L PREPRINT - To appear at VLSID 20244.5cm,1cm1.001.8

# Experimental Validation of Memristor-Aided Logic Using 1T1R
TaO$\rm{}_{\textbf{x}}$ RRAM Crossbar Array

Ankit Bende1§, Simranjeet Singh15§, Chandan Kumar Jha3, Tim Kempen6, Felix
Cüppers6,
Christopher Bengel9, André Zambanini10, Dennis Nielinger10, Sachin Patkar5,
Rolf Drechsler38, Rainer Waser16, Farhad Merchant2, Vikas Rana6 {a.bende,
si.singh, t.kempen, a.zambanini, d.nielinger, v.rana<EMAIL_ADDRESS>{simranjeet<EMAIL_ADDRESS>
<EMAIL_ADDRESS>{chajha<EMAIL_ADDRESS><EMAIL_ADDRESS>1PGI-7, 6PGI-10, & 10ZEA-2, Forschungszentrum
Jülich GmbH, Germany, 9IWE-2 RWTH Aachen, Germany,
5Indian Institute of Technology Bombay, India, 3University of Bremen, Germany,
8DFKI GmbH, Germany, 2Newcastle University, UK

###### Abstract

Memristor-aided logic (MAGIC) design style holds a high promise for realizing
digital logic-in-memory functionality. The ability to implement a specific
gate in a MAGIC design style hinges on the SET-to-RESET threshold ratio. The
TaOx memristive devices exhibit distinct SET-to-RESET ratios, enabling the
implementation of OR and NOT operations. As the adoption of the MAGIC design
style gains momentum, it becomes crucial to understand the breakdown of energy
consumption in the various phases of its operation. This paper presents
experimental demonstrations of the OR and NOT gates on a 1T1R crossbar array.
Additionally, it provides insights into the energy distribution for performing
these operations at different stages. Through our experiments across different
gates, we found that the energy consumption is dominated by initialization in
the MAGIC design style. The energy split-up is 14.8%, 85%, and 0.2% for
execution, initialization, and read operations respectively.

###### Index Terms:

MAGIC, RRAM, logic-in-memory, fabrication

§§footnotetext: Equal contribution

##  I Introduction

Memristive devices, such as resistive random access memory (RRAM), offer a
solution to the von Neumann bottleneck by implementing operations within
memory itself [1, 2]. One approach to implementing operations in memory is
designing digital logic gates exploiting two distinct states - the high
resistive state (HRS) and low resistive state (LRS) of an RRAM. These states
are correspondingly mapped to logic “0” and logic “1” respectively. Depending
on the input combination stored as a resistive state, the output memristor
state can switch from one state to another, representing a logical output
value. Several methods for achieving digital logic-in-memory (LiM) have been
suggested in the literature. Various stateful and non-stateful logic
techniques have been presented in the literature such as IMPLY [3], FELIX [4],
majority logic [5], and memristor-aided logic (MAGIC) [6]. Amongst all the
techniques, MAGIC stands out as a popular choice because it stores the output
in the form of the memristor’s state itself, representing stateful logic.

Experimental validation of MAGIC gates has recently been achieved using
fabricated valence change memory (VCM) devices [7, 8]. However, this study
specifically focuses on passive crossbar architectures, which suffer from
sneak path currents and scalability challenges [9]. The passive crossbars also
encounter difficulties during device formation, requiring significant initial
current. To address these issues and enhance forming capabilities, a solution
involves incorporating a transistor in series with the memristive device. This
configuration creates a 1T1R cell, effectively mitigating sneak path problems.
The 1T1R cell enables precise current control at the individual device level,
offering enhanced control for the MAGIC operations. Despite the increase in
physical footprint, the 1T1R configuration renders the overall system more
scalable and allows for better control [9].

Figure 1:  (a) 1T1R cell schematic (b) schematic cross-section of CMOS
integrated memristive device, (c) SEM image of fabricated memristive devices,
and (d) I-V switching characteristics of a 1T1R cell after forming.

The MAGIC design style offers the potential for a variety of logic gates.
However, the availability of specific logic gates is contingent upon the SET
and RESET switching thresholds, which are directly influenced by the material
stack used in the fabrication process [10]. The RRAM device based on
Pt/TaOx/W/Pt stack offers implementation of OR, NIMP, and 2-cycled XOR gates
and has been demonstrated using 1R passive devices [7]. Additionally, in this
specific stack configuration, the output device is initialized to the HRS
(logic “0”) state, and the input combination, along with execution voltage,
determines its transition to the LRS (“1”) from the HRS state.

To the best of our knowledge, for the first time, this paper demonstrates the
implementation of MAGIC gates on a _1T1R TaO x RRAM crossbar array_ and
illustrates the realization of both OR and NOT gates, which can be effectively
combined to implement any Boolean operation as they are functionally complete.
Additionally, the paper offers a detailed breakdown of energy consumption
during each operation phase within the MAGIC design style. Given the potential
application of the MAGIC design style in creating general-purpose processing
units, it becomes imperative to examine its design from an energy consumption
perspective [11]. Such an analysis is crucial for gaining insights into its
viability in future technologies. The following are the contributions of this
paper:

  * •

Fabrication of TaOx RRAM devices and their integration with CMOS to make the
1T1R crossbar array.

  * •

Experimental validation of OR and NOT gates based on MAGIC design style on
fabricated 1T1R crossbar array.

  * •

Comprehensive energy assessment and breakdown of energy consumption during the
MAGIC operations.

The rest of the paper is organized as follows. We provide a background of the
used technique in Section II. Section III discusses the experimental methods
used for logic implementation and energy estimations. The results obtained
from the study are explained in Section IV. Finally, we conclude the paper in
Section V.

Figure 2: Logic gates mapping on fabricated 8x4 1T1R array. (a) SEM image of
the fabricated 8x4 array (array orientation rotated by 90°), (b) Schematic of
8x4 1T1R array, (c) schematic of two input OR gate implemented in the array,
(d) schematic of NOT gate implemented in the array.

##  II Background and Related Work

###  II-A Memristive Devices

Memristive devices have emerged as a significant advancement in non-volatile
memory technology. Initially proposed as a concept by Professor Leon Chua in
1971 [12], RRAM has gained prominence due to its unique ability to store data
by modulating resistance states [13]. The resistance modulation is achieved by
applying a voltage across the terminals of the RRAM. In response, the
resistance of the devices changes based on the magnitude and direction of the
current flow. Remarkably, the memristor preserves its resistance value even
when devoid of power, safeguarding its data until a new voltage is applied,
firmly establishing its status as a non-volatile memory element [1].

These memristive devices can be interconnected to form a crossbar structure.
However, when individual memristive devices are connected in a passive
crossbar configuration, issues related to forming and sneak-path currents can
arise. To mitigate these concerns, memristive devices are fabricated with a
CMOS transistor in series, resulting in what is known as a 1T1R cell. In Fig.
2 (a), the SEM image of the fabricated 1T1R cell is shown, with multiple cells
interconnected in an 8x4 (rows $\times$ columns) crossbar structure
(fabrication detailed is discussed in Section III). Fig. 2 (b) provides a
schematic layout of the 1T1R crossbar array. The word lines (WLs) from 1 to 4
are linked to the gates of the devices connected in columns, while the source
lines (SLs) from 1 to 8 are connected in a row-wise fashion, shorted to all
the source pins of the transistors within the same row. The bit lines (BLs)
from 1 to 4 are connected to the top electrode (TE) pin of each memristor
within a column, as illustrated in Fig. 2 (b).

###  II-B MAGIC Design Style

MAGIC represents a stateful logic methodology that employs crossbar-connected
memristive devices to execute logic operations. Each memristor is programmable
to two distinct states, HRS and LRS, which are subsequently mapped to logic
“0” and “1” respectively. An initialization step is necessitated to facilitate
the MAGIC operations, configuring the output memristor to its initial state.
The range of achievable gates within these devices is contingent upon the SET-
to-RESET switching threshold ratio. The SET-to-RESET voltage ratio for the
device stack used in our study made logic OR, NIMP, and NOT gates attainable,
diverging from the original NOR and NOT gates proposed in [6]. The output
memristor ($y_{out}$) is always initialized to the HRS state rather than the
LRS state.

In the OR operation, an execution voltage ($V_{exe}$) is applied to the input
memristors ($x_{1}$ and $x_{2}$). Simultaneously, the output memristor,
initially set to the HRS state, is grounded, as depicted in Fig. 2 (c).
Furthermore, the NOT operation necessitates three memristors but with
different voltage values compared to the OR operation. In this context, one of
the inputs is consistently initialized to LRS (designated as $x_{1}$ for
clarity) in conjunction with the output memristor as shown in Fig. 2 (d).
Additionally, distinct execution voltages ($V_{exe}$ and $V_{exe}/{3}$) are
employed for the $x_{1}$ and $x_{in}$ memristors, respectively.

###  II-C Related Work

The experimental validation of MAGIC design style on TaOx RRAM devices using
_passive crossbars_ has been demonstrated in existing literature [7].
Nonetheless, passive crossbars are plagued by the issue of sneak-path
currents, which can disrupt the accurate reading of the final state.
Additionally, forming processes in passive crossbars poses challenges. Non-
stateful logic (e.g., scouting and majority logic) has been demonstrated using
1T1R cells [14]. This paper represents the pioneering demonstration of
_stateful logic on a 1T1R TaO x RRAM crossbar_.

Prior studies have indicated that energy consumption in the MAGIC design style
is predominantly influenced by initialization energy [15]. However, it’s
essential to note that these findings are primarily derived from simulation
studies. Furthermore, the presented energy figures are based on simulation
models and pertain specifically to NOT and NOR gates based on the MAGIC design
style. This paper takes strides towards calculating the energy consumption of
OR and NOT gates in fabricated TaOx RRAM devices.

##  III Experimental Methods

###  III-A Device Fabrication

For the experimental validation of MAGIC gates, 1T1R-based active memristive
arrays were fabricated and integrated with CMOS 180 nm technology provided by
X-FAB. The dimensions of the memristive devices in the arrays are 100 $nm$ x
100 $nm$ and consist of Pt/TaOx/W/Pt stack. Fig. 1 (b) shows the schematic
vertical cross-section of the fabricated device. Firstly, the W plugs from the
processed wafers were exposed, and the 25 nm thick Pt layer was deposited as
the bottom electrode (BE) with DC sputtering. The BE layer was then patterned
using electron beam lithography and back etching using reactive ion etching
(RIE). A 7 nm thick TaOx layer was then deposited by RF sputtering in Ar (77%)
and O2 (23%) gas mixture at 236W RF power followed by deposition of 13 nm
thick W electrode using the DC sputtering. Subsequently, a 25 nm thick Pt
layer was deposited as TE using the DC sputtering. Finally, the deposited
switching oxide and TE stack were patterned using electron beam lithography
and RIE-based back etching. Fig. 1 (c) shows the SEM image of the fabricated
1T1R TaOx RRAM device.

###  III-B Electrical Characterization Setup

The electrical characterization of the 1T1R memristive devices was carried out
using Keithley 4200 SCS. Fig. 1 (a) shows the schematic of the fabricated 1T1R
memristive device with different terminals. The memristive device in its
pristine state needs a one-time forming step, which involves the creation of a
conductive filament in the switching oxide by applying a positive voltage
across the memristor. The current through the memristor during the
electroforming process is controlled by applying an appropriate DC gate
voltage to avoid permanent breakdown of the oxide. To realize digital (logic
in memory) LiM using these devices, four distinct operations are necessary for
any logic operation, which are discussed below.

  * •

SET Operation: The SET operation involves changing the device’s state from HRS
to LRS. This is achieved by applying a positive ramp voltage from 0 to 1.8V at
the TE electrode while maintaining a constant DC voltage of 1.6V on the gate
terminal to limit the current flowing through the memristive to 500$\mu A$.
The drain and bulk pins of the transistor are connected to the ground.

  * •

RESET Operation: The RESET operation switches the device from LRS to HRS by
applying a positive ramp voltage from 0 to 2V at the source terminal, while
keeping TE and bulk grounded. A minimum current requirement is crucial to
dissolve the filament, and if not met, the device remains in LRS. Achieving
this current requirement entails applying the maximum allowed voltage of 5V at
the NMOS gate, opening the channel for current conduction, and affecting the
minimum transistor size usable with memristors for SET and RESET processes.

  * •

Execution Operation: During the execution operation, specific voltage
configurations are applied to memristors for executing the OR and NOT
operations. Once the input memristors are loaded with the input, the execution
voltage applied across them decides the logic operation being executed on
them. During this operation, the output memristor is connected to the ground.
The detailed voltage value is discussed in Section IV-B and IV-C.

  * •

Read Operation: The read operation serves to determine the current state of
the memristor. A read voltage of 0.5V is applied at the TE and the source and
the bulk terminals are pinned to the ground. A gate voltage of 3.3V is applied
at the gate to open the NMOS channel during read operation.

Figure 3: Electrical characterization of TaOx RRAM devices. (a) Forming and
median I-V curve for 100 switching cycles. (b) Cycle-to-cycle variability for
100 switching cycles. (c) Device-to-device variability for 17 devices with 100
switching cycles each.

Through skillful control of these operations, it becomes possible to design
any arbitrary logic configuration on the crossbar. In this specific instance,
these voltage sequences are combined to generate OR and NOT gates on the
crossbar. However, these voltage patterns can also be harnessed to execute
complex circuits sequentially or implement architectures resembling single
instruction multiple data. Subsequently, we look into the outcomes achieved by
sequentially applying these voltages to achieve the desired operations.

##  IV Results and Discussion

Within this section, we unveil the outcomes derived from our experimental
examinations and analyses, offering valuable insights into the energy
consumption associated with the implementation of MAGIC OR and NOT gates.

###  IV-A 1T1R TaOx RRAM Switching Characteristics

The initial forming and subsequent 100 switching cycles are shown in Fig. 3
(a). The memristive devices exhibit counterclockwise switching
characteristics. Fig. 3 (b), shows the CDF plot of HRS and LRS states obtained
by switching the 1T1R devices 100 times. The device exhibits low cycle-to-
cycle (C2C) variation with a consistent HRS/LRS ratio of 10. For evaluating
device-to-device (D2D) variability, around 17 devices were switched 100 times
each as shown in Fig. 3 (c).

Figure 4: Execution operation in OR gate for (a) input “00”, (b) input “01”,
(c) input “10” and (d) input “11”. For input “00” there is no sharp change in
Iout implying no change of yout state. For the rest of the input combinations,
yout undergoes the SET process by drawing the majority of the current through
the input memristors that are in the LRS state.  Figure 5: Logic OR operation.
Output read currents for the different combinations of input read currents.

From the C2C and D2D variability plots, it is evident that the HRS state
exhibits higher variability with resistances distributed over two orders of
magnitude ranging from around 100K$\Omega$ to 1M$\Omega$. The high variability
in HRS can be attributed to the stochastic or uncontrolled breaking of
filament during the RESET process [16]. Although the devices exhibit
variability in the HRS state, a _consistent HRS/LRS ratio of 10_ is obtained
across all the tested devices and suffices for the implementation of logic
gates. Next, we will discuss the execution of logic OR and NOT operations on
these devices.

Figure 6: Execution operation in NOT gate for (a) input “1”, (b) input ”0”. TABLE I: Truth table for OR and NOT logic gate operation OR | NOT
---|---
$x_{1}$ | $x_{2}$ | $y_{init}$ | $y_{out}$ | $x_{1}$ | $x_{in}$ | $y_{init}$ | $y_{out}$
0 | 0 | 0 | 0 | - | - | - | -
0 | 1 | 0 | 1 | - | - | - | -
1 | 0 | 0 | 1 | 1 | 0 | 0 | 1
1 | 1 | 0 | 1 | 1 | 1 | 0 | 0

###  IV-B MAGIC OR Implementation

Fig. 2 (c) shows the schematic to implement the logic OR gate on the crossbar.
To implement OR gate only three memristors are required so this needs to be
mapped to the fabricated crossbar with a size of 8x4 as shown in Fig. 2 (c).
To map the OR gate on the crossbar, three memristive devices sharing a common
WL and a BL in the array are used to implement the gates. Firstly, the inputs
of the OR gate are stored as the resistance state in $x_{1}$ and $x_{2}$.
Subsequently, the output memristor ($y_{out}$) is initialized to the “0”
state. Next, an execution voltage ($V_{exe}$) sweep from 0 to 3.3V at the TE
terminal of $x_{1}$ and $x_{2}$ and the current at $y_{out}$ is monitored.
During this cycle, the BL shared by three $x_{1}$, $x_{2}$ and $y_{out}$ is
kept floating. While WL/$V_{G}$ is connected to a DC voltage of 3.3V and the
SL of the output memristor is grounded. All the other unused WLs, SLs, and BLs
are kept floating. Post-execution cycle, the state of $y_{out}$ is obtained by
performing a READ operation. The memristor output currents during execution
cycles for different inputs are shown in Fig. 4.

The truth table for the OR gate with the input and output states of the
memristor are shown in Table I. In Table I, $y_{init}$ column shows the
initialization state of the output memristor before the execution cycle. It
can be inferred from the truth table that for successful OR gate operation,
the $y_{out}$ changes its state for all combinations of inputs except for the
input “00” during the execution cycle. Fig. 4 shows the execution cycles for
different input combinations. For the input “00” case, both $x_{1}$ and
$x_{2}$ are in HRS state represented by the current value $I_{in1}$ and
$I_{in1}$, respectively. When an execution voltage is applied at the inputs,
the current through the $y_{out}$ is limited by the parallel combination of
$x_{1}$ and $x_{2}$. This current is not sufficient to drive the $y_{out}$ to
the SET state. This is evident from the fact that no sharp change in output
current ($I_{out}$) w.r.t applied $V_{exe}$ is observed in the $y_{out}$ as
shown in Fig. 4 (a).

On the other hand, for the input combination of “01”, “10”, and ”11”, either
one or both of the input memristors are in LRS states. This allows a sharp
increase in $I_{out}$ during the execution cycle, contributing to the change
of state of $y_{out}$ as shown in Fig. 4 (b), Fig. 4 (c) and Fig. 4 (d),
respectively. In Fig. 5, the read currents of output memristors for different
combinations of input memristors are displayed, showing a successful logic OR
operation.

Figure 7:  Logic NOT operation. Output read currents for the different combinations of input read currents.  TABLE II: Number of SET and RESET operations required during initialization for OR gate implementation | No. of initialization steps
---|---
Input | RESET | SET
00 | 3 | 0
01 | 2 | 1
10 | 2 | 1
11 | 1 | 2

###  IV-C MAGIC NOT Implementation

The schematic of the MAGIC NOT gate is shown in Fig. 2 (d). For performing NOT
logic operation, memristor $x_{1}$ and $y_{out}$ are initialized to LRS and
HRS state, respectively. Input memristor $x_{in}$ is initialized in accordance
with the input. An execution ramp voltage of $V_{exe}$ from 0 to 1.5V is
applied at the source terminal of $x_{1}$ whereas 1/3 $V_{exe}$ voltage is
ramped at the $x_{in}$. Table I shows the truth table for NOT gate with
different input combinations. For the successful operation of NOT gate,
$y_{out}$ memristor should change its state when input “0” is applied and
should remain in “0” state for input “1”. The currents in the output memristor
during the execution cycle for different inputs are shown in Fig. 6.

The input voltage applied at $x_{in}$ is 1/3rd of the voltage applied at
$x_{1}$. Because of this, during the input “0”, the potential difference
across the $y_{out}$ memristor (currently in HRS state) is higher as compared
to the $x_{in}$. This higher potential across $y_{out}$ allows it to undergo a
transition from RESET to SET through $x_{1}$ (currently in LRS) state. This
can be seen as a sharp rise in output current as depicted in Fig. 6 (b). On
the other hand, for the case of input bit “1”, the $y_{out}$ memristor path
becomes higher in resistance as compared to $x_{in}$ path and does not receive
enough current to drive it into SET from RESET state as shown in Fig. 6 (a).
This allows $y_{out}$ to change its state only when $x_{in}$ is in the HRS
state. Fig. 7 summarizes the output read currents for different input read
current combinations and successfully demonstrates NOT gate.

TABLE III: Energy consumption in SET, RESET, and READ operations for full voltage ramp cycle Operation | Voltage(sweep) | Pulse duration ($\mu$ s) | Energy(nJ)
---|---|---|---
SET | 0 to 2 V | 4 ms | 1300
RESET | 0 to 1.8 V | 3.6 ms | 312
LRS Read | 0 to 0.3 V | 0.6$\mu$ s | 5.4
HRS Read | 0 to 0.3V | 0.6$\mu$ s | 0.056
Figure 8: I,V-t curves for calculating the energy consumption of (a) SET
process (b) RESET process (c) Logic OR ”11” input execution.

###  IV-D Energy Calculations

The energy consumption of logic operations is heavily dependent on the
initialization as well as execution energies. In earlier works, researchers
have typically calculated the energy consumption of a log-in memory system
through a coarse-grained approach by multiplying the average energy of
operations by the number of operations [17]. However, this method has been
found to underestimate the actual energy consumption as it ignores the
initialization energy involved during the operations [15, 18]. However, these
results are in the simulation and with different memristor models. Therefore,
in the current study, a similar approach has been considered for real devices,
for calculating the energy consumption of logic operations by taking execution
as well as initialization energy into account.

TABLE IV:  Comparison of initialization energy to the total energy consumed during different input combinations for logic OR operations. OR | Full Voltage Ramp cycle | Optimal
---|---|---
Input | Initialization | Execution | Read | % Initialization | % Read | Initialization | Execution | Read | % Initialization | % Read
| Energy (nJ) | Energy (nJ) | Energy (nJ) | Energy | Energy | Energy (nJ) | Energy (nJ) | Energy (nJ) | Energy | Energy
00 | 3900 | 139 | 0.1 | 97 | 0.002 | 696 | 8 | 0.07 | 99 | 0.003
01 | 2912 | 2455 | 5.4 | 54 | 0.1 | 738 | 108 | 2.8 | 87 | 0.1
10 | 2912 | 2300 | 5.4 | 56 | 0.1 | 738 | 73 | 2.8 | 91 | 0.1
11 | 1924 | 3531 | 10.8 | 35 | 0.2 | 780 | 134 | 5.6 | 85 | 0.2

During OR operation, the different initialization steps involve numerous SET
and RESET operations as mentioned in Table III. In the study, triangular wave
sweep voltages are used to perform, SET, RESET, and READ as well as execution
operations. The energy is obtained by multiplying the voltage waveforms with
the sensed current and then integrating the product over the measurement time
as per $\int_{0}^{t}v(t)\times i(t)\,dt$, where $t$ is the pulse time
considered for energy calculations. The I,V-t curves corresponding to median
SET and RESET operations are shown in Fig. 8. Two approaches have been used
for calculating the energy: a) using full voltage ramp cycle time and b) using
optimum times. In the first technique, energy is calculated for the whole
triangular voltage sweep. In contrast, in the case of optimum energy
calculation, the SET, RESET, and execution times have been derived from the
experimental data, and energy is calculated for the same as shown in Fig. 8.

The former technique gives actual energy consumption numbers but is an
overestimate. Therefore, for more realistic energy values, the latter
technique is used. Table IV summarises the energy consumption for OR and NOT
gate operation through both techniques. It is quite evident that for both the
optimal and full voltage ramp cycle-based energy calculations, depending on
the inputs of the OR operation, around 35-85% of the energy consumption is
constituted by the initialization energy, execution energy constitutes around
64.8-14.8% while read energy only constitutes around 0.2% of energy Evidently,
as the accuracy of energy calculation increases, the gap between total energy
and execution energy also increases. Therefore, for accurate calculation of
energy consumption for logic operations using the MAGIC technique, the
initialization energy must be taken into consideration.

##  V Conclusion

In this paper, TaOx RRAM devices were fabricated and integrated on a 180 nm
CMOS substrate to create a 1T1R crossbar array. The fabricated devices
consistently exhibited an HRS/LRS ratio of approximately 10, making them
eligible for the implementation of MAGIC gates. Subsequently, we demonstrated
the implementation of logic OR and NOT gates, and along with energy
consumption values. Energy consumptions for logic OR and NOT operations were
calculated by evaluating both the initialization and execution energies. It
was found that the initialization energy played a significant contributing
role in the overall energy consumption during logic implementation, similar to
the trends observed in the simulation study.

## Acknowledgments

This work was supported in part by the Federal Ministry of Education and
Research (BMBF, Germany) in the project NEUROTEC II under Project 16ME0398K,
Project 16ME0399, German Research Foundation (DFG) within the Project PLiM (DR
287/35-1, DR 287/35-2) and through Dr. Suhas Pai Donation Fund at IIT Bombay.
Special thanks go out to Dr. Michael Schiek for his management work in the
NEUROTECH II project.

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##  0.0.1 Fundamentals and Applications of Isotope Effect in Modern
Technology.

V.G. Plekhanov.

Fonoriton Science Lab., Garon Ltd., P.O. Box 2632, Tallinn, 13802, ESTONIA

$<$e-mail$>$<EMAIL_ADDRESS>

Different crystals (semiconductors and insulators) with varying isotopic
composition have been recently grown. I discuss here the effect of isotopic
mass and isotopic disorder on the properties (vibrational, elastic, thermal
and optical) of different crystals. The main applications of the stable
isotopes are included self-diffusion, neutron transmutative doping (NTD) of
different semiconductors, optical fibers, isotope-based quantum computers,
etc. Because of space limitations this discussion will not exhaustive. I hope
however, to give sufficient references to published work so that the interest
reader can easily find the primary literature sources to this rapidly
expanding field of solid state physics.

Phonons, excitons, isotope-mixed crystals, laser materials, quantum
information, isotope-based quantum computers.

It is well-known that the presence of randomly distributed impurities in a
crystal can give rise to significant variations of its mechanical, electrical,
thermal, and optical properties with respect to those of the pure solid. All
these properties are, more or less, directly related to the structure of the
manifold of phonon states and any variation induced in this structure by the
presence of the impurities, will produce a corresponding alteration of the
physical properties of the material. Of particular interest is the case in
which the impurity species is of the same chemical nature, but with a
different mass, i.e. the case of isotopic impurities. The mechanisms by which
the impurities (isotopes) perturb the phonon distribution will depend on the
mass difference between the host and guest species [1-3]. Phonons are the
crystal excitations most directly related to the isotopic masses. In monatomic
crystals (like C, Ge, Si., etc.), and within the harmonic approximation, all
phonon frequencies scale like the square root of the average isotopic mass.
Namely, this feature can be used for the nondestructive isotopic
characterization investigated materials. The isotopic effect can be classified
into two categories: 1) The first type is caused by the variation of the
phonon frequencies with the average isotopic mass. To this type belongs the
isotope effect in superconductors, which plays an important role in the search
for the mechanism of high Tc superconductivity (see, e.g. [4]). The effect of
changing the atomic mass M is to change the phonon frequencies $\omega$
according to:

$\omega$ = $\sqrt{\frac{\alpha}{\text{M}}}$, (1)

where $\alpha$ is a force constant characteristic of the phonon under
consideration. The change in atomic mass implies, at low temperatures (see
below), a change in the average atomic displacement for each phonon mode. In
the case of one atom per primitive cell the mean squared phonon amplitude
$\langle$u${}^{2}\rangle$ is given by [1;2]:

$\langle$u${}^{2}\rangle$ =
$\langle\frac{\hbar^{2}}{\text{4M}\omega}\left[1\text{ +
2n}_{B}\text{(}\omega\text{)}\right]\rangle$ =
$\langle\frac{\hbar}{\text{4M}^{1/2}\alpha^{1/2}}\left[1\text{ +
2n}_{B}\text{(}\omega\text{)}\right]\rangle$, (2)

where nB($\omega$) is the Bose - Einstein statistical factor, $\omega$ is the
frequency of a given phonon and $\langle$…$\rangle$ represents an average over
all phonon modes. The average in r.h.s. of (2) is often simplified by taking
the value inside $\langle$…$\rangle$ at an average frequency $\omega_{D}$
which usually turns out to be close to the Debye frequency. We should
distinguish between the low temperature ($\hbar\omega$ $>$$>$ kBT) and the
high temperature ($\hbar\omega$ $<$$<$ kBT) limits and see:

($\hbar\omega$ $>$$>$ kBT), $\langle$u${}^{2}\rangle$ =
$\frac{\hbar}{\text{4M}\omega_{D}}$ $\sim$ M-1/2 independent of T and

($\hbar\omega$ $<$$<$ kBT), $\langle$u${}^{2}\rangle$ =
$\frac{\text{k}_{B}\text{T}}{\text{2M}\omega^{2}}\sim$ T independent of M (3).

Using Eq. (1) we can find from last equations that $\langle$u${}^{2}\rangle$,
the zero-point vibrational amplitude, is proportional to M-1/2 at low
temperatures: it thus decrease with increasing M and vanishes for
M$\longrightarrow\infty$. For high T, however, we find that
$\langle$u${}^{2}\rangle$ is independent of M and linear in T (details see [3]
and references therein).

Another type of isotope effects is produced by the isotopic mass fluctuations
about the average mass $\langle$M$\rangle$. These fluctuations perturb the
translational invariance of a crystal and lift, at least in part, k \- vector
conservation. The most striking effect of this type is observed in the thermal
conductivity which has a maximum at a temperature TM $<$$<$ $\Theta_{D}$(here
$\Theta_{D}$ is Debye temperature, TM = 80 K for diamond, TM = 20 K for
silicon (see, also Figs. 64 - 66 in [3]). Reduction of the concentration of
13C from the standard 1% (against 99% of 12C) by a factor of ten increases the
thermal conductivity of diamond by about a factor of two, a fact that leads to
amplifications in situations where a large amount of generated heat has to be
driven away (e.g. as substrates for high power electronic devices [5]). As is
well-known this maximum represents the transition from boundary scattering to
the phonon unklapp scattering regime and its value Km is determined by the
isotopic fluctuation parameter g (mass variance):

g = $\frac{\left\langle\text{M}^{2}\right\rangle}{\left\langle
M\right\rangle^{2}}$ \- 1, (4)

the larger g - the smaller Km [6].

It is known that materials having a diamond structure are characterized by the
triply degenerate phonon states in the $\Gamma$ \- point of the Brillouin zone
(k = 0). These phonons are active in the Raman scattering (RS) spectra, but
not in the IR absorption ones (see, e.g. [7]). First - order Raman light -
scattering spectrum in diamond crystals includes one line with the maximum at
$\omega_{LTO}$($\Gamma$) = 1332.5 cm-1. In Fig. 1a, the first-order scattering
spectrum in diamond crystals with different isotope concentration is shown
[8]. As was shown, the maximum and the width of the first-order scattering
line in isotopically-mixed diamond crystals are nonlinearly dependent on the
concentration of isotopes x (see also [7]). The maximum shift of this line is
52.3 cm-1, corresponding to the limiting values of x = 0 and x = 1.

Fig. 1b demonstrates the dependence of the shape and position of the first-
order line of optical phonons in germanium crystal on the isotope composition
at liquid nitrogen temperatures [9]. The coordinate of the center of the
scattering line is proportional to the square root of the reduced mass of the
unit cell, i.e. M-1/2. It is precisely this dependence that is expected in the
harmonic approximation (details see [3]). An additional frequency shift of the
line is observed for the natural and enriched germanium specimens and is
equal, as shown in Refs. [7, 9] to 0.34$\pm$0.04 and 1.06$\pm$0.04 cm-1,
respectively (see also Fig. 7 in Chap. 4 of Ref. [10]). Detailed calculation
of the shape of the lines in RS of semiconductors have been performed by
Spitzer et al. [11]. In their paper a quantitative agreement with the
experimental data on diamond and germanium has been obtained. Comparing the
half-widths of the scattering lines in first-order RS in diamond and germanium
(see Fig. 1), it is easy to see that the observed line broadening due to
isotopic disorder in diamond is much greater than that in germanium. The
reason for this is that the k = 0 point is not the highest point in the
diamond dispersion curve (see Fig. 10b in Ref. [7]), whereas in the case of
germanium it is the highest point [12]. This shift of the maximum from the
$\Gamma$ \- point (k = 0) leads to a much larger density of states in the
vicinity of $\omega_{LTO}$ in comparison with the normal one calculated by the
formula:

Nd $\sim$ Re($\omega_{LTO}$ \- $\omega$ \+
i$\left[\frac{\Delta\omega_{LTO}}{\text{2}}\right]$)1/2 (5).

(for more details see Ref. [12]). The density of states in diamond is
asymmetric with respect to $\omega_{LTO}$, causing asymmetry in the shape of
the scattering line [7]. This asymmetry also leads to the asymmetric
concentration dependence of the half-width of the scattering line. As was
shown early (see, e.g. [3] and references therein), in the case of a weak
potential of isotopic scattering of phonons, their self-energy
$\varepsilon\left(\omega\right)$ does not depend on q (- phonon quasiimpuls).
This is precisely the situation observed for C and Ge. Indeed, if we express
the mass fluctuation $\Delta$M/$\overline{\text{M}}$ ($\overline{\text{M}}$ is
the mean mass of all isotopes) in the form of the variation of the phonon band
width $\Delta\omega_{0}$ = 12 cm-1 at q = 0 and compare it with the width of
the band of optical phonons in Ge equals to $\approx$ 100 cm-1, we will see
that the variations very small. Under this conditions the localization of
optical phonons in Ge is naturally, absent, and as observed in experiment,
they stay delocalized (see below, however opposite case in LiHxD1-x crystals).
Moreover, direct measurements of the phonon lifetime in Ge show that, in the
case of anharmonic decay, it is two orders of magnitude shorter than the
lifetime that is due to the additional scattering by isotopes, i.e.
$\tau_{anharm}$ = $\tau_{disord}\cdot$ 10-2[13]. Therefore, the contribution
of anharmonicity to the half-width of the first-order light scattering line in
Ge is two orders of magnitude greater than that caused by the isotopic
disorder in crystal lattice. In conclusion of this part of our report we
should mention that analogous structure of first-order RS and their dependence
on isotope composition has by now been observed many times, not only in
elementary Si and $\alpha$-Sn, but also in compound CuCl, CuBr, ZnSe, GaN
semiconductors (details see Ref. [3]).

In Fig. 2 (curve 1) the spectrum of second-order RS of light in pure LiD
crystal is shown [7]. In spite of the fact, according to the nomogram of
exciton states [14], the crystal studied should be considered to be pure, its
RS spectrum contains a clear high-frequency peak around 1850 cm-1. The
observed peak does not have an analogue in RS of pure LiH (Fig. 2, curve 4)
and has already been observed earlier in the second-order RS and has been
interpreted (see [7] and references therein) as a local vibration of the
hydrogen in LiD crystals. Further we note that as the concentration grows
further (x $>$ 0.15) one observes in the spectra a decreasing intensity in the
maximum of 2LO($\Gamma$) phonons in LiD crystal with a simultaneous growth in
intensity of the highest frequency peak in mixed LiHxD1-xcrystals (Fig. 2,
curve 3). The origin of the last one is in the renormalization of LO($\Gamma$)
vibrations in mixed crystals [7]. Comparison of the structure of RS spectra
(curves 1 and 2 in Fig. 2) allows us, therefore, to conclude that in the
concentration range of 0.1 $<$ x $<$ 0.45 the RS spectra simultaneously
contain peaks of the LO($\Gamma$) phonon of pure LiD and the LO($\Gamma$)
phonon of the mixed LiHxD1-xcrystal. Thus, the second-order RS spectra of
LiHxD1-xcrystals have one- and two-mode character for LO($\Gamma$) phonons,
and also contain a contribution from the local excitation at small values of
x. Moreover, we should add that an additional structure in RS spectra on the
short-side of the 2LO($\Gamma$) peak (see Fig. 21 in Ref. [7]) was observed
relatively ago in mixed LiHxD1-xcrystals and, very recently, in isotopically
mixed crystals of diamond, germanium and $\alpha$-Sn (details see [3, 11]).
These effects caused by isotopic disorder in the crystal lattice of
isotopically mixed crystals [3]. The observation of two-mode behavior of the
LO($\Gamma$) phonons in RS spectra of LiHxD1-xcrystals contradicts the
prediction of the CPA [15], according to which the width W of optical
vibration band should be smaller than the frequency shift ($\Delta$) of
transverse optical phonon. However, as was shown early (see, e.g. [7] and
references therein) in LiHxD1-x mixed crystals, the reverse inequality is
valid, i.e. W $>$ $\left|\Delta\right|$. According [16], this discrepancy
between experimental results and theory based on CPA [15] is mainly explained
by the strong potential of scattering of phonons, caused by a large change in
the mass upon substitution of deuterium for hydrogen. Once more reason of the
discrepancy between theory and results of the experiment may be connected with
not taking into account in theory the change of the force-constant at the
isotope substitution of the smaller in size D by H ion. We should stress once
more that among the various possible isotope substitution, by far the most
important in vibrational spectroscopy is the substitution of hydrogen by
deuterium. As is well-known, in the limit of the Born-Oppenheimer
approximation the force-constant calculated at the minimum of the total energy
depends upon the electronic structure and not upon the mass of the atoms. It
is usually assumed that the theoretical values of the phonon frequencies
depend upon the force-constants determined at the minimum of the adiabatic
potential energy surface. This leads to a theoretical ratio
$\omega\left(\text{H}\right)$/$\omega\left(\text{D }\right)$of the phonon
frequencies that always exceed the experimental data. Very often anharmonicity
has been proposed to be responsible for lower value of this ratio. In isotope
effect two different species of the same atom will have different vibrational
frequencies only because of the difference in isotopic masses. The ratio p of
the optical phonon frequencies for LiH and LiD crystals is given in harmonic
approximation by:

p = $\frac{\omega\left(\text{H}\right)}{\omega\left(\text{D }\right)}$ =
$\sqrt{\frac{\text{M}\left(\text{LiD}\right)}{\text{M}\left(\text{LiH}\right)}}\simeq$
$\sqrt{\text{2 }}$ (6)

while the experimental value (which includes anharmonic effects) is 1.396
$\div$ 1.288 (see Table in Ref. [17]). In this Table there are the
experimental and theoretical values of p according to formula (6), as well as
the deviation $\delta$ = $\frac{\text{P}_{Theory}\text{ -
p}_{\exp}}{\text{p}_{theory}}$ of these values from theoretical ones. Using
the least squares method it was found the empirical formula of ln($\delta$%)
$\sim$ f(ln[$\frac{\partial\text{E}}{\partial\text{M}}]$) which is depicted on
Fig.3. As can be seen the indicated dependence has in the first approximation
a linear character:

ln($\delta$%) = -7.5 + 2ln($\frac{\partial\text{E}}{\partial\text{M}}$). (7)

From the results of Fig. 3, it can be concluded that only hydrogen compounds
(and its isotope analog - deuterium) need to take into account the force-
constant changes in isotope effect. It is also seen that for semiconductor
compounds (on Fig. 3 - points, which is below of Ox line) the isotope effect
has only the changes of the isotope mass (details see [3, 7]).

The dependence of the band gap energy on isotopic composition (via mechanism
of electron-phonon interaction) has already been observed for insulators (Fig.
4) and lowest (indirect - direct) gap of different semiconductors ([3] and
references therein). It has been shown to result primarily from the effect of
the average isotopic mass on the electron-phonon interaction, with a smaller
contribution from the change in lattice constant. It was the first paper [19]
where the exciton binding energy EB was found to depend on the isotopic
composition. It was shown further that this change in EB was attributed to the
exciton-phonon interaction (originally with LO phonons) (see, also [3]). At
present time such dependence of EB $\sim$ f(x) (x- isotope concentration) was
found for different bound excitons in semiconductors [20 - 21]. The simplest
approximation, in which crystals of mixed isotopic composition are treated as
crystals of identical atoms having the average isotopic mass is referred to as
virtual crystal approximation (VCA) [15]. Going beyond the VCA, in
isotopically mixed crystals one would also expect local fluctuations in the
band-gap energy from statistical fluctuations in local isotopic composition
within some effective volume, such as that of an exciton (see, e.g. Fig. 2 of
Ref. [18]). Using the least-squares method it was found the empirical
dependence of ln$\left(\frac{\partial\text{E}_{g}}{\partial\text{M}}\right)$
$\sim$ f$\left(\text{lnE}_{g}\right),$ which is presented on Fig. 5. As can be
seen the mentioned dependence has a parabolic character:

ln$\left(\frac{\partial\text{E}_{g}}{\partial\text{M}}\right)$ =
6.105$\left(\text{lnE}_{g}\right)^{2}$ \- 7.870$\left(\text{lnE}_{g}\right)$
\+ 0.565. (8)

From this figure it can be concluded also that the small variation of the
nuclear mass (semiconductors) causes the small changes in Eg also. When the
nuclear mass increases it causes the large changes in Eg (C, LiH, CsH, etc.)
(details, see [18, 3]).

Detail analyze the process of self-diffusion in isotope pure materials and
hetero-structures was done in [5]. Interest in diffusion in solids is as old
as metallurgy or ceramics, but the scientific study of the phenomenon may
probably be dated some sixth-seven decades ago. As is well-known, the measured
diffusion coefficients depends on the chemistry and structure of the sample on
which it is measured. In cited paper [5] it was shown to use the stable
isotopes for the study of diffusion process in different semiconducting
structures (bulk, hetero-structures etc.).

Chapter 6 indicated book [5] describes the new reactor technology - neutron
transmutative doping (NTD). Capture of thermal neutrons by isotope nuclei
followed by nuclear decay produces new elements, resulting in a very number of
possibilities for isotope selective doping of solids. The importance of NTD
technology for studies of the semiconductor doping as well as metal-insulator
transitions and neutral impurity scattering process is underlined. The low-
temperature mobility of free carriers in semiconductors is mainly determined
by ionized- and neutral-impurity scattering. The ionized-impurity scattering
mechanism has been extensively studied (see e.g. [5] and references therein),
and various aspects of this process are now quite well understood. Scattering
by neutral impurities is much less than by ionized centers, i.e., its
contribution is significant only in crystals with low compensation and at very
low temperatures where most of the free carriers are frozen on the impurity
sites. The availability of highly enriched isotopes of Ge which can be
purified to residual dopant levels $<$ 1012 cm-3 has provided the first
opportunity to measure neutral impurity scattering over a wide temperature
range. In paper [22] three Ge isotopes transmute into shallow acceptors (Ga),
shallow donors (As) and double donors (Se) (see also above):

${}_{32}^{70}$Ge + n $\rightarrow$
${}_{32}^{71}$Ge${}_{EC(t_{1/2}=11.2days)}\rightarrow$ ${}_{32}^{71}$Ga +
$\nu_{e},$

${}_{32}^{74}$Ge + n $\rightarrow_{32}^{75}$Ge ${}_{\beta^{-}(t_{1/2}\text{ =
}82.2min)}$ $\rightarrow$ ${}_{32}^{75}$As + $\beta^{-}$ \+ $\bar{\nu}_{e}$,

${}_{32}^{76}$Ge + n $\rightarrow$
${}_{32}^{77}$Ge${}_{\beta^{-}\text{(t}_{1/2}}$ = 11.3 ${}_{h\text{)
}}\rightarrow$ $\beta^{-}$ \+ $\bar{\nu}_{e}$ \+ ${}_{32}^{77}$As
${}_{\beta^{-}\text{(t}_{1/2}}$ = 38.8 ${}_{h\text{)}}$ $\rightarrow$
${}_{32}^{77}$Se + $\beta^{-}$ \+ $\bar{\nu}_{e}$. (9)

The isotopes 72Ge and 73Ge are transmuted into the stable 73Ge and 74Ge
respectively. Controlling the ratio of 70Ge and 74Ge in bulk Ge crystals
allows fine tuning of the majority- as well as the minority carrier
concentration. Currently, this is the best method to vary the free-carrier
concentration independently from compensation ratio. As opposed to other
doping methods, NTD yields a very homogeneous, perfectly random distribution
of the dopants down to the atomic levels [5]. Thus isotopically controlled
crystals offer a unique possibility to study systematically the scattering
mechanism of the charge carriers in semiconductors. Extensive Hall-effect and
resistivity measurements from room temperature down to 4.2K yielded very
accurate free-carrier concentrations and mobilities as a function of
temperature and doping level were done in paper [5]. Itoh et al. [22] have
performed temperature-dependent Hall measurements on four different p-type and
two-different n-type Ge crystals (Fig. 6). Fig. 6 shows the relative strength
of the scattering from the ionized and the neutral impurities. There is only a
relatively small temperature region in which the scattering from the neutral
impurities dominates. This range extends to higher temperatures as the free-
carrier concentration is increased. The calculated ”transition temperatures”
above which the ionized impurities are the main scattering centres compare
very well with experimental results of Itoh et al [22] (see also Fig. 6.31 in
Ref. [5]). In order to demonstrate the importance of the homogeneous dopant
distribution, Itoh et al. have performed the same study on samples cut from Ge
: Ga crystals grown by the conventional Czochralski method, where Ga
impurities were introduced to Ge melt during the crystal growth. These authors
observed deviations of the measured mobility from the theoretical
calculations, which are most likely due to inhomogeneous Ga impurity
distributions in melt-doped Ge. Only the use of NTD semiconductors with
randomly distributed dopants allows for an accurate test of the neutral
impurity-scattering models (details, see [5]).

Another application of isotope pure and isotope mixed crystals that will be
discussed here is related to the possibility of using an isotopically mixed
medium (e.g. LiHxD1-x or 12Cx 13C1-x) as an oscillator of coherent radiation
in the ultraviolet spectral range. To achieve this, the use of indirect
electron transitions involving, say, LO phonons was planned [23]. The
detection of LO phonon replicas of free - exciton luminescence in wide - gap
insulators attracted considerable attention to these crystals (see e.g. [10;
23]). At the same time it is allowed one to pose a question about the
possibility of obtaining stimulated emission in UV (VUV) region (4 - 6 eV) of
the spectrum, where no solid state sources for coherent radiation exist yet.
In the first place this related to the emitters working on the transitions of
the intrinsic electronic excitation (exciton). The last one provides the high
energetical yield of the coherent emission per unit volume of the substance.

In this part we will discuss the investigation results of the influence of the
excitation light density on the resonant secondary emission spectra of the
free - exciton in the wide - gap insulator LiHxD1-x (LiH1-xFx) crystals. The
cubic LiH crystals are typical wide - gap ionic insulator with Eg = 4.992 eV
[10] with relatively weak exciton - phonon interaction however:
EB/$\hbar\omega_{LO}$ = 0.29 where E${}_{B\text{ }}$and $\hbar\omega_{LO}$ are
exciton binding energy and longitudinal optical phonon’s energy ,
respectively. Besides it might be pointed out that the analogous relation for
CdS, diamond and NaI is 0.73; 0.45 and 12.7, respectively . In the insert of
Fig. 7 depicts the luminescence of 1LO and 2LO phonon replicas in LiH
crystals. An increase in the density of the exciting light causes a burst of
the radiation energy in the long-wave wing of the emission of the 1LO and 2LO
repetitions (see Fig. 7) at a rate is higher for the 1LO replica line [23]. A
detailed dependence of the luminescence intensity and the shape of the 2LO
phonon replica line are presented in Fig. 7. The further investigations have
shown [5] that with the increase of the excitation light intensity at the
beginning a certain narrowing can be observed, followed by widening of the
line of 2LO phonon replica with a simultaneous appearance of a
characteristics, probably mode structure (see Fig. 8.11 in Ref. [5]). From
this Fig. it can be seen that the coupling between longwavelength luminescence
intensity and excitation light intensity is not only linear, but, in fact, of
a threshold character as in case of other crystals . A proximity of the
exciton parameters of LiH and CdS (ZnO) crystals allowed to carry out the
interpretation of the density effects in LiH on the analogy with these
semiconducting compounds. Coming from this in the paper [23] it was shown that
for the observed experimental picture on LiH crystals to suppose the exciton-
phonon mechanism of light generation [5] is enough the excitons density about
1015 cm-3. This is reasonable value, if the high quality of the resonator
mirrow - the crystal cleavage ”in situ” and relatively large exciton radius (r
= 40 Å [10] )is taken into account. To this light mechanism generation must be
also promoting a large value of the LO phonon energy
$\left(\hbar\omega_{LO}\text{ = 140 meV }\right).$ Owing to this the radiative
transition is being realized in the spectral region with a small value of the
absorption coefficient, and thus with a small losses in resonator (details see
[5]).

In conclusion of this section we should underlined that if the observable mode
structure is really caused by the laser generation it may be smoothly tuned in
the region of energies 4.5 $\pm$ 5.1 eV owing to smooth transition of the line
emission energy in the LiHxD1-x (LiHxF1-x; LiDxF1-x) mixed crystals as well as
in the range 5.35 - 5.10 eV in 12Cx 13C1-x mixed crystals (see also [10]).

Concluding our report we should be paid your attention to the reports of
Professors Schoven, Weston, Wendt as well as Dr. Chai of our conference which
are devoted in the first step of radioactive isotope applications.

Figure Captions.

Fig. 1. a) First-order Raman spectra of 12C${}_{x}^{13}$C1-x diamonds with
different isotope compositions. The labels A,B, C, D, E and F correspond to x
= 0.989; 0.90; 0.60; 0.50; 0.30 and 0.01 respectively. The intensity is
normalized at each peak (after [8]); b) First-order Raman scattering spectra
in Ge with different isotope contents (after [13]).

Fig. 2. Second-order Raman spectra of LiHxD1-x crystals at room temperature:
(1); (2); (3) and (4) x = 0; 0.42; 0.76 and 1, respectively (after [7]).

Fig. 3. The dependence of ln($\delta$%) $\sim$
f(ln[$\frac{\partial\text{E}}{\partial\text{M}}]$): points are experimental
values and continuous line - calculation on the formula (7) (after [17]).

Fig. 4. Mirror reflection spectra of crystals: LiH, curve 1; LiHxD1-x, curve 2
and LiD, curve 3 at 4.2 K. Light source without crystals, curve 4 (after
[18]).

Fig. 5. The dependence of
ln$\left(\frac{\partial\text{E}_{g}}{\partial\text{M}}\right)$ $\sim$
f$\left(\text{lnE}_{g}\right)$: points are experimental date and continuous
line - calculation on the formula (8) (after [18]).

Fig. 6. Temperature dependence of the carrier mobility of a) p - type and b) n
- type NTD Ge crystals. c) Temperature dependence of relative contributions to
the mobility. Note that the mobility is dominated by neutral impurity
scattering below 20 K (70Ge:Ga $\sharp$ 2 crystal) (after [22]).

Fig. 7. The dependence of the intensity in the maximum (1) and on the long-
wavelength side (2) of 2LO replica emission line of LiH crystals on the
excitation light intensity. In insert: luminescence spectra of free excitons
in LiH crystals in the region of the emission lines of 1LO and 2LO phonon
repetitions at 4.2 K for low (1) and high (2) density of excitations of 4.99
eV photons (after [23]).

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Table. Values of the coefficients dE/dM (mev, cm-1) for the optical phonons
and the experimental and theoretical values of p as well as deviation
$\delta$% of these values from theoretical ones.

Substances | Frequencies | pexp | p${}_{\text{theory}}$ |  $\delta$% = $\frac{\text{p}_{\text{theory}}\text{ - p}_{\text{exp}}}{\text{p}_{\text{theory}}}$
---|---|---|---|---
LiH/LiD | 140(meV)/104(meV)[10,24] | 1.288-1.346 |  $\sqrt{\text{2}}$ = 1.414 | 4.8 - 8.9
SiH4/SiD${}_{\text{4}}$ | 2186.87/1563.3(cm-1)[10] | 1.399 |  $\sqrt{\text{2}}$ = 1.414 | 1.5
12C/13C | 1332,5/1280(cm-1)[10;25] | 1.041 |  $\sqrt{\frac{\text{13}}{\text{12}}}$ = 1.041 | 0.001
70Ge/76Ge | 309.8/297.7(cm-1)[10,26,27] | 1.041 |  $\sqrt{\frac{\text{76}}{\text{70}}}$ = 1.042 | 0.096
28Si/30Si | 524.8/509.8(cm-1)[28] | 1.029 |  $\sqrt{\frac{\text{30}}{\text{28}}}$ = 1.035 | 0.58
64Zn76Se/68Zn80Se | 213.2/207.4(cm-1)[10] | 1.028 |  $\sqrt{\frac{\mu_{1}}{\mu_{2}}}$ = 1.029 | 0.097
$\alpha-^{112}$Sn/$\alpha$\- 124Sn | 206.5/196.8(cm-1)[10,30] | 1.049 |  $\sqrt{\frac{\text{124}}{\text{112}}}$ = 1.052 | 0.30
Ga14N/Ga15N | 535/518(cm-1)[10,31] | 1.033 |  $\sqrt{\frac{\text{15}}{\text{14}}}$ = 1.035 | 0.19
63Cu35Cl/65Cu37Cl | 174.4/171.6(cm-1)[10,32] | 1.016 |  $\sqrt{\frac{\mu_{1}}{\mu_{2}}}$ = 1.022 | 0.59