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Dependence of the polaron binding energy and effective mass in a crystal layer on its thickness
Solid State Communications, Vo1.39, pp.273-277. 1981. Printed in Great RPergamon Press Ltd.
DEPENDENCE
00361098/81/260273-05$02.00/O
OF THE POLARON BINDING ZNERGY AND EFFECTIVE MASS IN A CRYSTAL LAYER
ON ITS
TEKlXKNE;SS
A.V.Shexman Institute of Physics, Estonian SSR Academy of Sciences, Tartu
202400,
USSR
The effective mass and module of the binding energy of a strong-coupling large polaron in a crystal layer are calculated as functions of the layer thickness. It is shown that these values may considerably exceed the corresponding quantities for an infinite crystal at a layer thickness of the order of the polaron radius already. The dependence of the binding energy module on the thickness is not monotonous: with the increase of the thickness the binding energy module first increases and then decreases up to the value of an infinite crystal.
As was shown in [I], the module of the binding energy IEI (the absolute value of the energy of the polaron at rest counted from the bottom of the lower size- uantized conduction subband) and the eBfective mass M of the polaron in a crystal layer surrounded by vacuum may considerably exceed the corresponding values for an infinite crystal, if the layer thickness d is much less than the polaron radius r,and high-fre uency dielectric constant of the cry&a P &=I. An analogous enhancement of these values will also take place when the layer is surrounded by a medium with the dispersionless dielectric constant&'s> E. The reason of the effect may be understood in the following manner. The polarization appearing in an ionic crystal under the-action of an external electric field mav be divided into two parts-the infrared polarization connected with displacements of rigid ions from their eouilibrium oositions, and the optical -&larization caused by the deformations of the electron shells of the ions. The second polarization type leads, on one hand, to the screening of the Coulomb interaction between an electron and an infrared polarization, on the other hand, to the renormalization of the electron masse The screening vanishes to a considerable extent in the crystal layer at dSr &=l(further, for simplicity, the 1ayerP)isc nsidered to be surrounded by vacuum)[2!i . That is just what leads to the enhancement of the binding energy module and the effective mass of the polaron in the layer, as compared with a large crystal, in spite of the considerable polaron
deformation at d"r,, Note that the effective mass of the electron, in which the interaction with the optical polarization is included 131, in a layer does not differ notabiy-from its value in an infinite crvstal. if the laver is macroscopically thick:.d*a (a is"the lattice constant), as the radius of the corresponding polaron (the electron in t~;lflasmon cloud) is microscopically In*this paper,the binding energy and effective mass of a strong-coupling large polaron are calculated for an arbitrary layer thickness. The applicability conditions of the stron -coupling large polaron model are (see731) 1E 1d151wo, dua, r=*a, (1) P where o. is the frequency of the Lo phonons, rp=l~2~~me2,&~1=&-1-&~1, E, is the static dielectric constant of the crystal, m and -e are the effective mass and the charge of an electron. If conditions (1) are fulfilled, the film may be considered as a dielectrically polarizable continuum and adiabatic ap$?oximation can be used. General formulas for the oolaron binding ener y and effective-mass were obtained in fI] with the help of the variational procedure. We quote part of them used in the calculations. Under the condition v==oorpthe energy of the polaron moving along the layer with the velocity Tmay be written with the accuracy up to the terms proportional to 2 v: 273
DEPENDCNCE
274
Ev = E + Mv2/2,
OF THE PCLARON
(2)
(31
(4) cyis the electron wave function minimizing functional (3). g = e
BINDING
ENERGY
Vol. 39, No. 2
of size-quantized levels (hWd)2/ er 2mg9the electron penetration in vacuum may be neglected), the polaron state is built up mainly from bulk electron states, and in an infinite crystal the dispersion 1a.wof the conduction band bottom has a form E(6,3=h2(k2+q2)/2m EL-i,312, so that for a fixed vector i; there are only two degenerate states with the z components of wave vector equalling q and -9. In such case an envelope functioncf,entering Eqs. (2)-(7) must satisfy the following boundary conditions yq,
z =?d/2) = 0.
(8)
In this connection the probe function is chosen in the form
is the electric field produced by an electron in the layer with the dielectric constant E immersed in vacuum, V(?*F') = (2RE) .exp[iIT.(F-JD], (6)
f(k,z,z') = exp(-k/z-z'1)-2 ch[k(z-z')] + aech[k(z+z')]exp(kd) I- Z2exp(2kd) is an electrostatic potential at the point i: in such system, produced by the unitary point charge sited at ?*. The coordinate system has been chosen so that the equations of the layzr surfaces have the form z&d/2, l?,p, and p, are two-dimensional vectors parallel to the surfaces (two latter are the projections of the vectors I' and ?'),ae= =(&+?)/(&-,I). Eo 'E ir= -e4fl
grad~3r*]~(~1)/2Vo(?,F') (7) is the vector of the infrared polarization brought about by rigid ion displacements only this definition differs from that of 33, where the electron shell polarization caused b the displacements is included in‘i3 3 , V, is defined by Eq. (6 with the re lacement of E by E. and by Eq. (7P with the replacement of &, by E(U) (also in V,). The approximation &(G.))=&&~(Q$-u~)/ is used for the dielectric (E+&03) constant with time dispersion taken into account, the derivatives of w2 are taken at J=O. The effective mass a proximation used in Eqs. (2) and (3 P is verified in case the second condition of (1) is fulfilled, the electron affinity is much larger than the characteristic en-
y.~= y(z)cy(p),y(y)
= a$Ze'~~,
yl(z).C(z2-d2/4)e-B(21, /d3r$
= 1,
,_, iYI
C is a normalization constant. In confirmitg with Ritz's variational procedure the poleron binding energy is found by the minimization of functional (3) on the variables a and p . The values of the variables obtained are then used for the computation of the effective mass (4). The results of the computations are shown in Figs. l3, where the values of the parameters are also indicated. Fig. 1 demonstrates the dependence of the binding energy module on the thickness of the crystal layer, expressed in the units of the polaron radius for two sets of..parameters. For the first set the Frohlich's coupling constant dF=6.64 and for the commutation of the nolaron binding energy in an infinite-crystal the s&ongcoupling model [ 31 may be used. Therefore this value, I$1 =0.0544me4/(fGp)2, is accepted as the unit of energy for curve 1. As is seen from Fig. 1,the binding energy module in the layer considerably exceeds its uantity in an infinite crystal alrea8y at d=2r (for this set of parameters r-669). W%th a further increase of the Tayer thickness IEi becomes less than lE$ and tends to the value IE+I=o.04Cme4/(h&p)2 (in Fig. 1 shown with the dashed line). This indicates that function (9) is worse probe function for an i i-ite crystalthan the Pekar's oneYP3 and for the polaron at d=3r,anothe$ approximation must be used. The comments in the beginning of the paper suggest that for such thicknesses functional (3) probably has the extremum on the functions of the typey(z)=C(z2-d2/4)chaz orcy(z)= &(z2-d2/4)shaz with the maxima of electron density near the surfaces, not on the E?.(g)-type functions, where this maxrmum is in the depth of the la-
Vol. 39, No. 2
DEPENDENCE OF THE POLARON BINDING ENERGY
275
2- ~=5.4,~,=35.l,tq’m,=O.l8, h~,=14.3 meV(TI Br)
1
’
2
I
I
4
6
I
8
d/rp
Pig.l.The dependence of the polaron binding energy module \El on the thickness of the crystal layer d for two indicated sets of parameters. m. is the mass of an electron. For curve 1 the energy module is expressed in the units of the poleron binding energy module of an infinite crystal in the model of strong coupling IEp(; for curve 2, in the model of intermediate coupling ISpI. The thickness is expressed in the units of the polaron radius. The binding energy module obtained by the probe function accepted at d--is shown by the dashed line, yer. Some of the investigations on the surface polaron support this point of view for the strong-coupling case (see, e.g. [4] ). On the other hand, it ten be seen in Fig. 1 that the approach to the asymptotical value IE$l occurs comparatively slow: the difference between IEI and &,l is visible as far as d='lOrp It probably indicates that the polsron "senses" the presence of surfaces already at such great distances. From the nhvsical noint of view it is ouite understandable: the interaction%etween an electron and phonons considered is a long-range Coulomb one. Curve 2 corresponds to the set of parameters of TlBr crystal [ ], for which dp=2.06 and therefore Evl=$$oothe binding energy module of the polai ron in an infinite crsstal~in the intermediate coupling model [6]- is chosen as the unit of egergy. In the region of d=-0.5r ( -1qOA) the accepted model of strongcoE pling is useless. Its applicability is doubtful also in the region immediately bordering from the left the point d=O.!jr,,even though IEP - IEEl* However, the quantity found for /El is a lower valuation of a true quan tity of the value. Therefore,it may be asserted that in this case the enhancement effect takes place at least at d= CO.% Fi&e 2 demonstrates the dependence of the polaron effective mass on the layer thickness for the same sets of
parameters. ME and ME are the values of the effective-mass in an infinite crystal in the models of strong and intermediate couplings, respectively [3,6] . As is seen,the enhancement effect is more manifest in this dependence: with the increase of the binding energy module bs some times the effective mass may ch"&ge by dozens of times. Therefore, for the detection of the effect the mobility measurement is a more sensitive experiment than for example,the measurement of the poiaron absorption spectrum. Since another probe function is applied than in [31, at d-oocurve I tends-to 0.53, not to i. With the decrease of the laver thickness the binding energy module reaches the maximum at d=$ and then decreases. For sets 1 and 2 in Fig. 1 tge thicknesses dm are of some &ngstroms and therefore the maxima are not shown. However, the parameters may be chosen so that this maximum falls into the region of macroscopically thick layers. The corresponding situation is shown in Fig.3. The reason for the aooeerence of the maximum is the existence-of two competing processes taking place with a reduction of the layer thickness: the enhancement of the Coulomb interaction between an electron and polarization and a cutting of the growing part of the polarization cloud by layer surfaces. At d<$ the growth of the Coulomb interaction compensates only partly the loss of the polarization cloud. For all
276
DEPENDENCE OF THE POLARON BINDING ENERGY
30.
Vol. 39, No. 2
3
.I
L
&T
PIS>
1
1
2
0
I-i 1
0
2
:
d/r, Fig.2. The dependence of the polaron effective mass on the layer thickness for the sets of parameters shown in Fig.1, MF and Mp are the polaron effective masses of an infinite crystal in the models of intermediate and strong couplings, respectively.
2
3 dinm)
4
Fig.3. The nonmonotonic dependence of the polaron binding energy module on the layer thickness. The corresponding dependence of the effective mass is also shown.
sets of parameters considered an analogous maximum in the dependence M(d) appears in the region of microscopically thin layers. As has been pointed out in [I], the difference in the character of the dependences of E(d), obtained in Ref. 7 and here, is probably due to the neglection of the interaction of an electron with surface phonons in [73 in contrast to our consideration.
The energy of F centre may be calculated analogously and due to the same reasons a considerable enhancement of its module near surfaces and in a crystal layer is to be expected. Acknowledgement - The author is thankful to Dr. V.V.Hixhnyakov for useful discussions, to Dr. G.S.Zavt and B.V. Shulichenko for the help in computations.
Vol. 39, No. 2
DEPENDENCE OF THE POLARON BINDING ENERGY
REFERENCES 1. A.Sherman,Izv. Acad. Nauk Est. SSR, Fiz., Nlatem.a, 382 (1980); Phys. Status Solidi, to be published. 2. L.V.Keldysh,Zh. Eksp. Teor. Fiz. (Pis'ma v red.) 3, 716 (1979). 3e S.I.Pekar,Investigations on the Electronic Theory of Crystals. GITTL, !.1oscow-Leningrad (1951); Usp. Fiz. Nauk 50, 197 (1953).
4. S.I.Beril& E.P.Pokatilov,Fiz. Tverd. Tela 1 1627 (1977). 5. S.D.Mahanti $ h.M.Varma,Phys. Rev. B6, 2209 (1972). 6. %Appel,Solid State Phys. a, 193 (1968). 7. J.J.Licari,Solid State Commun. a, 625 (1979).
277
DEPENDENCE
00361098/81/260273-05$02.00/O
OF THE POLARON BINDING ZNERGY AND EFFECTIVE MASS IN A CRYSTAL LAYER
ON ITS
TEKlXKNE;SS
A.V.Shexman Institute of Physics, Estonian SSR Academy of Sciences, Tartu
202400,
USSR
The effective mass and module of the binding energy of a strong-coupling large polaron in a crystal layer are calculated as functions of the layer thickness. It is shown that these values may considerably exceed the corresponding quantities for an infinite crystal at a layer thickness of the order of the polaron radius already. The dependence of the binding energy module on the thickness is not monotonous: with the increase of the thickness the binding energy module first increases and then decreases up to the value of an infinite crystal.
As was shown in [I], the module of the binding energy IEI (the absolute value of the energy of the polaron at rest counted from the bottom of the lower size- uantized conduction subband) and the eBfective mass M of the polaron in a crystal layer surrounded by vacuum may considerably exceed the corresponding values for an infinite crystal, if the layer thickness d is much less than the polaron radius r,and high-fre uency dielectric constant of the cry&a P &=I. An analogous enhancement of these values will also take place when the layer is surrounded by a medium with the dispersionless dielectric constant&'s> E. The reason of the effect may be understood in the following manner. The polarization appearing in an ionic crystal under the-action of an external electric field mav be divided into two parts-the infrared polarization connected with displacements of rigid ions from their eouilibrium oositions, and the optical -&larization caused by the deformations of the electron shells of the ions. The second polarization type leads, on one hand, to the screening of the Coulomb interaction between an electron and an infrared polarization, on the other hand, to the renormalization of the electron masse The screening vanishes to a considerable extent in the crystal layer at dSr &=l(further, for simplicity, the 1ayerP)isc nsidered to be surrounded by vacuum)[2!i . That is just what leads to the enhancement of the binding energy module and the effective mass of the polaron in the layer, as compared with a large crystal, in spite of the considerable polaron
deformation at d"r,, Note that the effective mass of the electron, in which the interaction with the optical polarization is included 131, in a layer does not differ notabiy-from its value in an infinite crvstal. if the laver is macroscopically thick:.d*a (a is"the lattice constant), as the radius of the corresponding polaron (the electron in t~;lflasmon cloud) is microscopically In*this paper,the binding energy and effective mass of a strong-coupling large polaron are calculated for an arbitrary layer thickness. The applicability conditions of the stron -coupling large polaron model are (see731) 1E 1d151wo, dua, r=*a, (1) P where o. is the frequency of the Lo phonons, rp=l~2~~me2,&~1=&-1-&~1, E, is the static dielectric constant of the crystal, m and -e are the effective mass and the charge of an electron. If conditions (1) are fulfilled, the film may be considered as a dielectrically polarizable continuum and adiabatic ap$?oximation can be used. General formulas for the oolaron binding ener y and effective-mass were obtained in fI] with the help of the variational procedure. We quote part of them used in the calculations. Under the condition v==oorpthe energy of the polaron moving along the layer with the velocity Tmay be written with the accuracy up to the terms proportional to 2 v: 273
DEPENDCNCE
274
Ev = E + Mv2/2,
OF THE PCLARON
(2)
(31
(4) cyis the electron wave function minimizing functional (3). g = e
BINDING
ENERGY
Vol. 39, No. 2
of size-quantized levels (hWd)2/ er 2mg9the electron penetration in vacuum may be neglected), the polaron state is built up mainly from bulk electron states, and in an infinite crystal the dispersion 1a.wof the conduction band bottom has a form E(6,3=h2(k2+q2)/2m EL-i,312, so that for a fixed vector i; there are only two degenerate states with the z components of wave vector equalling q and -9. In such case an envelope functioncf,entering Eqs. (2)-(7) must satisfy the following boundary conditions yq,
z =?d/2) = 0.
(8)
In this connection the probe function is chosen in the form
is the electric field produced by an electron in the layer with the dielectric constant E immersed in vacuum, V(?*F') = (2RE) .exp[iIT.(F-JD], (6)
f(k,z,z') = exp(-k/z-z'1)-2 ch[k(z-z')] + aech[k(z+z')]exp(kd) I- Z2exp(2kd) is an electrostatic potential at the point i: in such system, produced by the unitary point charge sited at ?*. The coordinate system has been chosen so that the equations of the layzr surfaces have the form z&d/2, l?,p, and p, are two-dimensional vectors parallel to the surfaces (two latter are the projections of the vectors I' and ?'),ae= =(&+?)/(&-,I). Eo 'E ir= -e4fl
grad~3r*]~(~1)/2Vo(?,F') (7) is the vector of the infrared polarization brought about by rigid ion displacements only this definition differs from that of 33, where the electron shell polarization caused b the displacements is included in‘i3 3 , V, is defined by Eq. (6 with the re lacement of E by E. and by Eq. (7P with the replacement of &, by E(U) (also in V,). The approximation &(G.))=&&~(Q$-u~)/ is used for the dielectric (E+&03) constant with time dispersion taken into account, the derivatives of w2 are taken at J=O. The effective mass a proximation used in Eqs. (2) and (3 P is verified in case the second condition of (1) is fulfilled, the electron affinity is much larger than the characteristic en-
y.~= y(z)cy(p),y(y)
= a$Ze'~~,
yl(z).C(z2-d2/4)e-B(21, /d3r$
= 1,
,_, iYI
C is a normalization constant. In confirmitg with Ritz's variational procedure the poleron binding energy is found by the minimization of functional (3) on the variables a and p . The values of the variables obtained are then used for the computation of the effective mass (4). The results of the computations are shown in Figs. l3, where the values of the parameters are also indicated. Fig. 1 demonstrates the dependence of the binding energy module on the thickness of the crystal layer, expressed in the units of the polaron radius for two sets of..parameters. For the first set the Frohlich's coupling constant dF=6.64 and for the commutation of the nolaron binding energy in an infinite-crystal the s&ongcoupling model [ 31 may be used. Therefore this value, I$1 =0.0544me4/(fGp)2, is accepted as the unit of energy for curve 1. As is seen from Fig. 1,the binding energy module in the layer considerably exceeds its uantity in an infinite crystal alrea8y at d=2r (for this set of parameters r-669). W%th a further increase of the Tayer thickness IEi becomes less than lE$ and tends to the value IE+I=o.04Cme4/(h&p)2 (in Fig. 1 shown with the dashed line). This indicates that function (9) is worse probe function for an i i-ite crystalthan the Pekar's oneYP3 and for the polaron at d=3r,anothe$ approximation must be used. The comments in the beginning of the paper suggest that for such thicknesses functional (3) probably has the extremum on the functions of the typey(z)=C(z2-d2/4)chaz orcy(z)= &(z2-d2/4)shaz with the maxima of electron density near the surfaces, not on the E?.(g)-type functions, where this maxrmum is in the depth of the la-
Vol. 39, No. 2
DEPENDENCE OF THE POLARON BINDING ENERGY
275
2- ~=5.4,~,=35.l,tq’m,=O.l8, h~,=14.3 meV(TI Br)
1
’
2
I
I
4
6
I
8
d/rp
Pig.l.The dependence of the polaron binding energy module \El on the thickness of the crystal layer d for two indicated sets of parameters. m. is the mass of an electron. For curve 1 the energy module is expressed in the units of the poleron binding energy module of an infinite crystal in the model of strong coupling IEp(; for curve 2, in the model of intermediate coupling ISpI. The thickness is expressed in the units of the polaron radius. The binding energy module obtained by the probe function accepted at d--is shown by the dashed line, yer. Some of the investigations on the surface polaron support this point of view for the strong-coupling case (see, e.g. [4] ). On the other hand, it ten be seen in Fig. 1 that the approach to the asymptotical value IE$l occurs comparatively slow: the difference between IEI and &,l is visible as far as d='lOrp It probably indicates that the polsron "senses" the presence of surfaces already at such great distances. From the nhvsical noint of view it is ouite understandable: the interaction%etween an electron and phonons considered is a long-range Coulomb one. Curve 2 corresponds to the set of parameters of TlBr crystal [ ], for which dp=2.06 and therefore Evl=$$oothe binding energy module of the polai ron in an infinite crsstal~in the intermediate coupling model [6]- is chosen as the unit of egergy. In the region of d=-0.5r ( -1qOA) the accepted model of strongcoE pling is useless. Its applicability is doubtful also in the region immediately bordering from the left the point d=O.!jr,,even though IEP - IEEl* However, the quantity found for /El is a lower valuation of a true quan tity of the value. Therefore,it may be asserted that in this case the enhancement effect takes place at least at d= CO.% Fi&e 2 demonstrates the dependence of the polaron effective mass on the layer thickness for the same sets of
parameters. ME and ME are the values of the effective-mass in an infinite crystal in the models of strong and intermediate couplings, respectively [3,6] . As is seen,the enhancement effect is more manifest in this dependence: with the increase of the binding energy module bs some times the effective mass may ch"&ge by dozens of times. Therefore, for the detection of the effect the mobility measurement is a more sensitive experiment than for example,the measurement of the poiaron absorption spectrum. Since another probe function is applied than in [31, at d-oocurve I tends-to 0.53, not to i. With the decrease of the laver thickness the binding energy module reaches the maximum at d=$ and then decreases. For sets 1 and 2 in Fig. 1 tge thicknesses dm are of some &ngstroms and therefore the maxima are not shown. However, the parameters may be chosen so that this maximum falls into the region of macroscopically thick layers. The corresponding situation is shown in Fig.3. The reason for the aooeerence of the maximum is the existence-of two competing processes taking place with a reduction of the layer thickness: the enhancement of the Coulomb interaction between an electron and polarization and a cutting of the growing part of the polarization cloud by layer surfaces. At d<$ the growth of the Coulomb interaction compensates only partly the loss of the polarization cloud. For all
276
DEPENDENCE OF THE POLARON BINDING ENERGY
30.
Vol. 39, No. 2
3
.I
L
&T
PIS>
1
1
2
0
I-i 1
0
2
:
d/r, Fig.2. The dependence of the polaron effective mass on the layer thickness for the sets of parameters shown in Fig.1, MF and Mp are the polaron effective masses of an infinite crystal in the models of intermediate and strong couplings, respectively.
2
3 dinm)
4
Fig.3. The nonmonotonic dependence of the polaron binding energy module on the layer thickness. The corresponding dependence of the effective mass is also shown.
sets of parameters considered an analogous maximum in the dependence M(d) appears in the region of microscopically thin layers. As has been pointed out in [I], the difference in the character of the dependences of E(d), obtained in Ref. 7 and here, is probably due to the neglection of the interaction of an electron with surface phonons in [73 in contrast to our consideration.
The energy of F centre may be calculated analogously and due to the same reasons a considerable enhancement of its module near surfaces and in a crystal layer is to be expected. Acknowledgement - The author is thankful to Dr. V.V.Hixhnyakov for useful discussions, to Dr. G.S.Zavt and B.V. Shulichenko for the help in computations.
Vol. 39, No. 2
DEPENDENCE OF THE POLARON BINDING ENERGY
REFERENCES 1. A.Sherman,Izv. Acad. Nauk Est. SSR, Fiz., Nlatem.a, 382 (1980); Phys. Status Solidi, to be published. 2. L.V.Keldysh,Zh. Eksp. Teor. Fiz. (Pis'ma v red.) 3, 716 (1979). 3e S.I.Pekar,Investigations on the Electronic Theory of Crystals. GITTL, !.1oscow-Leningrad (1951); Usp. Fiz. Nauk 50, 197 (1953).
4. S.I.Beril& E.P.Pokatilov,Fiz. Tverd. Tela 1 1627 (1977). 5. S.D.Mahanti $ h.M.Varma,Phys. Rev. B6, 2209 (1972). 6. %Appel,Solid State Phys. a, 193 (1968). 7. J.J.Licari,Solid State Commun. a, 625 (1979).
277