Coherent motion and anomalous transport properties of exciton and hole polarons with intrinsic vibrationa
It is shown that intrinsic vibrational degrees of freedom, inherent in two-atom exciton and hole polarons, drastically affect their transport properties in wide-band dielectrics (rare-gas solids and alkali halides). A fast excitonic energy transport and a comparatively high hole mobility, experiment...
Збережено в:
| Дата: | 2003 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2003
|
| Назва видання: | Физика низких температур |
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/128813 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Coherent motion and anomalous transport properties of exciton and hole polarons with intrinsic vibrationa / A.M. Ratner // Физика низких температур. — 2003. — Т. 29, № 3. — С. 237-249. — Бібліогр.: 22 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-128813 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1288132025-02-09T16:50:06Z Coherent motion and anomalous transport properties of exciton and hole polarons with intrinsic vibrationa Ratner, A.M. Electronically Induced Phenomena: Low Temperature Aspects It is shown that intrinsic vibrational degrees of freedom, inherent in two-atom exciton and hole polarons, drastically affect their transport properties in wide-band dielectrics (rare-gas solids and alkali halides). A fast excitonic energy transport and a comparatively high hole mobility, experimentally observed and attributed to two-site polarons tightly bound with the lattice, cannot be explained by the conventional theory of small-radius polarons that predicts their negligibly weak diffusion, exponentially small in the ratio of the binding energy to temperature. The theory, developed below with allowance for the intrinsic vibrational structure of two-site polarons describes qualitatively a large relevant set of experimental data which seem anomalous from the viewpoint of the conventional theory. The author is thankful to A.N. Ogurtsov and E.V. Savchenko for helpful discussions. This work was carried out within the project DFG No. 436 UKR 113/55/0. 2003 Article Coherent motion and anomalous transport properties of exciton and hole polarons with intrinsic vibrationa / A.M. Ratner // Физика низких температур. — 2003. — Т. 29, № 3. — С. 237-249. — Бібліогр.: 22 назв. — англ. 0132-6414 PACS: 71.38.+i https://nasplib.isofts.kiev.ua/handle/123456789/128813 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Electronically Induced Phenomena: Low Temperature Aspects Electronically Induced Phenomena: Low Temperature Aspects |
| spellingShingle |
Electronically Induced Phenomena: Low Temperature Aspects Electronically Induced Phenomena: Low Temperature Aspects Ratner, A.M. Coherent motion and anomalous transport properties of exciton and hole polarons with intrinsic vibrationa Физика низких температур |
| description |
It is shown that intrinsic vibrational degrees of freedom, inherent in two-atom exciton and hole polarons, drastically affect their transport properties in wide-band dielectrics (rare-gas solids and alkali halides). A fast excitonic energy transport and a comparatively high hole mobility, experimentally observed and attributed to two-site polarons tightly bound with the lattice, cannot be explained by the conventional theory of small-radius polarons that predicts their negligibly weak diffusion, exponentially small in the ratio of the binding energy to temperature. The theory, developed below with allowance for the intrinsic vibrational structure of two-site polarons describes qualitatively a large relevant set of experimental data which seem anomalous from the viewpoint of the conventional theory. |
| format |
Article |
| author |
Ratner, A.M. |
| author_facet |
Ratner, A.M. |
| author_sort |
Ratner, A.M. |
| title |
Coherent motion and anomalous transport properties of exciton and hole polarons with intrinsic vibrationa |
| title_short |
Coherent motion and anomalous transport properties of exciton and hole polarons with intrinsic vibrationa |
| title_full |
Coherent motion and anomalous transport properties of exciton and hole polarons with intrinsic vibrationa |
| title_fullStr |
Coherent motion and anomalous transport properties of exciton and hole polarons with intrinsic vibrationa |
| title_full_unstemmed |
Coherent motion and anomalous transport properties of exciton and hole polarons with intrinsic vibrationa |
| title_sort |
coherent motion and anomalous transport properties of exciton and hole polarons with intrinsic vibrationa |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2003 |
| topic_facet |
Electronically Induced Phenomena: Low Temperature Aspects |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/128813 |
| citation_txt |
Coherent motion and anomalous transport properties of exciton and hole polarons with intrinsic vibrationa / A.M. Ratner // Физика низких температур. — 2003. — Т. 29, № 3. — С. 237-249. — Бібліогр.: 22 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT ratneram coherentmotionandanomaloustransportpropertiesofexcitonandholepolaronswithintrinsicvibrationa |
| first_indexed |
2025-11-28T03:39:33Z |
| last_indexed |
2025-11-28T03:39:33Z |
| _version_ |
1850003874186788864 |
| fulltext |
Fizika Nizkikh Temperatur, 2003, v. 29, No. 3, p. 237–249
Coherent motion and anomalous transport properties
of exciton and hole polarons with intrinsic vibrational
structure
A.M. Ratner
B. Verkin Institute for Low Temperature Physics and Engineering
of the National Academy of Science of Ukraine, 47 Lenin Ave., Kharkov, 61103, Ukraine
E-mail: Ratner@ilt.kharkov.ua
Received June 25, 2002
It is shown that intrinsic vibrational degrees of freedom, inherent in two-atom exciton and hole
polarons, drastically affect their transport properties in wide-band dielectrics (rare-gas solids and
alkali halides). A fast excitonic energy transport and a comparatively high hole mobility, experi-
mentally observed and attributed to two-site polarons tightly bound with the lattice, cannot be ex-
plained by the conventional theory of small-radius polarons that predicts their negligibly weak
diffusion, exponentially small in the ratio of the binding energy to temperature. The theory, devel-
oped below with allowance for the intrinsic vibrational structure of two-site polarons describes
qualitatively a large relevant set of experimental data which seem anomalous from the viewpoint
of the conventional theory.
PACS: 71.38.+i
Introduction
In dielectrics with broad exciton and hole bands,
formed by a strong exchange interaction, the most sta-
ble excitonic and hole states are known to be of a
two-site type (two neighboring atoms, strongly
brought together, form a quasi-molecule with the ex-
change binding growing sharply with a decrease of the
interatomic distance). Such two-atom electronic exci-
tations with an intrinsic vibrational degree of freedom
are inherent and distinctly observed in the electronic
spectra of rare-gas solids [1] and alkali-halide crystals
[2] (these classes of dielectrics have common features
of electronic structure, outlined at the beginning of
Sec. 1).
Such two-atom excitations are usually treated as
polaronic states tightly bound with the lattice. In the
theory of polarons, the energy of this binding is identi-
fied with the separation � of the polaron level from
the bottom of the corresponding band. For the classes
of dielectrics mentioned, the separation � is of the
scale of 1eV.
However, the transport properties of such two-site
excitations cannot be satisfactorily described within
the conventional polaronic theory [3�6]. For a strong-
ly bound polaron, the latter predicts an exponentially
small diffusion coefficient
D =D0 exp (�Uact/Teff), (1)
where the activation energy Uact is close to � and Teff
stands for an effective temperature (expressed in
energy units) which at low temperatures makes al-
lowance for the lattice zero-vibration energy. This in-
ference of the theory drastically contradicts experi-
mental evidence (analyzed in Secs. 5 and 6) of a fast
transport of charge and especially energy by two-site
polarons.
The physical reason of such discrepancy consists in
the following. The conventional theory considers the
hopping of a polaron devoid of intrinsic structure.
Such a polaron, staying near the lattice site A, causes
lattice deformation around it and a corresponding
lowering of the polaron level by an amount � � Uact.
The electron or hole, localized at the site A at a deep
level EA, can go over to a neighboring site B only un-
der a fluctuation of its lattice surroundings strong
enough to lower the level of the electronic state, loca-
lized at the site B, down to EA. The energy of such a
lattice deformation is found to be close to � which is
© A. M. Ratner, 2003
described by Eq. (1). Such a simple notion is not ap-
plicable to a two-site polaron: its intrinsic degree of
freedom enables it to move continuously together with
the deformation cloud maintaining the polaronic sta-
te. In other words, the lattice deformation around a
new position of a two-site polaron is mainly produced
not by thermodynamic fluctuations but by the conti-
nuous motion of the polaron itself.
In the present paper, the theory of the continuous
motion and transport properties of two-site polarons is
developed and qualitatively compared with experi-
ment as applied to rare-gas and alkali-halide crystals.
First, in Sec. 1, it is shown that a fast transport in
these dielectrics is conditioned by some features of
their electronic spectrum, viz., the exchange nature of
the exciton and hole bands and their anisotropic struc-
ture, entailing one-dimensional translational motion.
As a result, the energy barrier, impeding the trans-
lational motion, is sharply diminished as compared to a
structureless polaron with the same binding energy �.
In the next Sections, 2, 3, and 4, such one-dimen-
sional motion of two-site excitations is explored, de-
pending on the main factors, dictating their character,
viz., the vibrational energy of a two-atom quasi-mole-
cule and the height of the «residual» energy barrier.
In the case of a high vibrational energy, much excee-
ding the barrier height and temperature, the trans-
lational motion has a coherent directional character
(Sec. 3), but under thermodynamic equilibrium (Sec. 4),
the translational motion gains diffusive character and
transport is slowed down by several orders of magni-
tude compared to the former case. In Secs. 5 and 6
a qualitative comparison with experiment is carried
out as applied to these two cases.
1. Continuous one-dimensional motion
of polarons with intrinsic degrees of freedom
This manner of translational motion is to a large de-
gree conditioned by a feature of the electronic spec-
trum of rare-gas or alkali-halide crystals. In the case of
a rare-gas crystal, the system of excitonic bands occu-
pies the upper part of the dielectric gap (more than
10 eV wide) and originates from the Rydberg atomic
excited states ns2np5(n + 1)s. A significant bandwidth
of about 1 eV is due to a rather strong exchange inter-
action between an excited atom and adjacent
ground-state ones. On the other hand, this strong ex-
change results in the existence of two-site excitonic
states. A two-site exciton is formed on adjacent atoms
which are brought together, providing a much stron-
ger attractive exchange interaction compared to regu-
lar lattice sites. Such a two-atom quasi-molecule with
a bond energy of about 1 eV is quite similar (judging
from spectroscopic data) to the corresponding excimer
molecule in the gas phase [1]. In a crystal, the vibra-
tional levels of such a quasi-molecule turn to narrow
subbands practically covering the energy extent of all
excitonic bands [7,8].
The dispersion law of excitons or holes is deter-
mined by the interatomic overlap of excited states
ns2np5(n + 1)s, which mainly coincides with the over-
lap of the np-hole states ns2np5 centered at adjacent
sites. This overlap is largest in the direction of the
p-state axis and gives rise to a sharply anisotropic
excitonic dispersion law [9]. Actually, a free exciton
or hole moves in the direction of the minimal effective
mass, which coincides with the axis of the np-hole and
dictates the axis direction of a two-site exciton or hole
polaron.
The outlined picture qualitatively holds for alkali
halide crystals with large anions and small cations
(e.g., NaI, NaCl, and KI). The exciton and hole states
(free or self-trapped), formed in the sublattice of
closed-shell anions, are similar to those of rare-gas so-
lids. These electronic excitations, associated with the
anion sublattice, cannot be noticeably interfered by
small cations with a very high ionization potential.
The translational motion of two-site excitons and
holes is described by the same equations and has the
same qualitative features (a distinction in their beha-
vior, caused by essentially different lifetimes, will be
considered in Secs. 3 and 4). So far this distinction
does not manifest itself, we will speak; for definite-
ness, about holes.
Let us trace qualitative features of the motion of a
two-site self-trapped hole (called also two-atom ioni-
zed quasi-molecule) in a rare-gas crystal. The molecu-
lar ion consists of two identical rare-gas atoms A and
B with a common shared px-hole in their outer shell
(the x-axis coincides with the axis of the atomic P
state). These atoms are brought together by a strong
exchange interaction conditioned by the hole. The ex-
change potential, mainly proportional to the overlap
of the px-states centered at the points A and B,
strongly diminishes with an increase of the angle be-
tween the x axis and the direction AB [9]. So, the
hole, forming the quasi-molecule AB, is polarized in
the direction x coincident with the quasi-molecule
axis AB. The translational motion of the hole is condi-
tioned by the exchange interaction between the atoms
A, B and other atoms. Since this exchange is signifi-
cant only for adjacent atoms lying in the same axis x,
the motion of a two-site hole is of one-dimensional
character (which is evidenced experimentally; see
Sec. 6, Item 2). Below we will consider the motion of
such hole along the atomic chain, allowance being
made for its static three-dimensional surroundings.
The same relates to two-site excitons as well.
238 Fizika Nizkikh Temperatur, 2003, v. 29, No. 3
A. M. Ratner
The nontrivial phenomena under consideration are
conditioned by a non-pair exchange interaction of
atoms among which a hole is distributed [9,10]. Such
an exchange is described by an usual exchange Hamil-
tonian
H � ��
�
�E a an
n
N
n0
1
� � ��
�
�
�
� �
��V x x a a a an n n
n
N
n n n( )( )1
1
1
1 1 . (2)
Here E0 is the site hole level, an
+ is the creation ope-
rator for a hole on the nth atom, �V(x) is the nega-
tive exchange energy strongly dependent on the inter-
atomic distance x, and xn is the coordinate of the nth
atom counted along the chain. The hole is generally
distributed among several atoms, as described by the
eigenfunction of the Hamiltonian (2)
� � �� �c cn n
n
n
n
� , | |2 1, (3)
where |cn|
2 is the portion of the hole at the nth site,
and
� n na� ��ground (4)
is the corresponding site state of the crystal (�ground
is its ground state).
The same Hamiltonian (2) also describes an
exciton, with the sole difference that the operator an
�
in Eqs. (2), (4) creates an atomic excitation instead a
hole at the nth site. Therefore, all of the inferences,
drawn below in this Section and in Sec. 2, are related
to holes and excitons to the same degree.
The lowest eigenvalue of the Hamiltonian (2) for
arbitrary fixed positions of the atoms is
W E V x x c c cn n
n
n n n
n
� � �
�
��
�
�
��
�� �� �0 1 1
2 1max ( ) , | |
(5)
(energy is minimized with respect to the set {cn}). If
the occupation numbers cn were fixed, the quantity
(5) as a function of atomic coordinates would be the
sum of pair potentials V(xn+1 � xn) multiplied by
fixed coefficients. But in fact the occupation numbers
cn depend substantially on V(xn+1 � xn) and, hence,
on xn; therefore, the hole energy (5) cannot be re-
duced to the sum of pair potentials. This circumstance
is of fundamental importance: it results in a substan-
tial lowering of the energy barrier that impedes
translational motion (within the approach of pair po-
tential, this barrier is found to be much higher and
the translational motion much slower [7,8]).
For a periodic chain, consisting of N atoms, the
minimum of energy (5) is achieved at cn = N �1/2,
which corresponds to the lowest band state.
We consider the opposite case of a tightly bound
polaronic state with a hole localized near two adjacent
atoms brought together (they are numbered, for defi-
niteness, by indices n = 2 and 3). For this pair of
atoms, the exchange multiplier V(x3 � x2) in (5)
strongly exceeds the rest of multipliers V(xn+1 � xn);
hence, the energy (5) is minimized at c2 = c3 � 2�1/2,
the rest of the coefficients cn being much less. Such a
state labeled by A is shown in Fig. 1, where the areas
of the circles denote the portion of the hole, |cn|
2, lo-
calized on atoms. Let us trace the change from the
state A to a similar state D with the hole, shifted by
one chain period, first keeping to the traditional no-
tion and then with allowance for the nonpair exchange
interaction.
Within the traditional notion [3�6], the state A
turns immediately to the state D: the hole hops by one
lattice period from the atomic pair (2,3) to the pair
(3,4). To make such a hop possible, an adjacent atom
4 (devoid of a hole until the hop occurs) must be
moved towards atom 3 strongly enough to reduce the
distance (4,3) down to the distance (3,2). The energy
of such a deformation nears the binding energy, so
that the hopping rate is described by Eq. (1).
If the nonpair exchange interaction is taken into ac-
count, the picture becomes quite different: the initial
state A turns to the final state D through a continuous
sequence of intermediate states (B, C...) in the fol-
lowing way. To minimize energy (5), atom 4, when
Coherent motion and anomalous transport properties of exciton and hole polarons
Fizika Nizkikh Temperatur, 2003, v. 29, No. 3 239
0
0
D
C
B
A
1
1
2
2
3
3
4
4
5
5
Positions of atoms
Fig. 1. Translational motion of a two-site self-trapped
hole via its continuous redistribution among atoms shown
by circles (the areas of the circles indicate the portion of
the hole on the atoms). The hole sequentially passes the
states A, B, C, D without overcoming a significant bar-
rier. This scheme relates to a two-site exciton as well.
moving towards the pair (2,3), gets a portion of hole
that increases with decreasing separation (4,3) (as
shown by the varying areas of the circles). Thus, the
distribution of the hole among atoms follows their mo-
tion continuously (not by a jump as within the con-
ventional scheme). This sharply diminishes the energy
barrier B to be overcome when changing from the ini-
tial state A to the final state D.
Nevertheless, the height of the «residual» energy
barrier essentially influences the translational motion
of a two-site hole that, in view of its long lifetime,
moves in thermodynamic equilibrium with the lattice
and occupies rather low vibrational levels comparable
with the barrier height. Unlike a hole, a two-site
exciton, during its short lifetime, comparable with the
time of vibrational relaxation, occupies high vibra-
tional levels. As long as the vibrational energy much
exceeds the barrier height, a two-site exciton moves in
a coherent directional way practically irrespective of
the barrier.
For this reason, the barrier is of great importance
only for the motion of holes. In the next Section, the
barrier is investigated as applied to two-site holes, but
the obtained results are related to two-site excitons as
well.
2. Energy barrier for translational motion
To obtain the barrier, let us introduce the total adi-
abatic potential of an N-atom chain containing one
hole:
W x x W x xN Ntot ( ,..., ) ( ,..., )1 1� �
� � � ��
�
�
�
� ��u x x un n
n
N
n
n
N
0 1
1
1
0
1
( ) (| | )r R
R
. (6)
The total potential consists of the hole energy (5)
(first term on the right-hand side of Eq. (6)) and the
sum of the pairwise ground-state interatomic poten-
tials u0 taken along the chain (second term) as well as
between every chain atom (with coordinates rn) and
the immovable lattice atoms not belonging to the
chain (with coordinates R). To find the barrier, one
has to continuously change from the state A to the
state D (see Fig. 1) via reducing the distance (4,3),
the rest of interatomic distances being adjusted to the
minimum of the total potential (6) at a given separa-
tion (4,3). Such a trajectory in the space (x1,..., xN)
inevitably passes a symmetric configuration C (Fig. 1)
that provides an extremum of Wtot. Usually a two-site
hole state A or D is assumed to be stable, that is
Wtot(C) > Wtot(A). (7)
If the condition (7) is met, the barrier height should
be defined as
B = Wtot(C) � Wtot(A) = Wtot(C) � Wtot(D). (8)
Below, the barrier definition (8) will be used with-
out restriction (7), a negative B being understood in
the meaning of the energy separation between the sta-
ble three-atom state C and unstable two-atom state A
or D.
To calculate the barrier (8) for a real crystal, it is
necessary to reconstruct the total potential (6). It con-
tains, besides the ground-state potential u0 which is
known to good accuracy [9], the resonance contribu-
tion (5) defined through the exchange potential V(x),
mainly proportional to the interatomic overlap inte-
gral J(x) (x is interatomic distance). The dependence
J = J(x) can be approximately derived from the
ground-state repulsion potential that, on one hand, is
in main proportional to J(x)2 and, on the other hand,
is known to decrease with increasing x nearly as
exp (�12x/l) (l is the interatomic distance in the
ideal lattice) [9]. Thus, the exchange potential V can
be approximated in the form
V x V q
l x
l
l x
l
( ) exp�
�
�
��
�
�
�
�
�
�
��
�
�
��
0
2
� , q � 6. (9)
Here a small quadric term, influencing the barrier, is
allowed for. The preexponential V0 (close to 0.24 eV
for solid krypton) can be derived from the width of
the hole band formed in the ideal lattice (V =V0 at
x = l).
Below, the parameters q, � and V0 = 0.24 eV are
taken for solid krypton. For hole states, there are no
spectroscopic data to derive the exchange interaction
parameters q and �. But q and � can be derived for the
lowest two-site excitonic state via fitting the corre-
sponding luminescence spectrum (energy position and
halfwidth) with experiment. Note that the exchange
interaction, resulting from a p-hole in the valence
shell, cannot noticeably differ for the hole and
exciton: the p-electron is completely removed in the
former case or carried over to a large-radius Rydberg
state in the latter. Making use of this, let us fit the re-
quired parameters q and � with the two-site exciton lu-
minescence band of solid krypton. The fitting can be
carried out with nearly the same accuracy with q vary-
ing in the range 6 0 5� . and the corresponding values
of � given in Table 1 as a function of q.
For every set (q, �), Table 1 presents the barrier
(8) calculated with the exchange potential (9) for two
lattice parameters 0.5812 and 0.5850 nm related to the
temperature of 80 and 110 K, respectively [11]. (Note
that the lattice parameter, appearing in the last term
of Eq. (6), essentially influences the barrier: it grows
with a decrease of the distance between chain atoms
and the nearest lattice atoms lying beyond the chain.)
240 Fizika Nizkikh Temperatur, 2003, v. 29, No. 3
A. M. Ratner
Table 1 demonstrates a high sensitivity of the barrier
to the exchange potential parameters as well as to the
lattice period (its thermal increase by 0.7 % lowers the
barrier by about 100 K). The point is that the poten-
tial (6), whose maximum makes up the barrier, is
formed by the sum of positive and negative quantities
with a scale of 1 eV, which exceeds the characteristic
height of the barrier by about two orders of magni-
tude. A slight variation of a separate summand causes
a significant relative variation of the barrier.
As will be shown below, the barrier height influ-
ences the hole motion in an essential and nontrivial
way (the specific way of realizing the barrier is of
much less importance).
Song [12] has estimated the barrier B for a two-site
hole quasi-molecule in solid argon within the approxi-
mation of pair potentials. The interaction between the
atoms of the quasi-molecule (atoms 2, 3 in Fig. 1,
state A) and between every atom of the quasi-mole-
cule and an adjacent atom not belonging to it (the
pairs 1,2 and 3,4) was described by the same potential
that was assumed to be independent of the hole distri-
bution between these atoms. As a result, it was found
that the barrier height does not exceed 0.05 eV =
= 600 K, which agrees in order of magnitude with
Table 1.
3. Directional translational motion of a two-site
polaron (exciton) on high vibration levels
Due to its short lifetime, comparable with the vi-
brational relaxation time, a two-site exciton dwells on
high vibrational levels during a significant part of its
existence. The directional translational motion, possi-
ble only on high vibrational levels, manifests itself in
an anomalously fast energy transport.
Even within the one-dimensional model (grounded
in Sec. 1), the rigorous quantum-mechanical descrip-
tion of such a complicated phenomenon seems unrea-
listic. Below, only the electronic subsystem will be
considered in a quantum-mechanical way; the lattice
motion will be described classically in the adiabatic
approximation. The latter is justified since the separa-
tion between the lowest level of the Hamiltonian (2)
and the rest of its levels much exceeds both T and the
vibrational frequencies of the chain (expressed in
energy units). Within this model, the lattice is repre-
sented by an N-atom chain (its length is chosen large
enough that the calculation results are independent of
N). The rest of the lattice atoms are assumed to be
placed strictly at their sites and are taken into account
by the last term in the total potential (6).
Keeping to the adiabatic scheme within the one-di-
mensional model, let us first consider the lowest
excitonic state formed at arbitrary fixed positions of N
atoms. The energy of this state W is given by Eq. (5).
According to a standard procedure, the partial deriva-
tives of the right-hand side of Eq. (5), added to the
sum W �cn
2, should be taken with respect to the coef-
ficients cn and equated to zero. This results in the sys-
tem of linear homogeneous equations
V x x c Wc
V x x c V x x c Wcn n n n n n
( ) ,
( ) ( )
2 1 2 1
1 1 1 1 1
0� � �
� � � �� � � � � � �
� � �� �
0 1
01 1
( ),
( ) .
n N
V x x c WcN N N N
(10)
Let D be the N-order determinant of this equation
system. The excitonic energy W(x1,...,xN) at arbi-
trary fixed positions, x1, ..., xn, of the chain atoms can
be obtained from the equation
D W x xN( , ,..., )1 0� . (11)
Of N roots of this determinant, the lowest one cor-
responds to the self-trapped excitonic state and is se-
parated by a large gap from all the other roots, which
belong to a slightly distorted excitonic band. The least
root is identified with W. Substituting W into (6),
Coherent motion and anomalous transport properties of exciton and hole polarons
Fizika Nizkikh Temperatur, 2003, v. 29, No. 3 241
Table 1
Barrier height (8) for solid krypton calculated using the
exchange potential (9) with different q and � for lattice
periods 0.5812 and 0.5850 nm related to the temperature 80
and 110 K, respectively
q �
B, K
T=80 K T=110 K
5.66 0.07 473 410
5.76 0.06 386 308
5.865 0.05 288 196
5.97 0.04 168 60
6.08 0.03 38 �84
6.22 0.02 �62 �199
6.36 0.01 �192 �343
6.51 0 �335 �500
we obtain the total adiabatic potential Wtot of the
N-atom system with a self-trapped exciton.
The classical motion of the system of N atoms with
the non-pair potential (6) is described by the equation
M
d x
dt
W x x
x
n N
n
0
2
2
1 0�
�
�
�tot ( ,..., )
(12)
with the atom mass M0.
Equation (12) was solved numerically for solid
krypton under the following conditions. At the initial
time t = 0, a two-site quasi-molecule with the vibra-
tional energy Evib arose on two adjacent atoms arbi-
trarily chosen inside the atomic chain. In the course of
motion, the electronic excitation is distributed among
4 atoms (as shown in Fig. 1). The nonequilibrium vi-
brational energy, concentrated within this four-atom
complex, is maintained at the given level Evib via a
weak continuous input of energy. The rest of atoms are
in equilibrium with the lattice at a given constant
temperature T. In the course of motion, the total ave-
rage kinetic energy of all atoms of the chain (except
the above four-atom complex) was maintained at the
level (N�4)T/2 via a very slow injection or carrying
away of heat. Simultaneously the dispersion of this ki-
netic energy as a function of time was maintained at a
proper level of (N�4)1/2T/2 (to that end, every atom
of the chain received weak random augmentations to
its momentum with the frequency of lattice vibra-
tions).
The computer solution of Eqs. (11),(12) exhibits
some general features. The strongest resonance bond,
formed between two atoms with the numbers L and
L+1, comprises more than half the electronic excita-
tion (this means that c cL L
2
1
2 0 5� �� . ); on four atoms
(L�1, L, L+1, L+2) the excitation is concentrated
practically completely. The vibrational energy of this
four-atom complex decreases slowly (by about 1 % per
period) in the course of thermal relaxation. Note that
in the corresponding quantum-mechanical system the
vibrational relaxation occurs substantially (by two or-
ders of magnitude) more slowly in view of the rele-
vant conservation laws and complicated phase rela-
tions between different vibrational wave functions
[9]. This circumstance was taken into account by a
weak steady input of energy into the four-atom com-
plex, which maintained its vibrational energy at the
initial level.
The character of the motion is determined by the
parameters Evib and T. Numerical analysis of the
Eqs. (11),(12) shows that the translational motion of
the four-atom complex can occur either in a diffusive
way (through random translational shifts, one of
which is shown in Fig. 1) or in a coherent directional
way (through sequential shifts agreed in phase with
one another). The directional motion takes place at a
high vibrational energy Evib and a low temperature; in
the opposite case, the self-trapped exciton moves in a
random diffusive way. The picture is practically inde-
pendent of the barrier height B as long as B <<
<< Evib.The boundary between these types of motion
in the Evib- versus-T plane is shown in Fig. 2.
Figure 3 shows the motion of the exciton center of
weight
x c x cc n n
n
N
n
n
N
� �
�
�
�
�
�
�
�
�
� �
� �| | | |2
1
2
1
1 (13)
at different points of the Evib- versus-T plane. The
figure demonstrates that the motion of a two-site
exciton changes its character in a similar way with in-
creasing vibrational energy at a constant temperature
(Fig. 3,a) or with decreasing temperature at a con-
stant vibrational energy (Fig. 3,b). The directional
motion of an exciton is quite stable at the right-hand
side from the boundary indicated in Fig. 2, whereas
at the left-hand side directional motion can be rea-
lized during short time intervals in a random way due
to fluctuations.
At any temperature, the stability of the coherent
translational motion of the self-trapped exciton de-
pends strongly on its vibrational energy. This seems
quite natural since this translational motion is coupled
with the vibrations of the four-atom complex.
The directional motion velocity, V = dxc/dt, is
found to be of the scale of the sound velocity in the
crystal. This is easy to understand: a longitudinal elas-
242 Fizika Nizkikh Temperatur, 2003, v. 29, No. 3
A. M. Ratner
0 0.1 0.2 0.3 0.4 0.5
100
80
60
40
20
Vibration energy, eV
T
e
m
p
e
ra
tu
re
, K
Directional
motion
Diffusive
motion
Fig. 2. Regions of the diffusive and directional motion of
a two-site exciton in the plane temperature versus vibra-
tional energy.
tic wave, coupled with electronic excitation, propa-
gates in the crystal mainly in the same way as ordinary
sound. As can be seen from Fig. 3,a, V grows with
Evib. The directional motion velocity was found to be
proportional to E /
vib
1 2 and practically independent of
temperature and barrier height (at Evib = 0.63 eV, V
coincides with the longitudinal sound velocity in solid
krypton, equal to 1.37·105 cm/s ).
4. Diffusive motion of two-site polaron (hole)
in thermodynamic equilibrium with lattice
Unlike short-lived excitons (with lifetime of about
1 ns), two-site hole polarons during measurement of
their mobility have a long lifetime (more than 1 �s)
[13], exceeding the vibrational relaxation time by 3 or
4 orders of magnitude. Therefore, a self-trapped hole
moves under conditions of thermodynamic equilib-
rium with the lattice. Such motion was investigated
by solving Eqs. (11),(12) in a way described in
Sec. 3, but without the input of energy into the
four-atom complex to maintain its vibrational energy
at a given level. Under such conditions the hole
polaron motion is of a random diffusive character, so
that the hole mobility � is connected with the hole dif-
fusion coefficient D by the known Einstein relation
� � �
�
�De
T
e
Tt
x j
t tj
2
2( )� , (14)
where e is the electron charge, T is temperature in
energy units, �xj stands for the random shift of the
center of weight of the hole at the time moment tj;
the summation extends over all statistically inde-
pendent shifts occurring during time t.
To evaluate the diffusion coefficient D with suffi-
cient accuracy, the diffusional motion of a hole was
traced for a long enough time of about 5 ns; during
this time, about 700 squared chain periods are accu-
mulated in the sum (14) [11].
Unlike an exciton occupying a high vibrational
level, the equilibrium motion of a self-trapped hole es-
sentially changes its character depending on the bar-
rier height B, as is demonstrated in Fig. 4. As can be
seen from the figure, for a high barrier B = 386 K =
= 4.8 T (upper curve in Fig. 4), a hole participates in
the motion of two sharply different types: high-fre-
Coherent motion and anomalous transport properties of exciton and hole polarons
Fizika Nizkikh Temperatur, 2003, v. 29, No. 3 243
0.1
0.2
0.3
0.6
0.2
50
45
40
35
30
25
20
15
10
5
0
– 5
C
o
o
rd
in
a
te
o
f
e
xc
ito
n
, c
h
a
in
p
e
ri
o
d
s
0 5 10 15 20 25
Time, ps
a
60 K
30 K
1 K
90 K
50
45
40
35
30
25
20
15
10
5
0
0
– 5
C
o
o
rd
in
a
te
o
f
e
xc
ito
n
, c
h
a
in
p
e
ri
o
d
s
5 10 15 20 25
Time, ps
b
Fig. 3. Typical fragments of the motion of the exciton center of weight calculated by solving Eqs. (11), (12). The exciton
coordinate in units of the chain period (ordinate axis) and time (abscissa axis) are counted from an arbitrary origin. The
figure shows how the character of motion changes: (a) � with increasing vibrational energy (indicated in the figure in eV)
at a constant temperature equal to 60 K; (b) � with decreasing temperature (indicated in the figure) at a constant vibra-
tional energy equal to 0.2 eV. The directional motion is quite stable on the right of the boundary presented in Fig. 2 and
arises in a random fluctuation way on the left of the boundary.
quency vibrations with a small amplitude (merged
into a thick line on the reduced scale of the figure)
and random translational shifts by one chain period,
which look like hops on the scale used. Note that in
fact such «hops» occur in a continuous way, as shown
in Fig. 1. For a lower barrier, «hops» by several (M =
= 2, 3...) chain periods become possible (the meaning
of such «multiple hops» will be explained below). In
Fig. 4, every «hop» is marked by the number M indi-
cating the distance of hopping in units of the chain pe-
riod or, more briefly, the hopping multiplicity.
As the barrier lowers, the distinction between
translational shifts («hops») of the hole and its vibra-
tional motion is blurred. Along with usual high-fre-
quency vibrations, low-frequency ones appear with
amplitudes enhanced up to the chain period l. Such
low-frequency vibrations can result (but not inevita-
bly) in a translational shift according to the scheme of
Fig. 1; their large period (up to 10 ps) corresponds to
a small variation of the adiabatic potential Wtot when
changing between the states B and C near the top of a
low barrier.
It is easy to understand why the vibrational motion
is well pronounced only in the case of a low barrier.
Of interest are vibrations of a few atoms (the atoms 2,
3, 4 in Fig. 1) among which the hole is distributed.
Such vibrations are accompanied by overcoming the
barrier in both directions. For a high barrier (B >> T),
a noticeable motion of the hole is made possible by a
strong short-lived thermodynamic fluctuation during
which the system has enough time to pass the barrier
only in one direction; such motion looks like a series of
random independent hops (quite distinct at B =
= 386 K = 4.8 T). As the barrier is lowered somewhat
(B = 288 K = 3.6 T), the fluctuation lifetime becomes
sufficient to pass the barrier one time sequentially in
both directions, which results in single-period vibra-
tions with an amplitude near the chain period; such vi-
brations do not contribute to the mobility (14). When
the barrier decreases and disappears (B = 85 and 5 K),
the vibrational motion of the hole becomes most de-
veloped and occurs in the form of a long sequence of
aperiodic oscillations interrupted by comparatively in-
frequent «hops». In the absence of the barrier, the vi-
brational motion predominates over hopping; this is
the reason why the mobility at a given T achieves its
maximum value not in the absence of the barrier but at
B � 1.5 T (see Figs. 5, 6).
In order to compare the theory with experiment
[13], performed for solid krypton in the interval
80 K < T < 110 K, computer calculations of the mo-
bility with the use of Eqs. (11),(12) were carried out
for solid krypton at the temperatures T = 80 and 110 K
(the analysis of these results, made below, permits one
to extend their temperature range). For these fixed
values of temperature, Fig. 5 presents the mobility �
(in units of cm2·s�1 ·V�1) as a function of barrier
height B. It should be recalled (see Sec. 2) that B va-
ries noticeably with temperature due to thermal ex-
pansion. In Fig. 5, this is allowed for by two scales of
barrier, related to 80 and 110 K. Nearly the same re-
sults (but with a large statistic straggling) were ob-
tained via direct calculation of the mobility with the
use of Eq. (12) complemented by an electrostatic term
describing the hole charge in a weak external electric
field.
Along with this, it is interesting to trace how mo-
bility depends on B/T irrespective of the way of vary-
ing the barrier (by varying the interatomic potentials
or the lattice period). To that end, Fig. 6 presents the
same results as in Fig. 5 in the coordinate � versus re-
duced variable B(T)/T for T = 80 K and 110 K. As
can be seen from Fig. 6, in the region B > T, where the
«hopping» has an activational character, the mobility
244 Fizika Nizkikh Temperatur, 2003, v. 29, No. 3
A. M. Ratner
0
5
10
15
20
25
0 50 100 150
25
20
15
10
5
0 50 100 150
B = 386 K
1
1 1
288
1 1
2
85 2
1
3
5
1 1
1
–192
1
2
Time, ps
C
e
n
te
r
o
f
w
e
ig
h
t
co
o
o
rd
in
a
te
o
f
h
o
le
, c
h
a
in
p
e
ri
o
d
s
Fig. 4. Typical fragments of the diffusional motion of the
center of weight of a hole in thermodynamic equilibrium
with the lattice at T = 80 K, calculated for different barri-
ers (whose height is indicated in Kelvins by the large nu-
merals). Random translational shifts (hole «hops») are
marked by small numerals indicating the hopping distance
in units of the chain period. As the barrier B is lowered in
the region B < T, the vibrational motion of the hole de-
velops at the expense of its translational motion (there-
fore, as can be seen from Fig. 5, diffusion is not monoto-
nously enhanced as the barrier is lowered).
actually depends only on the reduced variable B/T.
Thus, the part of the curve to the right of the maxi-
mum gives the temperature dependence of the mobi-
lity in the range of lower temperatures for a positive
barrier.
The absolute values of the mobility, presented in
Fig. 5, agree in order of magnitude with experiment
[13] (see Sec. 6) but are systematically underesti-
mated by almost a factor of four in comparison with
the experimental data. The physical reason for this
quantitative discrepancy, to all appearance, is con-
nected with underestimation of the multiple hopping
of self-trapped holes.
As was shown in Sec. 3, a two-site polaron on a
high vibrational level simultaneously with vibrational
motion participates in the directional translational
motion which is realized through sequential shifts cor-
related in phase. A short-time motion of such type can
occur also under conditions of thermodynamic equilib-
rium with the lattice when a strong fluctuation brings
the vibrational energy of the four-atom complex (on
average equal to 4T) to a critical level necessary for
the directional translational motion. Such fluctuation
enhancement of Evib gives rise to a short-lived cohe-
rent elastic wave coupled with the hole. To all
appearance, the model of one-dimensional chain un-
derestimates the lifetime of such a fluctuational elas-
tic wave. Indeed, the lattice atoms, not belonging to
the chain but adjacent to it, are assumed to be immov-
able, whereas in fact they are involved in this elastic
wave, thus enhancing the total mass of the atoms in-
volved and the stability of the wave. The underestima-
tion of the hopping multiplicity within the one-dimen-
sional model entails a corresponding underestimation
of the mobility (quadric in the multiplicity). This
seems to be a reason for the above mentioned quantita-
tive discrepancy between the theory and experiment.
Let us consider multiple hops in more detail and in-
troduce the hopping multiplicity M, equal to the total
length of a hop in units of chain periods. As can be
seen from Fig. 4, where all hops are marked by num-
bers indicating their length M, multiple hops with
M = 2 or 3 occur comparatively often. Hops with
greater M (up to 10) occur more seldom but they no-
ticeably contribute to the hole mobility (14). If the
number of M-fold hops is denoted by �M, their contri-
bution to mobility is proportional to wM = �MM 2.
The mean multiplicity of hopping should be defined
via averaging M over hops with the weights wM:
Coherent motion and anomalous transport properties of exciton and hole polarons
Fizika Nizkikh Temperatur, 2003, v. 29, No. 3 245
0
0.002
0.004
0.006
0.008
0.01
0.012
–– 200200 00 200200 400400
Barrier, K at T = 80 K
Barrier, K at T = 80 K
T=80 K
T=110 K
– 200 0 200 400
M
o
b
ili
ty
, c
m
· s
·
V
2
–
1
–
1
Fig. 5. Two-site hole mobility calculated for solid krypton
as a function of the barrier height B at T = 80 and 110 K
(B varies noticeably with temperature due to thermal ex-
pansion, which is allowed for by two scales of barrier, re-
lated to 80 and 110 K).
0
0.002
0.004
0.006
0.008
0.010
0.012
3 2 ––– 1 0 1 2 3 4 5
T=80 K
T = 110 K
M
o
b
ili
ty
, c
m
· s
·
V
2
–
1
–
1
B (T)/ T
Fig. 6. The same results as in Fig. 3 presented versus the
reduced variable B(T)/T for T = 80 and 110 K. Note that
in the region B > T, where the «hopping» has an activa-
tion character, mobility actually depends only on the re-
duced variable B/T.
M
w M
w
M
M
M
M
M
M
M
M
M
M
� �
�
�
�
�
�
�
3
2
. (15)
The mean multiplicity of hopping, calculated ac-
cording to (15), is shown in Fig. 7 for two tempera-
tures as a function of B(T)/T. Note that multiple
hopping, due to its fluctuational nature, is rapidly en-
hanced with increasing temperature.
5. Qualitative comparison with experiment.
Energy transport by two-site excitons
Let us, first, adduce the available spectroscopic ev-
idence for a predominant role of two-site excitons in
energy transfer to impurity centers in wide-band di-
electrics and, second, explain experimental data on
fast energy transport realized by two-site excitons.
Energy transfer from excitons to neutral impurity
centers has been observed by many authors in dielec-
trics of various types. The character of the excitonic
state, responsible for energy transfer, depends on the
specific band structure of the dielectric crystal. Expe-
rimental data for wide-band dielectrics of the type
considered provide evidence that energy transport is
realized not by free excitons but by two-site excitonic
polarons.
5.1. Energy transport in alkali halide crystals.
The assumption of energy transfer by free excitons
should be discarded for the following reason. Usually
impurity levels lie below the bottom Ebot of the low-
est exciton band. An exciton, before it can transfer its
energy to such an impurity center, first must achieve
Ebot via thermal relaxation. The quantum yield of cre-
ating excitons at the band bottom can be estimated
from the quantum yield of the free-exciton lumines-
cence, Yfree (in the crystals of the type considered, the
radiative decay of excitons is possible only at the band
bottom, where the exciton quasi-momentum goes to
zero). For all the alkali halides examined [17], Yfree is
of the order of 10�2. Such a low quantum yield of the
photoproduction of free excitons cannot provide effi-
cient energy transfer to the impurity centers observed
spectroscopically [14�16]. But even after an exciton
has reached the band bottom, the transfer of its energy
to an impurity atom must be preceded by localization
in a shallow potential well near the impurity center;
this is impeded by the rather small effective mass of a
free exciton [9].
On the other hand, an efficient energy transfer to
thallium impurity centers, present in a very low con-
centration of 10�6 to 10�5, was observed in KI(Tl),
RbI(Tl), and NaI(Tl) crystals under excitation below
the bottom of the lowest excitonic band [14�16]. In
this excitation energy region, two-site excitons are
created with a high efficiency, but free excitons can-
not be generated at all (which is confirmed by the zero
quantum yield of free-exciton luminescence measured
for pure KI, RbI, and NaI under excitation below Ebot
[17]). Moreover, the efficiency of the energy transfer
to the impurity did not grow as the excitation fre-
quency was changed from a value below Ebot (where
Yfree = 0) to any value lying inside the band (where
Yfree � 0.01) [14].
Consider now the rate of energy transport realized
by two-site excitons. The efficiency of the energy
transfer to impurity centers can be characterized by
the ratio Yimp/Yhost, where Yimp is the intensity of
impurity luminescence and Yhost is the luminescence
intensity of two-site excitons formed in the host crys-
tal. The ratio Yimp/Yhost was measured spectroscopi-
cally for the alkali halide crystals KI, RbI, and NaI
weakly doped with thallium [14�16]. According to
Refs. 14�16, efficient energy transfer to the impurity
(Yimp/Yhost � 1) at T = 5 K is achieved for KI(Tl),
RbI(Tl), and NaI(Tl) at the concentrations n =
= 1.5·1016, 3·1016, and 5·1017 cm�3, respectively. Let
us show that the coherent directional motion of
self-trapped excitons with the velocity V, investigated
in Sec. 3, provides such an efficient energy transfer to
the impurity, described by the relation
246 Fizika Nizkikh Temperatur, 2003, v. 29, No. 3
A. M. Ratner
1
2
3
4 T=110 K
T=80 K
5
3 22 ––– 1 0 1 2 3 4 5
M
e
a
n
m
u
lt
ip
lic
ity
o
f
h
o
p
p
in
g
, c
h
a
in
p
e
ri
o
d
s
B (T)/ T
Fig. 7. Mean distance of hopping in units of the chain pe-
riods (hopping multiplicity), calculated at T = 80 K and
110 K.
Yimp/Yhost �Vt� n � 1. (16)
Here t � 1 ns stands for the vibrational relaxation
time and is the cross section of the exciton localiza-
tion near an impurity ion; the velocity V of the direc-
tional motion of two-site self-trapped excitons is
taken equal to the longitudinal sound velocity in the
crystal, near 2·105 cm/s.
The value of is determined by the interaction of
an exciton with an impurity ion, described (in atomic
units) by the van der Waals potential
U R r R( ) � � �! imp
2 6, (17)
where r2 is the squared state radius of the site
exciton, and !imp stands for the positive difference
between the polarizability of the impurity atom (mo-
lecule) and that of the substituted lattice atom [9]. A
two-site exciton, moving along the anion chain, can
be localized near an impurity atom lying at a distance
R from the chain if |U(R)| > T. This gives the esti-
mate (in atomic units):
" "
!
� �
�
�
�
�
�
�
�
�
�
�
R
r
T
/
2
2
1 3
imp
. (18)
With realistic values of the parameters: r2 150� ,
!imp= 40 [9,18] at T=5 K, the estimate (18) gives
= 2300 a. u. = 6.3·10�14 cm2. For KI(Tl), RbI(Tl),
and NaI(Tl), the right-hand side of Eq. (16) takes on
the values 0.4, 0.6 and 6, respectively. Thus, the fast
energy transfer to impurity, observed in Refs. 14�16,
is explained qualitatively by directional motion of
two-site excitons. (In view of the rather weak de-
pendence of the trapping cross section (18) on the pa-
rameters, their choice cannot influence this conclu-
sion.)
Experiment shows [14�16] that the ratio
Yimp/Yhost decreases rapidly with increasing tempe-
rature, especially above 30 K. The observed tempera-
ture dependence of Yimp/Yhost is too strong to be de-
scribed by Eqs. (16) and (18) with a constant V and
should be attributed mainly to the temperature de-
struction of the coherent directional regime of the
two-site exciton motion (see Sec. 3).
5.2. Energy transport in rare-gas crystals
For rare-gas crystals, the free-exciton mechanism of
energy transfer should be discarded in view of the fol-
lowing experimental fact. The quantum yield Yimp of
the luminescence of a very weak impurity sharply
grows with an increase of the excitation photon energy
E near the point Eg of the dielectric gap width [19,
20]. This energy dependence of Yimp cannot mirror the
E dependence of the free-exciton photoproduction ef-
ficiency, which is of opposite character (a photon with
E < Eg inevitably generates a free exciton, whereas an
electron and hole, generated at E > Eg, do not neces-
sarily recombine in the form of a free exciton [21,8]).
However, this feature of the excitation spectrum
can be easily understood in terms of the vibrational
spectrum of two-site excitons, assuming that they are
responsible for the energy transfer to impurities [8,20].
Indeed, a photon with E > Eg generates an elec-
tron—hole pair. The hole is very rapidly self-trapped,
turning into a two-site molecular ion. The latter, after
recombination with an electron, becomes a two-site
exciton related to a high excited atomic state (close to
the ionization level) with a very large state radius r
[9]. Due to the large r , this excited state is strongly
attracted to impurity centers by its van der Waals po-
tential (17). On the other hand, a two-site exciton,
occupying a high vibrational level (extended to a nar-
row subband), is a very heavy band particle [7]; the
large r in combination with a large effective mass
provides a high probability for a two-site exciton to be
localized near an impurity center even at a very low
concentration of impurity. A two-site exciton, local-
ized on a high vibrational level, remains pinned at the
same impurity center during vibrational relaxation,
after which the exciton energy is transferred to the im-
purity. Under such conditions, a fast directional mo-
tion of two-site excitons provides an extremely effi-
cient energy transfer to impurities. Note that the high
motion velocity of two-site excitons, derived in Sec. 3
within the classical approach with allowance for a
non-pair exchange potential, does not contradict the
large effective mass of the two-site exciton subbands
derived quantum-mechanically within the approach of
a pairwise exchange potential (this seeming contradic-
tion was elucidated in Ref. 10).
Quite a different relaxation picture takes place
when a photon with E<Eg turns to a free exciton
(such optical transitions are partly allowed in a real
crystal with lattice defects [22]). The generated exci-
ton persists in a free state during the relaxation
through the upper part of the exciton band where the
free and two-site states are not mixed [9]. Unlike a
two-site exciton, which can be localized on any vibra-
tional level near an impurity center, a free exciton
cannot be trapped inside the band on a impurity level
lying below the band bottom, since this electronic
transition requires a too strong jump-wise heat re-
lease. Only in the lower part of the band the free
exciton is mixed with two-site states [7] and can be lo-
calized near an impurity center; but the localization
probability is low because of a comparably small state
radius.
Coherent motion and anomalous transport properties of exciton and hole polarons
Fizika Nizkikh Temperatur, 2003, v. 29, No. 3 247
6. Qualitative comparison with experiment.
Mobility of two-site holes
6.1. Rare-gas crystals
The mobility of two-site holes in rare-gas crystals
was measured experimentally [13] in a vicinity of the
triple point, where the structural perfection of the
samples was high enough to make such measurements
possible. Table 2 presents the temperature change of
the mobility of self-trapped holes in rare-gas crystals
measured in the temperature interval T1 < T< T2 [13].
Both the experimental values of mobility and its tem-
perature behavior observed strongly contradict the
law (1) with activation energy Uact � � � 1 eV. The
conventional equation (1) predicts a negligibly small
mobility and its highly sharp dependence on T. The
observed mobility is comparatively high and its tem-
perature variation is found to be weak (and even prac-
tically absent for krypton). This can be agreed with
(1) only by substituting a very small binding energy
Eb [13]. These values of Eb (given in the next-to-last
column of Table 2) strongly differ from the true bind-
ing energy estimated from spectroscopic data and
given in the last column. (The spectroscopic estimate
of Eb [1] is related to the corresponding two-site
excitonic state; it turns into the self-trapped hole
after ionization, which can only enhance the binding
energy [9].)
As shown in Sec. 4, these contradictions are natu-
rally eliminated by taking into account the intrinsic
vibrational structure of two-site hole polarons. Figu-
re 5 shows that, depending on the barrier height, the
calculated mobility can either grow or diminish some-
what with increasing temperature. This inference, as
well as the absolute values of the mobility obtained,
are in a qualitative agreement with experimental data
listed in Table 2.
6.2. Alkali-halide crystals
As was already mentioned in Sec. 1, the anion
sublattice of an alkali-halide crystal has a spectrum of
electronic excitations similar to that of the corre-
sponding rare-gas crystal (the electronic configuration
of the anion, e.g., I � or Cl�, is identical with that of
the corresponding rare-gas atom Xe or Ar, respec-
tively). The spectrum of electronic excitations of the
cation (Na+, K+ etc.) sublattice lies much higher and
cannot interfere with the spectrum of the anion
sublattice. In particular, a stable self-trapped state of
an anion hole has the form of a two-site quasi-mole-
cule similar to that formed in rare-gas crystals.
However, as regards the manner of motion of
self-trapped holes, alkali-halide crystals exhibit a dis-
tinction from rare-gas solids. As was mentioned in
Sec. 2, lattice atoms, lying beyond the chain, to some
degree hamper the motion of the chain atoms, so that
the lattice surroundings of the chain heighten the bar-
rier. In alkali-halide crystals, the effect of the lattice
surroundings on the barrier depends on the separation,
dAC, of an anion from the nearest cation, or, more ex-
actly, on the ratio of dAC to the distance dAA between
adjacent anions within the chain. The barrier height
grows with a decrease of the ratio dAC/dAA and, quite
naturally, with an increase of the cation-to-anion size
ratio.
Such an effect of the lattice surroundings on the
barrier B is illustrated by Table 3, based on the experi-
mental data of Ref. 2. The lowest barrier of about
0.1 eV was observed for the case of the largest anion I�
and the smallest cation Na+. Although Cs+ signifi-
248 Fizika Nizkikh Temperatur, 2003, v. 29, No. 3
A. M. Ratner
Table 2
Brief summary of experimental data on the self-trapped hole mobility � in rare-gas crystals measured in the temperature
interval T1 < T < T2 near the triple point [13]. The next-to-last column gives the polaron binding energy Eb found in Ref. 13
by fitting Eq. (1) with the experimental dependence � = �(T); the last column presents an independent realistic estimate of
Eb from spectroscopic data [1]. Note that the conventional theory, making no allowance for the intrinsic structure of a
self-trapped hole [3�6], cannot explain a weak temperature dependence of �, drastically contradicting to Eq. (1) with the
true binding energy Eb � 1 eV.
Crystal
T
1 T
2
�(T
1
) �(T
2
) E
b
, eV
K in units 0.01 cm2·V�1·s�1 from �(T) [13] spectroscopically
Ne 18 25 0.2 1.05 0.024 >1.9
Ar 71 83 1.2 2.3 0.07 >1.2
Kr 84 113 4.0 � 0.5 4.0 � 0.5 0.04 >0.9
Xe 110 160 3.2 1.8 0.005 to 0.02 >0.7
cantly exceeds Na+ in size, the barrier remains almost
the same for CsI due to a greater ratio dAC/dAA =
= 31/2/2 for the body-centered cell (for a rare-gas
crystal, the corresponding ratio equals unity). The
barrier gradually grows, due to a decrease of the anion
size, in the sequence CsI � CsBr � CsCl and rises
sharply (despite a decrease of the cation size) when
changing to the KCl crystal, with a smaller dAC/dAA
ratio. The difference in B between KCl and NaI ma-
nifests itself in a sharp difference in mobility
(10�3 cm2·s�1·V�1 for NaI and 10�10 cm2·s�1·V�1 for
KCl) [2].
Table 3
Height of the barrier for translational motion B and axis
reorientation Breor of two-site self-trapped holes in alkali
halides according to the experimental data of Ref. 2. dAC is
the separation of an anion from the nearest cation and dAA
is that between adjacent anions.
Crystal NaI CsI CsBr CsCl KCl
d
AC
/d
AA
0.707 0.866 0.866 0.866 0.707
B, eV 0.1 0.13 0.20 0.24 0.57
B
reor
, eV 0.18 0.20 0.37 � 0.54
Along with this, Table 3 contains data related to
the reorientation of the hole axis, i.e., changing the
hole polarization direction [2]. These data provide evi-
dence for a one-dimensional character of the two-site
hole motion (stated in Sec. 1). The motion of a hole
along the anion chain is accompanied by conservation
of its polarization direction (parallel to the chain). A
hole changes its motion direction (escaping to another
chain intersecting the former one) simultaneously
with changing the polarization direction. So, the dis-
tance (in units of the chain periods), traveled by a
hole along the chain before hopping to another chain,
can be estimated as
P
P
B B
T
transl
reor
reor~ exp
��
�
�
�
�
�, (19)
where Ptransl and B denote the rate and barrier height
for the hole translational motion; Preor and Breor are
those for the hole axis reorientation.
As is seen from Table 3, Breor exceeds B by about
1000 K which provides a predominant motion of a hole
within the same chain with infrequent hops between
different chains. The difference Breor � B is negative
only for KCl that drops out of the developed scheme,
being nearer to the conventional notion; but even for
KCl, Ptransl was found to exceed Preor at least by an
order of magnitude [2]. In Ref. 2 a general conclusion
for alkali halides was reached that the number of
translational hops of a two-site hole greatly exceeds
the number of reorientation hops.
Acknowledgement
The author is thankful to A.N. Ogurtsov and E.V.
Savchenko for helpful discussions.
This work was carried out within the project DFG
No. 436 UKR 113/55/0.
1. I.Ya. Fugol’, Adv. Phys. 27, 1 (1978).
2. E.D. Aluker, D.Yu. Lusis, and S.A. Chernov, Electro-
nic Excitations and Radioluminescence of Alkali Halide
Crystals, Riga Zinatne, Latvia (1979) (in Russian).
3. T. Holstein, Ann. Phys. N.Y. 8, 343 (1959).
4.D. Emin and T. Holstein, Ann. Phys N.Y. 53, 439
(1969).
5. D. Emin, Phys. Rev. B3, 1321 (1971).
6. D. Emin, Phys. Rev. B4, 3639 (1971).
7. A.M. Ratner, J. Luminesc., 81, 271 (1999).
8. A.M. Ratner, Phys. Lett. A265, 411 (2000).
9. A.M. Ratner, Phys. Rep. 269, 197 (1996).
10. A.M. Ratner, Phys. Lett. A291, 165 (2001).
11. A.M. Ratner, Phys. Lett. A298, 422 (2002).
12. K.S. Song, Canadian J. Phys. 49, 26 (1971).
13. P.G. Le Comber, R.J. Loveland, and W.E. Spear,
Phys. Rev. B11, 3124 (1975).
14. M. Tomura, O. Fujii, and H. Nishimura, J. Phys. Soc.
Jpn. 41, 194 (1976).
15. H. Nishimura, T. Kubota, and M. Tomura, J. Phys.
Soc. Jpn. 42, 175 (1977).
16. H. Nishimura and M. Tomura, J. Phys. Soc. Jpn. 39,
390 (1976).
17. H. Nishimura and T. Yamano, J. Phys. Soc. Jpn. 51,
2947 (1982).
18. P. Gombas, Theorie und Losungsmethoden des
Mehrteilchenproblems der Wellenmechanik. Basel,
(1950).
19. M. Haevecker, M. Runne, and G. Zimmerer, Electron
Spectr. Related Phenomena 79, 103 (1996).
20. A.N. Ogurtsov, A.M. Ratner, E.V. Savchenko, V.
Kisand, and S. Vielhauer, J. Phys.: Condens. Matter
12, 2769 (2000).
21. D. Varding, I. Reimand, and G. Zimmerer, Phys.
Status. Solidi. B185, 301 (1994).
22. A.M. Ratner, Phys. Lett. A269, 245 (2000).
Coherent motion and anomalous transport properties of exciton and hole polarons
Fizika Nizkikh Temperatur, 2003, v. 29, No. 3 249
|