**2. Using injection spectroscopy for determining parameters of localized states in II-VI compounds**

#### **2.1. Theoretical background of the injection spectroscopy method**

500 Advanced Aspects of Spectroscopy

photoconvertors, etc.

chalcogen [13-14, 17-18].

of injection and optical spectroscopy.

**1.1. Defect classification in layers of II-VI compounds** 

monocrystalline layers or columnar strongly textured polycrystalline layers with low concentration of stacking faults (SF), dislocations, twins with governed ensemble of point defects (PD) [7-8]. However, an enormous number of publications points out the following features of these films: tend to departure of stioichiometric composition, co-existing two polymorph modifications (sphalerite and wurtzite), lamination morphology of crystalline grains (alternation of cubic and hexagonal phases), high concentration of twins and SF, high level of micro- and macrostresses, tend to formation of anomalous axial structures, etc. [2-3, 9]. Presence of different defects which are recombination centers and deep traps does not improve electro-physical and optical characteristics of chalcogenide layers. It restricts the application of the binary films as detector material, basic layers of solar energy

Thus, the problem of manufacturing chalcogenide films with controllable properties for device construction is basically closed to the governing of their defect structure investigated in detail. We will limit our work to the description of results from the examination of parameters of localized states (LS) in polycrystalline films CdTe, ZnS, ZnTe by the methods

Defects' presence (in the most cases the defects of the structure are charged) is an important factor affecting structure-depended properties of II-VI compounds [3, 5, 10]. Defects of the crystalline structure are commonly PD, 1-, 2-, and 3-dimensional ones [11- 12]. Vacancies (VA, VB), interstitial atoms (Ai, Bi), antistructural defects (AB, BA), impurity atoms located in the lattice sites (CA, CB) and in the intersites (Ci) of the lattice are defects of the first type. However, the antistructural defects are not typical for wide gap materials (except CdTe) and they appear mostly after ionizing irradiation [13-14]. The PD in chalcogenides can be one- or two-charged. Each charged native defect forms LS in the gap of the semiconductor, the energy of the LS is *∆Еi* either near the conduction band (the defect is a donor) or near the valence band (then the defect is an acceptor) as well as LS formed in energy depth are appearing as traps for charge carriers or recombination centers [15-16]. Corresponding levels in the gap are called shallow or deep LS. If the extensive defects are minimized the structure-depending properties of chalcogenides are principally defined by their PD. The effect of traps and recombination centers on electrical characteristics of the semiconductor materials is considered in [16]. We have to note that despite a numerous amount of publications about PD in Zn-Cd chalcogenides there is no unified theory concerning the nature of electrically active defects for the range of chalcogenide vapor high pressures as well as for the interval of high vapor pressure of

Screw and edge dislocations are defects of second type they can be localized in the bulk of the crystalites or they form low-angled boundaries of regions of coherent scattering (RCS). Grain boundaries, twins and surfaces of crystals and films are defects of the third type. Pores and precipitates are of the 4th type of defects. All defects listed above are sufficiently The LS in the gap of the semiconductor make important contribution to the function of the device manufactured from the material solar cells, phodetectors, -ray detectors and others), for example, carriers' lifetime, length of the free path, etc., thus making their examination one of them most important problems of the semiconductor material science [3-5, 8, 13, 14, 18].

There are various methods for investigation of the energy position (*Et*), concentration (*Nt*) and the energy distribution of the LS [21-23]. However, their applicability is restricted by the resistance of the semiconductors, and almost all techniques are suitable for low-resistant semiconductors. At the investigation of the wide gap materials II-VI the analysis of currentvoltage characteristics (СVC) at the mode of the space-charge limited current (SCLC) had appeared as a reliable tool [24-25]. The comparison of experimental and theoretical CVCs is carried out for different trap distribution: discrete, uniform, exponential, doubleexponential, Gaussian and others [26-36]. This method is a so-called direct task of the experiment and gives undesirable errors due to in-advance defined type of the LS distribution model used in further working-out of the experimental data. The information obtained is sometimes unreliable and incorrect.

Authors [37-40] have proposed novel method allowing reconstructing the LS energy distribution immediately from the SCLC CVC without the pre-defined model (the reverse task), for example, for organic materials with energetically wide LS distributions [41-42]. However, the expressions presented in [37-40], as shown by our studies [43-45], are not suitable for analysis of experimental data for mono- and polycrystalline samples with energetically narrow trap distributions. So, we use the principle 37-40 and obtain reliable and practically applicable expressions for working-out of the real experiments performed for traditional II-VI compounds.

Solving the Poisson equation and the continuity equation produces SCLC CVC for rectangular semiconductor samples with traps and deposited metallic contacts, where the source contact (cathode) provides charge carriers' injection in the material [24-25]:

$$\mathbf{j} = e\mu E(\mathbf{x})\mathbf{n}\_{\rangle}(\mathbf{x}),\tag{1}$$

$$\frac{dE\left(\mathbf{x}\right)}{d\mathbf{x}} = \frac{e\left[\left(\boldsymbol{n}\_{/}\left(\mathbf{x}\right) - \boldsymbol{n}\_{/o}\right) + \sum\_{j} \left(\boldsymbol{n}\_{i\_{j}}\left(\mathbf{x}\right) - \boldsymbol{n}\_{i\_{j0}}\right)\right]}{e\boldsymbol{\varepsilon}\_{o}}\tag{2}$$

where *j* current density passes through the sample; *е* electron charge; drift carrier mobility; *<sup>0</sup>* dielectric constant;  *permittivity of the material*

*E*(*x*) is an external electric field changing by the depth of the sample; this field injects free carriers from the source contact (cathode) (*x=*0) to the anode collecting the carriers (*x=d*);

*nf(x)* is the free carriers' concentration at the injection;

*nf0* is the equilibrium free carriers concentration;

( ) *<sup>j</sup> <sup>t</sup> n x* is the concentration of carriers confined by the traps of the *j*-group with the energy level *<sup>j</sup> Et* ;

*ntj0* is the equilibrium carriers concentration trapped by the centers of the *j*-group;

*ns(x)* is a total concentration of the injected carriers.

The set of equations (1), (2) is commonly being solved with a boundary condition *E*(0)=0. The set is soluble if the function from *nf* and *nt* is known. We assume that all LS in the material are at thermodynamic equilibrium with corresponding free bands, then their filling-in by the free carriers is defined by the position of the Fermi quasi-level *EF*. Using the Boltzmann statistics for free carriers and the Fermi – Dirac statistics for the localized carriers we can write [39-40]:

$$m\_f(\mathbf{x}) = N\_{c(V)} \exp\left(\frac{E\_{c(V)}(\mathbf{x}) - E\_r\left(\mathbf{x}\right)}{kT}\right) \tag{3}$$

$$m\_i(E, \mathbf{x}) = \frac{h\left(E, \mathbf{x}\right)}{1 + g \exp\frac{E\_i(\mathbf{x}) - E\_r(\mathbf{x})}{kT}},\tag{4}$$

where *Nc(v)* are states density in conduction band (valence band);

*Еc(v)* is energy of conduction band bottom (valence band top);

*k* is Boltzmann constant;

*T* is the temperature of measurements;

*EF*(*x*) is the Fermi quasi-level at injection;

*g* is a factor of the spin degeneration of the LS which depends on its charge state having the following values: –1/2, 1 or 2 (typically *g* = 1) [15, 39-40].

The zero reference of the trap energy level in the gap of the material will be defined relatively to the conduction band or valence band depending on the type (*n* or *p*) of the examined material: *EC(V)=*0.

The set of equations (1)–(2) can also be reduced to integral relations. Detailed determination of these ratios presented in [37].

$$\frac{1}{j} = \frac{1}{e\mu d} \frac{\varepsilon \varepsilon\_{\boldsymbol{\alpha}}^{\*}}{\boldsymbol{e}} \Big|\_{\boldsymbol{\alpha}}^{\*} \frac{d\boldsymbol{n}\_{\boldsymbol{\prime}}}{n\_{\boldsymbol{\beta}}^{2} [(\boldsymbol{n}\_{\boldsymbol{\beta}} - \boldsymbol{n}\_{\boldsymbol{\beta}}) + \sum\_{j} (\boldsymbol{n}\_{\boldsymbol{\varepsilon}\_{j}} - \boldsymbol{n}\_{\boldsymbol{\varepsilon}\_{j}})]} \equiv \boldsymbol{y}\_{\boldsymbol{\prime}} \tag{5}$$

$$\frac{\text{LI}}{\text{m}^{\circ}} = \frac{\varepsilon \varepsilon\_{o}}{c(e\mu)^{2}} \int\_{n\_{\text{f}}}^{n\_{\text{f}}} \frac{dn\_{\text{f}}}{n\_{\text{f}}^{3}[(n\_{\text{f}} - n\_{\text{f}\_{\text{0}}}) + \sum\_{j} (n\_{\text{i}\_{\text{j}}} - n\_{\text{i}\_{\text{i}\_{\text{j}}}})]} \equiv z\_{\text{\textdegree}} \tag{6}$$

where *j, U* are current density and voltage applied to the sample; d is the sample thickness;

502 Advanced Aspects of Spectroscopy

drift carrier mobility;

 *permittivity of the material*

*<sup>0</sup>* dielectric constant;

we can write [39-40]:

*k* is Boltzmann constant;

examined material: *EC(V)=*0.

*T* is the temperature of measurements; *EF*(*x*) is the Fermi quasi-level at injection;

*е* electron charge;

( )

level *<sup>j</sup> Et* ;

where *j* current density passes through the sample;

*nf(x)* is the free carriers' concentration at the injection;

*nf0* is the equilibrium free carriers concentration;

*ns(x)* is a total concentration of the injected carriers.

*E*(*x*) is an external electric field changing by the depth of the sample; this field injects free carriers from the source contact (cathode) (*x=*0) to the anode collecting the carriers (*x=d*);

*<sup>j</sup> <sup>t</sup> n x* is the concentration of carriers confined by the traps of the *j*-group with the energy

The set of equations (1), (2) is commonly being solved with a boundary condition *E*(0)=0. The set is soluble if the function from *nf* and *nt* is known. We assume that all LS in the material are at thermodynamic equilibrium with corresponding free bands, then their filling-in by the free carriers is defined by the position of the Fermi quasi-level *EF*. Using the Boltzmann statistics for free carriers and the Fermi – Dirac statistics for the localized carriers

( ) ( ) exp , *C V <sup>F</sup>*

, , , () () 1 exp

*g* is a factor of the spin degeneration of the LS which depends on its charge state having the

The zero reference of the trap energy level in the gap of the material will be defined relatively to the conduction band or valence band depending on the type (*n* or *p*) of the

*n Ex Ex E x*

*hEx*

*<sup>g</sup> kT* 

( )

(3)

(4)

*E x Ex*

*kT*

*t F*

*ntj0* is the equilibrium carriers concentration trapped by the centers of the *j*-group;

( )

*f C V*

*nx N*

*t*

where *Nc(v)* are states density in conduction band (valence band); *Еc(v)* is energy of conduction band bottom (valence band top);

following values: –1/2, 1 or 2 (typically *g* = 1) [15, 39-40].

*fc n* , *fa n* are free carriers' concentration in cathode and anode, respectively.

Equations (5) and (6) determine SCLC CVCs in parametric form for an arbitrary distribution of LS in the gap of the material.

At thermodynamic equilibrium the total concentration ( <sup>0</sup>*<sup>s</sup> n* ), the carriers concentration for those localized on the traps ( <sup>0</sup>*<sup>t</sup> n* ), and the free carriers' concentration in the semiconductor ( <sup>0</sup> *<sup>f</sup> n* ) are in the function written as follows: 000 *stf nnn* , де <sup>0</sup> 0 ( ) ( ) exp *CV F f CV E E n N kT* in case when *Eс – EF0* 3*kТ* (3*kТ*= 0,078 eV at the room

temperature; *EF0* is the equilibrium Fermi level. It must be emphasized that this charge limits the current flow through the sample and determines the form of the SCLC CVC.

The carriers' injection from the source contact leads to appearance of the space charge in the sample, formed by the free carriers and charge carriers localized in the traps, 0 0 0 ( ) [( ) ( )] *i ss t f t f en e n n e n n n n* , where *ni* is the concentration of injected carriers.

Under SCLC mode the concentration of injected carriers is considerably larger than their equilibrium concentration in the material and, at the same time, it is sufficiently lower than the total concentration of the trap centers ( *<sup>f</sup>* <sup>0</sup> *i t n nN* ) [24-25]. Thus, in further description we will neglect the second term in the expression written above (except some special cases). Then we have ~ *<sup>j</sup> s t j en x e n x* .

Using (5) and (6) we find the first and second derivatives of *z* from *y*:

$$z' = \frac{dz}{dy} = \frac{d\left(\mathbb{L}\mathbb{L}/j^2\right)}{d\left(\mathbb{1}/j\right)} = \frac{d}{e\,\mu m\_{\mu}},\tag{7}$$

$$z'' = \frac{d^2 z}{dy^2} = \frac{d}{d\left(1/j\right)} \frac{d\left(\mathbf{U}/j^2\right)}{d\left(1/j\right)} = \left|\frac{\rho\_s d^2}{\varepsilon x\_\circ}\right|. \tag{8}$$

As the SCLC CVC are commonly represented in double-log scale [24-25], equations (7), (8) are rewritten with using derivatives: 2 3 2 3 (ln ) (ln ) (ln ) , , . (ln ) (ln ) (ln ) *dj d j d j dU dU dU* 

Then we have

$$m\_{\mu} = \frac{\eta}{2\eta - 1} \frac{jd}{e\mu \mathcal{U}} = \frac{1}{\alpha} \frac{jd}{e\mu \mathcal{U}},\tag{9}$$

$$\frac{\rho\_e}{e} = \frac{2\eta - 1}{\eta} \frac{\eta - 1}{\eta} \left[ 1 - \frac{\eta'}{\eta \left( 2\eta - 1 \right) \left( \eta - 1 \right)} \right] \frac{ez\_v l I}{e d^2} = \alpha \beta \frac{ez\_v l I}{e d^2} \tag{10}$$

where 2 1 , 1 1 <sup>1</sup> 1 . 21 1 *B* 

Further we will neglect the index *а*.

As a result, the Poisson equation and the continuity equation give fundamental expressions for a dependence of the free carrier concentration in the sample *nf* (the Fermi quasi-level energy) and space charge density at the anode on the voltage *U* and the density of the current *j* flowing through the structure metal-semiconductor-metal (MSM).

Now let us consider the practical application of expressions (7) and (8) or (9) and (10) for reconstructing the trap distribution in the gap of the investigated material. We would restrict with the electron injection into n-semiconductor.

If the external voltage changes the carries are injected from the contact into semiconductor; at the same time, the Fermi quasi-level begins to move between the LS distributed in the gap from the start energy *EF0* up to conduction band. This displacement *EF* leads to filling-in of the traps with the charge carriers and, consequently, to the change of the conductivity of the structure. Correspondingly, under intercepting the Fermi quasi-level and the monoenergetical LS the CVC demonstrates a peculiarity of the current [24-25]. As the voltage and current density are in the function of the LS concentration with in-advanced energy position and the Fermi quasi-level value we obtain a possibility to scan the energy distributions. This relationship is a physical base of the injection spectroscopy method (ІS).

Increase of the charge carriers *dns* in the material at a low change of the Fermi level position is to be found from the expression:

$$\frac{1}{e}\frac{d\rho}{dE\_{\mathbb{F}}} = \frac{dn\_i}{dE\_{\mathbb{F}}} \approx \frac{dn\_s}{dE\_{\mathbb{F}}} \approx \frac{dn\_i}{dE\_{\mathbb{F}}}.\tag{11}$$

The carrier concentration on deep states can be found from the Fermi-Dirac statistics

$$\mathfrak{m}\_\* = \bigwedge\_{E\_1}^{E\_2} \mathfrak{m}\_\*(E) dE = \bigwedge\_{E\_1}^{E\_2} \mathfrak{h}(E) f(E - E\_{r'}) dE + \mathfrak{m}\_{f'} \tag{12}$$

where d*ns*(*E*)/d*E* is a function describing the energy distribution of trapped carriers; *h*(*E*)=d*N*t/d*E* is a function standing for the energy trap distribution; *E1, E2* are energies of start and end points for the LS distribution in the gap of the material.

It is assumed that the space trap distribution in the semiconductor is homogeneous by the sample thickness then *hEx hE* (,) () .

After substitution of (12) in (11) we obtain a working expression for the functions *d/dEF* and *h*(*E*)

$$\frac{1}{e}\frac{d\rho}{dE\_r} \approx \frac{dn\_\circ}{dE\_r} = \frac{d}{dE\_\varepsilon}\Big[n\_\circ(E)dE = \int\_\varepsilon h(E)\frac{d(f(E-E\_r))}{d(E-E\_r)} + \frac{n\_f}{kT}\tag{13}$$

Thus, at arbitrary temperatures of the experiment the task of reconstructing LS distributions reduces to finding function *h(E)* from the convolution (12) or (13) using known functions *ns*(*EF*) or *d/dEF*. The expression (12) is the most preferable [39-40]. In general case the solution is complex and it means determining the function *h*(*E*) from the convolution (12) or (13) if one of the functions *ns* or *dns/dEF* is known [43-45]. We have solved this task according the Tikhonov regularization method [46]. If the experiment is carried at low temperatures (liquid nitrogen) the problem is simplified while the Fermi-Dirac function in (13) may be replaced with the Heavyside function and, neglecting *nf* , we obtain

$$\frac{1}{e}\frac{d\rho}{dE\_{\mathbb{F}}} \approx \frac{dN\_{\mathbb{A}}}{dE\_{\mathbb{F}}} \approx h(E). \tag{14}$$

This equation shows that the function 1/*e d/dEF* - *EF* at low-temperature approximation immediately produces the trap distribution in the gap of the semiconductor. Using (7) and (8), we transform the expression (14) for practical working-out of the experimental SCLC CVC. As the free carrier concentration and the space charge density are to be written as follows:

$$m\_{\circ} = \frac{d}{e\mu} \frac{j}{2\text{UI} - \text{UI}'j},\tag{15}$$

$$\frac{\mathcal{P}}{e} = (\mathcal{U}'\mathbf{j}^2 - 2\mathcal{U}'\mathbf{j} + 2\mathcal{U}I)\frac{e\mathcal{E}\_0}{e d^2},\tag{16}$$

the expression (14) will be

504 Advanced Aspects of Spectroscopy

Then we have

where 2 1 

are rewritten with using derivatives:

the injection spectroscopy method (ІS).

is to be found from the expression:

Further we will neglect the index *а*.

energy) and space charge density at the anode

restrict with the electron injection into n-semiconductor.

21 1

 

current *j* flowing through the structure metal-semiconductor-metal (MSM).

0 0

, 1 1 <sup>1</sup> 1 .

 

*dy d jd j*

2

*z*

 

. 1 1

0

2 3

 

2 3

(9)

2 2

(11)

 

on the voltage *U* and the density of the

(10)

(8)

> (ln ) (ln ) (ln ) , , . (ln ) (ln ) (ln ) *dj d j d j dU dU dU*

> >

*B*

2 2 2

*<sup>a</sup> dz d dUj d*

As the SCLC CVC are commonly represented in double-log scale [24-25], equations (7), (8)

<sup>1</sup> , 2 1 *fa jd jd <sup>n</sup>*

> 

As a result, the Poisson equation and the continuity equation give fundamental expressions for a dependence of the free carrier concentration in the sample *nf* (the Fermi quasi-level

Now let us consider the practical application of expressions (7) and (8) or (9) and (10) for reconstructing the trap distribution in the gap of the investigated material. We would

If the external voltage changes the carries are injected from the contact into semiconductor; at the same time, the Fermi quasi-level begins to move between the LS distributed in the gap from the start energy *EF0* up to conduction band. This displacement

*EF* leads to filling-in of the traps with the charge carriers and, consequently, to the change of the conductivity of the structure. Correspondingly, under intercepting the Fermi quasi-level and the monoenergetical LS the CVC demonstrates a peculiarity of the current [24-25]. As the voltage and current density are in the function of the LS concentration with in-advanced energy position and the Fermi quasi-level value we obtain a possibility to scan the energy distributions. This relationship is a physical base of

Increase of the charge carriers *dns* in the material at a low change of the Fermi level position

<sup>1</sup> . *ist FF F F d dn dn dn e dE dE dE dE*

 

*eU eU*

21 1 1 , 21 1 *<sup>a</sup> U U e ed ed*

 

 

$$h(E) \approx \frac{1}{e} \frac{d\rho}{dE\_r} = \frac{1}{kT} \frac{e\varepsilon\_o}{ed^2} \frac{\mathcal{U}"\hat{j}^3(2\mathcal{U} - \mathcal{U}' \hat{j})}{(\mathcal{U}"\hat{j}^2 - 2\mathcal{U}' \hat{j} + 2\mathcal{U})}.\tag{17}$$

Using derivatives*, ', "* this expression is easily rewritten:

$$h(E) \approx \frac{1}{e} \frac{d\rho}{dE\_r} = \frac{1}{kT} \frac{\varepsilon \varepsilon\_0 \mathcal{U}}{e d^2} \frac{2\eta - 1}{\eta^2} \left[ 1 + \frac{(3\eta - 3)\eta\eta' - \eta\eta'' + 3\eta'^2}{\eta^2 \left( (2\eta - 1)(\eta - 1) - \eta'/2 \right)} \right].\tag{18}$$

The expression (18) is also can be written with the first derivative () only. Denote

$$\mathbf{C} = \frac{(3\eta - 3)\eta\eta' - \eta\eta'' + 3\eta'^2}{\eta^2 \left[ (2\eta - 1)(\eta - 1) - \eta' / 2 \right]} = \left( 2 - 3\eta \right) \mathbf{B} + \frac{d\ln\left( 1 + B \right)}{d\ln I} = \frac{\left( 2\eta - 1 \right) \mathbf{B} + \left( 3\eta - 2 \right) \mathbf{B}^2 + \frac{d\left[ \ln\left( 1 + B \right) \right]}{d\ln I}}{1 + \left( \eta - 1 \right)B},$$
 
$$\text{where } B = -\frac{1}{\eta'\eta - 1/\eta - 1} \frac{d\eta}{d\ln I}.$$

where <sup>1</sup> . (2 1)( 1) ln *d U* 

We obtained an expression used by authors [39-40] for analysis of energetically wide LS distributions in organic semiconductors.

$$h(\mathbf{E}) \approx \frac{1}{e} \frac{d\rho}{dE\_{\parallel}} = \frac{1}{kT} \frac{\varepsilon \varepsilon\_{\rm 0} lI}{ed^2} \frac{2\eta - 1}{\eta^2} (1 + \mathbf{C}) = \frac{a\beta}{kT} \frac{\varepsilon \varepsilon\_{\rm 0} lI}{ed^2} \frac{1}{\eta - 1} \left(\frac{1 + \mathbf{C}}{1 + B}\right). \tag{19}$$

To make these expressions suitable for the working-out of SCLC CVC for the semiconductors with energetically narrow trap distributions we write them with reverse derivatives 2 3 2 3 (ln ) (ln ) (ln ) , , . (ln ) (ln ) (ln ) *dU d U d U dj dj dj* 

As a result:

$$h(E) \approx \frac{1}{e} \frac{d\rho}{dE\_{\parallel}} = \frac{1}{kT} \frac{e\varepsilon\_{\flat}\mathcal{U}}{ed^{2}} \left[ \frac{(2\gamma - 3)\gamma' + \gamma''}{(2 - \gamma)(1 - \gamma) + \gamma'} + \gamma \right] (2 - \gamma). \tag{20}$$

Solving the set of equations (3) and (7) gives energetical scale under re-building deep trap distributions. Using various derivatives we obtain

$$E\_r = kT \ln \frac{e \, \mu \mathcal{N}\_{c(V)}}{d} + kT \ln \frac{2 \mathcal{U} - \mathcal{U} \dot{\mathcal{Y}}}{\dot{\mathcal{Y}}} = kT \ln \frac{e \, \mu \mathcal{N}\_{c(V)}}{d} + kT \ln \frac{\dot{\mathcal{Y}}}{\mathcal{U}} + kT \ln \frac{\eta}{2 \eta - 1} = 0$$

$$= kT \ln \frac{e \, \mu \mathcal{N}\_{c(V)}}{d} + kT \ln \frac{\dot{\mathcal{Y}}}{\mathcal{U}} + kT \ln \frac{1}{2 - \gamma}. \tag{21}$$

Using sets of equations (17) - (18) or (20) - (21) allows to find a function describing the LS distribution in the gap immediately from the SCLC CVC. To re-build the narrow or monoenergetical trap distributions (typical for common semiconductors) the most suitable expressions are written with derivatives. The first derivative defines the slope of the CVC section in double-log scale relative to the current axis, the defines the slope of the CVC section in double-log scale relative to the voltage axis. For narrow energy distributions this angle is too large, and under complete filling-in of the traps it closes to [24-25]. However, it means the slope to the current axis is very small allowing finding the first and higher order derivatives with proper accuracy [44, 45, 48]. It is important that the narrowest trap distributions give the higher accuracy under determination of the derivatives , , !

506 Advanced Aspects of Spectroscopy

*, ',* 

*"* this expression is easily rewritten:

 

2 2 2 1 1 21 33 3 ( ) 1 . *<sup>F</sup>* 21 1 2

*B B d B d U C B*

21 32 (3 3) 3 ln 1 ln 2 3 , (2 1)( 1) / 2 ln 1 1

We obtained an expression used by authors [39-40] for analysis of energetically wide LS

2 3

0 2

 

(ln ) (ln ) (ln ) , , . (ln ) (ln ) (ln ) *dU d U d U dj dj dj*

> *<sup>d</sup> <sup>U</sup> h E e dE kT ed*

> > *d j*

expressions are written with derivatives. The first derivative

section in double-log scale relative to the current axis, the

0 0 2 2 2 1 1 21 1 1 ( ) (1 ) . 1 1 *<sup>F</sup> d C U U h E <sup>C</sup> e dE kT ed kT ed B*

 

(19)

To make these expressions suitable for the working-out of SCLC CVC for the semiconductors with energetically narrow trap distributions we write them with reverse

1 1 (2 3) ( ) ] (2 ). (2 )(1 ) *<sup>F</sup>*

Solving the set of equations (3) and (7) gives energetical scale under re-building deep trap

( ) ln ln ln *C V e N <sup>j</sup> kT kT kT*

*C V e N <sup>j</sup> kT kT kT d U*

( ) <sup>1</sup> ln ln ln .

Using sets of equations (17) - (18) or (20) - (21) allows to find a function describing the LS distribution in the gap immediately from the SCLC CVC. To re-build the narrow or monoenergetical trap distributions (typical for common semiconductors) the most suitable

section in double-log scale relative to the voltage axis. For narrow energy distributions this

 

  

0

The expression (18) is also can be written with the first derivative (

*<sup>d</sup> <sup>U</sup> h E e dE kT ed* 

2

*d U* 

distributions. Using various derivatives we obtain

( ) <sup>2</sup> ln ln *C V*

*e N U Uj E kT kT*

 

 

 

distributions in organic semiconductors.

derivatives 2 3

 

 

*d U B*

 

*d U*

2

(21)

2

) only. Denote

2

ln 1

(20)

defines the slope of the CVC

defines the slope of the CVC

*d B*

(18)

 

 

 

Using derivatives

2

As a result:

*F*

 

 

where <sup>1</sup> . (2 1)( 1) ln *<sup>d</sup> <sup>B</sup>*

If the distributions in the semiconductor are energetically broadened all expressions (17), (18), and (20) can be used as analytically identical formulas.

As is seen from the expressions written above, in order to receive information about LS distribution three derivatives are to be found at each point of the current-voltage function in various coordinates. Due to experimental peculiarities we had to build the optimization curve as an approximation of the experimental data with it's further differentiation at the sites. The task was solved by constructing smoothing cubic spline 47. However, the numerical differentiation has low mathematical validity (the error increases under calculation of higher order derivatives). To achieve maximum accuracy we have used the numerical modeling with solving of direct and reverse tasks.

Under solving the direct task we have calculated the functions  *- EF* and 1/*e d/dEF* - *EF* on base of known trap distribution in the gap of the material (the input distribution) using the expressions (12) and (13). Then we have built the theoretical SCLS CVCs ((5), (6)). The mathematical operations are mathematically valid. To solve the reverse problem of the experiment CVCs were worked out using the differential technique based on expressions (17), (21), (18), (20). As a result we have again obtained the deep centers' distribution in the gap of the material (output distribution). Coincidence of the input and output trap distributions was a criterion of the solution validity under solving the reverse task. Further the program set was used for numerical working-out of the experimental CVCs [43-45, 48].

#### **2.2. Determination of deep trap parameters from the functions 1/***ed/dEF* **-** *EF* **under various energy distributions**

Now we determine how the energy position and the trap concentration under presence of the LS in the gap may be found for limit cases by the known dependence 1/*ed/dEF* - *EF*. In the case of mono-level the LS distribution can be written as ( ) ( ) *t F hE N E E* , where is a delta-function.

After substituting this relationship in (12), (13) we obtain

$$m\_\* \approx m\_\* = \frac{N\_\*}{1 + g \exp\left(\frac{E\_r - E\_t}{kT}\right)} \,\tag{22}$$

$$\frac{1}{e}\frac{d\rho}{dE\_{\mathbb{F}}} \approx \frac{dn\_{\circ}}{dE\_{\mathbb{F}}} = \frac{gN\_{\circ}\exp(\frac{E\_{\mathbb{F}}-E\_{\circ}}{kT})}{kT(1+g\exp(\frac{E\_{\mathbb{F}}-E\_{\circ}}{kT}))^{2}}\tag{23}$$

The value of the last function at the maximum (*EF=Et*) is <sup>2</sup> <sup>1</sup> , 1 *Fm t F E d gN e dE kT g* 

.

$$\text{for at } g = 1 - \left(\frac{1}{e}\frac{d\rho}{dE\_{\mathbb{F}}}\right)\_{E\_{\mathbb{F}\_0}} = \frac{N\_\iota}{4kT}.$$

Than

<sup>1</sup> 4 . *Fm t F E <sup>d</sup> N kT e dE* (24)

Thus, building the function 1/*ed/dEF* - *EF* and finding the maximums by using (24) gives the concentration of discrete monoenergetical levels. The energy position of the maximum immediately produces energy positions of these levels.

If the LS monotonically distributed by energy *h*(*E*)*=ANt=* const are in the gap of the material it is easy to obtain

$$N\_{\iota} = h(E) = \frac{1}{e} \frac{d\rho}{dE\_{\iota}}.\tag{25}$$

In other words the trap concentration in the sample under such distributions is immediately found from the function 1/*ed/dEF* - *EF*.

In general case when LS distribution in the gap of the material is described by the arbitrary function their concentration is defined by the area under the curve 1/*ed/dEF* - *EF* and at low temperatures can be found from the relationship 2 1 ( ) *E t E N h E dE* . Under reconstruction of such distributions from the SCLC CVC these distributions are energetically broadening depending on the temperature of experiment [43-45]. LS energy positions are again determined by the maximums of the curve.

The correct determination of the trap concentration from the dependence 1/*ed/dEF* - *EF* may be checked out by using the function  *- EF*. In case of the mono-level where the Fermi quasi-

level coincides with the LS energy position, it is easy to obtain from (22) - [ ] <sup>1</sup> *Fm t t E N n <sup>g</sup>* ,

$$\text{then } \operatorname{N}\_{\iota} = (1 + \operatorname{g}) [\iota\_{\iota}]\_{\mathbb{E}\_{\mathbb{E}u}}. \text{ If } \operatorname{g} = 1 \text{ then } \left[ \iota\_{\iota} \right]\_{\mathbb{E}\_{\mathbb{E}u}} = \operatorname{N}\_{\iota} \Big/ \operatorname{\mathcal{D}}\_{\iota}. \text{ } \operatorname{N}\_{\iota} = \left[ \mathcal{D} \iota\_{\iota} \right]\_{\mathbb{E}\_{\mathbb{E}u}}.$$

If the LS distribution is a Gaussian function ( 2 1 2 ( ) exp 2 2 *t t t t N EE h E* ) the relationship

for determination of *Nt* is analogous to that described above.

Earlier [43-45] we have described the effect of experimental factors on accuracy of determining parameters of the deep centers by IC method. In Ref. [44, 45, 48] it was shown that the neglecting third order derivative or even the second order derivative does not lead to considerable decrease of the accuracy in determination of the LS parameters. It was demonstrated that under neglecting the 3rd order derivative *''* in (20) the error in definition of the function *h*(*E*) at the point *EF*=*Et* is no more than 0.4%. At the same time this error is somewhat larger in the interval *EF-Et~kT* but is not larger than (4–7)%. Such a low error of the calculation of the LS parameters is caused by the interception of zero point and the derivative *''* near the point *EF=Et* (commonly in the range of 0,2 *E E kT F t* ). As a result (regarding the absence of accurate experimental measurement of the 3rd derivative) it does not affect the differential working-out of CVCs in the most important section where the Fermi quasi-level coincides with the LS energy position.

If the second order derivative in the working expressions is neglected the error of the defining the function *h*(*E*) in the most principal (*EFEt*) is about (30-40)%. In both cases the simplification of the expression (21) does not contribute errors to the definition of energy position of the traps' level. Remember that the traditional method of SCLC CVC gives 60- 100% error of the traps' concentration [24-25].
