 1at H e 1200 G
(the parameter ~ùcr reaches the value umty at much stronger fields, H e 1.7 T).
The experimental results for the magnetoconductance lia (H) at a temperature T = 4.2 K are
presented in figure 3 (curve 1). What actually was measured is the magnetoresistivity Ap(H) but, since ~ùcr is small, the equahty Aa(H) = à(1/p(H)) holds to a high accuracy for the considered fields. The three other curves in figure 3 have been calculated from equations (23),
using various degrees of approximation for the eigenvalues FN When B tends to zero, one
needs more and more terms to calculate the weak localization correction using equation (23).
However the negative magnetoresistance is accurately given by the dilferei~ce of equatioi~s (23)
and (12), with only a finite number of terms.
~2 f2 M p3
~ f ~
~'~~~
" ~@ Ii
~
i -§N ~ ~~~~ ~ j~~l fl~~~G~~ ~ ~~ ~ j~~ll l~~~
with
f(x) = ~In[1 x~~/~]
2
This number must be chosen fairly high to get a reasonable accuracy near the origin but is not a cut-off as in references [6]. The main dif§erence is that the former dues net depend on the
magnetic field while the latter does. Curve 2 is based on the "exact" expression, equation (22)
