The transient exclusion effect in intrinsic semiconductors

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Sep 2, 2002 - In the latter case Lex was defined as the distance between the p+–p contact and the base coordinate where n(x) = 0.5n0. 3. Experimental ...
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SEMICONDUCTOR SCIENCE AND TECHNOLOGY

Semicond. Sci. Technol. 17 (2002) 1058–1063

PII: S0268-1242(02)36261-8

The transient exclusion effect in intrinsic semiconductors V K Malyutenko1, V V Vainberg1, G I Teslenko1, O Yu Malyutenko1 and J Pultorak2 1 2

Institute of Semiconductor Physics, 03028 Kiev, Ukraine Institute of Electron Technology, 02-668 Warsaw, Poland

E-mail: [email protected]

Received 23 April 2002, in final form 19 July 2002 Published 2 September 2002 Online at stacks.iop.org/SST/17/1058 Abstract The process of formation and relaxation of charge-carrier exclusion in a structure with an h–l-junction has been studied both theoretically and experimentally. It is shown, for the first time, that temporal dependencies of the most important process characteristics, such as the exclusion length, the extent of integral charge-carrier depletion in the base, and the current establishment, may be expressed by simple analytical formulae. The modelling experiment has been carried out using Ge crystals with intrinsic conductivity (T  300 K). In order to ‘visualize’ the spatial distribution of the charge carriers, the transmission beyond the edge of fundamental absorption and non-equilibrium thermal emission of the structure base in the spectral range 8–12 µm have been investigated. It is shown that thermal generation of charge carriers by the ohmic contact and base surface plays a key role in the process of exclusion relaxation. The difference between process duration in the cases of establishment of a steady-state current and spatial-charge-carrier distribution is explained by the formation of a high electric field domain.

1. Introduction The exclusion effect in structures with an h–l junction (a junction between a heavily (h) and lightly (l) doped regions of semiconductor) [1] is interesting for the development of optoelectronic devices as a possible alternative to the injection effect in structures with a p–n-junction [2]. This effect enables us, in particular, to make use of the negative luminescence, radiative cooling and thermal emission modulation [2, 3], to enhance photodetector sensitivity [4] and characteristics of ionizing radiation detectors [5]. The steady-state non-equilibrium decrease of electron and hole concentrations in the structure base has been studied in considerable detail [2, 6, 7]. However, the effect is manifested, as a rule, at large densities of electric current, J. Therefore, it is observed under the pulse regime (to avoid heating the structure by the electric current). At the same time the transient characteristics of the exclusion effect were not studied very well. So, the establishment of electric current was studied in [8] and the relaxation of photocurrent under optical excitation was explored in [9]. Some results of the numerical simulation 0268-1242/02/101058+06$30.00 © 2002 IOP Publishing Ltd

of the transient spatial distribution of charge carriers in the symmetric structure p+–p–p+ were presented in [10]. These results cannot characterize to a sufficient extent the transient spatial distribution of charge carriers and, hence, they cannot be used in the evaluation of the transient-process parameters in the structure base. The temporal variation of transient profiles of the carrier concentration under the conditions of establishment (J > 0) and relaxation (J = 0) of the exclusion are still to be studied. This paper is aimed at an experimental and theoretical investigation of this variation in samples with a rectangular shape. The quantitative analysis is based on the solution of the one-dimensional (1D) transient continuity equation by numerical methods. For the first time analytical approximate expressions are derived for temporal dependencies of important parameters such as the exclusion length, the electric current in a sample, the deviation of the total number of charge carriers from equilibrium. In experimental studies along with the customary method of infrared (IR) probing we have used the thermal vision technique to ‘visualize’ the spatial distribution of charge carriers. The h–l structures made from Ge with a different

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The transient exclusion effect

1,0 7' 0,8 6'

0,6

5' 0,4 4' 0,2

3'

0,0 0,0

0,5

1,0

Figure 1. The distribution of reduced-electron concentration over the base (L = 10 mm, calculations) at different time moments after switching the voltage on (curves 1–7) and off (curves 3 –7 ), U = 40 V. t/τ : 1—0.01, 2—0.05, 3, 3 —0.2, 4, 4 —0.4, 5, 5 —0.6, 6, 6 —1.0, 7, 7 —3.0. AA , the replacement of electron distribution by a step function. Inset: the configuration of the sample under study and a general scheme of IR probing: 1, sample; 2, ohmic R-contact; 3, p+–p contact; 4, slit image; 5, IR probe; 6, modulated IR probe; 7, photodetector.

doping level of the base and completed with the second ohmic contact have been used as the model material. The ohmic contact enables us to avoid the accumulation of carriers in the base, and implement the case of total non-equilibrium depletion of the base that is of large importance in applications.

2. Theoretical model The structure under study is shown in the inset of figure 1. It is a rectangular slab of a p-type semiconductor with length L (0 < x < L) and conductivity close to intrinsic (n0 ∼ = p0 ∼ = ni ). The transverse dimensions of the sample are much larger than the diffusion length of the charge carriers, LD. Therefore the task is analysed as one-dimensional. It is assumed that the condition of quasi-neutrality is fulfilled in the sample bulk3 , p = n + Na

(1)

where Na is the concentration of acceptors, p and n are the concentrations of holes and electrons, respectively. One of the front edges of the base (x = L) is provided with a p+–p junction, the opposite one (x = 0) is provided with the ohmic contact. By definition, the p+–p junction is assumed to be non-penetrable for electrons. The ohmic contact is defined as the contact always keeping the concentration of electrons constant and equal to the equilibrium value n0 . Such a contact is referred to as the recombination type (R) contact. It is characterized by an infinite velocity of charge-carrier recombination (SR = ∞). The polarity of the applied voltage causes the drift of electrons (and the same number of holes) toward the ohmic contact where they recombine. Thus, unlike the symmetric structure p+–p–p+ with exclusion at one contact and accumulation at the opposite one, in our case the total depletion of the base may be implemented. 3

The validity of this condition for the case of exclusion in Ge was quantitatively proved in [11].

The temporal variation of the spatial distribution n(x, t) of non-equilibrium carriers in the structure base after switching on voltage (t > 0) is determined by the continuity equation, which is obtained under the following assumptions. Deviations of the electron and hole concentrations from the equilibrium are equal, n = p, which agrees with the quasineutrality assumption. This also implies equality of the electron and hole lifetimes, τ n = τ p = τ . Here for simplicity it is assumed that τ = constant. The continuity equation may be written then as ∂n n − n0 ∂ 2n ∂n = D 2 + µE Ex − , (2) ∂t ∂x ∂x τ where µn µp (p − n) Dn Dp (n + p) ; µE = ; D= nDn + pDp nµn + pµp (3) j − (Dn − Dp ) ∂n ∂x ; Ex = e nµn + pµp Dn, Dp, µn, µp are the diffusion coefficients and the mobilities of electrons and holes, respectively, n0 is the equilibrium concentration of electrons. The boundary conditions at the p+–p and R contacts are written, respectively, as dn j nDn ; =− jn (x = L) = 0 ⇒ D (4) dx e nDn + pDp n(x = 0) = n0 where j, jn are the densities of the total current (j = jn + jp ) and its electron component. If the transient process runs under a regime of constant voltage then the system of equations must be completed by the integral equation for the voltage drop across the base.   L j L dx U = const = Ex dx = e pµ p + nµn 0 0  L (Dn − Dp )∂n/∂x − dx. (5) pµp + nµn 0 The system of equations (2)–(5) has been used for computing transient characteristics by numerical methods with the set of known parameters for Ge listed, for example, in [12]. The spatial distribution of charge carriers along the base is shown in figure 1 as a family of electron concentration profiles n(x) at different moments after switching the voltage on and off. The curves 1–7 depict formation of the region Lex depleted with electrons (the high-field domain) and adjacent to the p+–p contact. They also show the extent of this depletion and temporal expansion of Lex (the exclusion length) deep into the base. The curves 3 –7 illustrate the process of disappearance of the exclusion region after switching off the voltage. This process is caused by the thermal generation of carriers. Figure 2 illustrates the relaxation of the electric current to its steady-state value and the temporal expansion of the exclusion region. It should be noted that the establishment of a steadystate current and the formation of an exclusion region are characterized by significantly different time constants. Now using a simplified model of exclusion we consider how to obtain analytical expressions that describe the process parameters. For this purpose equation (2) may be written in the following equivalent form   ∂n j Dn Dp Na ∂n n − n0 ∂n ∂ D + = − . (2 ) ∂t ∂x ∂x e (nDn + pDp )2 ∂x τ 1059

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The transient current–voltage characteristic, I –V , may, consequently, be written as √ B U j (t) =  ,  1 − exp − 2tτ  (µn + µp )µp Na n0 where B = e . (9) µn τ In the case where the equilibrium recovered after switching off the voltage applied across a sample, the current j becomes equal to 0. Hence from (6) we obtain  ℵ(t) =

d

[n(x, t) − n0 ] dx = ℵ0 exp(−t/τ );

0

Figure 2. The dynamics of the exclusion region expansion (curves 1, 1 ) and the current relaxation (curve 2) to the steady-state value in the base at U = 40 V. L = 10 mm, calculations. 1, 2, numerical calculations; 1 , fitting Lex(t) by formula (8).

In order to determine the deviation of the total number of electrons in the crystal, ℵ, from the equilibrium value, n0L, we integrate (2 ) along x from 0 to L. Let us restrict consideration by the case when the exclusion region does not overlap with the ohmic R contact (as shown in figure 1). Then the following relationships are valid at the R contact, x = 0. ∂n n = n0  Na , = 0, ∂x while at the p+–p contact, x = L: nDn ∂n j D . n  Na , =− ∂x e (Dn + Dp )n + Na Dp Also taking into account the fact that  L  L ∂ ∂ n dx = (n − n0 ) dx, ∂t 0 ∂t 0 the result of integrating (2 ) along the x-axis may be approximately written as  L ∂ℵ ℵ j µn (n − n0 ) dx. + =− where ℵ = ∂t τ e(µn + µp ) 0 (6) Let us replace the electron distribution in the base (figure 1) by a step function (line A–A ). On the lefthand side of AA the electron concentration is assumed to be at equilibrium (n = n0), while on the right-hand side the total depletion (n = −n0) is assumed to be over the entire exclusion region. Then the total number of electrons removed from the base may be evaluated as ℵ = −Lex n0 . Let us also assume that all the voltage applied, U, drops across the depletion region Lex, where n  Na , p ≈ Na . Therefore, the conductivity of the Lex region is determined only by residual holes with the concentration of Na. This gives j = U eµp Na /Lex , whence ∂Lex Lex µn µp Na b , b=U . (7) + = ∂t τ Lex (µn + µp ) n0 This equation has the following solution with the initial condition: Lex = 0 at t = 0     µn µp Na 2t . (8) Lex (t) = U τ 1 − exp − µn + µp n0 τ 1060

ℵ0 =

j0 τ ; e(1 + µp /µn )

(11)

where j0 is the steady-state current before switching off voltage. The accuracy of the approximate formulae obtained may be evaluated in figure 2, where the dependencies Lex(t) computed by equation (8) are shown along with the results of numerical calculations. In the latter case Lex was defined as the distance between the p+–p contact and the base coordinate where n(x) = 0.5n0.

3. Experimental details p–Ge samples with an acceptor impurity concentration of Na = 1 × 1012 cm−3 (structure 1) and 1 × 1014 cm−3 (structure 2) have been studied experimentally. The samples have a rectangular shape with length L ∼ 1 cm and transverse dimensions ∼0.5 × 0.5 cm. These dimensions exceed the diffusion length Ld ∼ 0.1 cm considerably. The p+–p contact was formed by alloying indium in hydrogen ambient at 500 ◦ C. The ohmic R contact was made by alloying pure Sn. In order to decrease the influence of surface generation–recombination the lateral faces were etched in H2O2. Measurements were carried out at T = 290 K for structure 1 and at T = 400 K for structure 2 (in both cases ni  Na ). The lifetime values for these samples were 140 and 450 µs, respectively. The longitudinal distribution n(x) between contacts was studied by the customary method of IR probing in the spectral range beyond the edge of intrinsic absorption of Ge (see the inset in figure 1). A narrow band (cut by a 100 µm wide slit diaphragm) of IR radiation from a tungsten helical filament (T = 2500 K) is focused by a BaF2 lens at the wide lateral face of the base. The radiation passed through the diaphragm was detected by a cooled Ge:Au photodetector. The probing spectral range (2 < λ < 10 µm) was determined by the fundamental absorption of the base material and the lens transmission spectrum. The intensity of spectrum component modulated by the applied electric field depended unambiguously on the free charge-carrier concentration in the transverse cross-section of the base. The results of the measurements are presented in figures 3 and 6. The data on IR probing were used to extract expansion dynamics of the depletion region Lex(t) in the course of the establishment of

The transient exclusion effect

Figure 3. The distribution of reduced-electron concentration over the base (experiment) at different time moments after switching on a voltage pulse U = 6 V. , , , , experimental data. Full curves 1–4, results of numerical calculations. t (µs): 1 and , 20; 2 and , 60; 3 and , 120; 4 and , steady-state distribution.





(a )

Figure 4. The dynamics of the exclusion region expansion and current relaxation to the steady-state value. and , experimental points of Lex(t) dependence; full curves 1 and 2, results of exact numerical calculation of Lex(t), dotted curves 1 and 2, approximate calculation of Lex(t) by (8). Curves 1 and : structure 2, U = 16 V. Curves 2 and : structure 1, U = 6 V. Curves 3–5, temporal dependencies of current establishment J(t) in the structure; 1, U = 6 V; 3, experiment; 4, calculation by (9); 5, exact calculation by numerical methods.

a steady-state charge-carrier distribution. These results are shown in figure 4 for both samples. Also shown in figure 4 are curves for the current relaxation to a steady state. In order to evaluate the validity of the one-dimensional approximation used in the interpretation of the experimental results we have studied a two-dimensional (2D) distribution of electrons over the structure base. For this purpose a twodimensional pattern of thermal emission from a wide face of the base was recorded using a thermal vision camera sensitive in the range 8–12 µm. This experimental technique is described in full detail in [13]. It is based on the fact that in an optically thin crystal kd < 1 (where k is the absorption coefficient and d is the base thickness) the intensity of thermal emission in the spectral range of absorption by free charge

,

(b)

Figure 5. (a) The two-dimensional distribution of non-equilibrium carriers under the conditions of exclusion. The grey scale corresponds to the percentage of concentration deficit (depletion with carriers) with regard to the equilibrium value n0. Structure 2, U = 16 V, T = 400 K. The p+–p contact is on the right-hand side; the ohmic contact is on the left-hand side. (b) Longitudinal distributions n(x) in the crystal centre along AA (curve 1) and near the surface along BB (curve 2).

carriers is proportional to their concentration [14]. The results of the measurements are shown in figure 5.

4. Results and discussion Let us consider, first of all, the cause of the significant difference between the processes of current establishment j(t) and the spatial redistribution of charge carriers Lex(t). 1061

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Figure 6. Recovery of the charge-carrier concentration to its equilibrium value in the base centre. Structure 2, T = 400 K, U = 16 V. Full curve, experiment; dotted curve, fitting by exponent with the time constant τ = 450 µs. The arrow indicates the moment when the voltage is switched off.

This difference is demonstrated by the consistent results of theoretical calculations (figure 2) and experiment (figure 4). Indeed, the exclusion length Lex in the long-base structure (L ∼ 10LD) is formed during the period comparable with the charge-carrier lifetime τ , while the current establishment process is considerably shorter (t  τ ) . The cause of this difference consists, from our point of view, in the following. The moment of current switching on is followed by the formation of a narrow region of initial depletion adjacent to the p+ contact. This region is characterized by high resistivity (high-field domain). The continuity equation does not account for this process correctly. However, we can evaluate its parameters approximately. The duration of the initial depletion region formation is comparable, by order of magnitude, to the Maxwell relaxation time. The region length along current lines is, in fact, a negligible shift of electrons during this period. The process causes a sharp decrease in current in the circuit: power supply-depletion region-unperturbed base. The higher is the ratio of n0/Na, the stronger is the decrease of current in the initial period. The subsequent process of bipolar drift of electron–hole pairs toward the R contact results in an expansion of the exclusion region and a decreasing slope of its moving front (the bipolar diffusion effect). This causes only a small decrease of current. The characteristic time of establishment of a steadystate distribution is the carrier lifetime τ . The depletion region length is the length of the bipolar drift of charge carriers LE ∼ = µE Eτ . Our calculations have shown that in a structure with a short base the duration of transient process to a steadystate value of Lex becomes significantly shorter. It is, in fact, duration of drift of electron–hole pairs to the R contact. For example, if L ∼ LD and U = 10 V the process duration is only t ∼ 0.1τ . It should be noted that the duration of the j(t) process remains unchanged. Recovery to the equilibrium state in the electron system of the structure base is caused by bulk-thermal generation of electron–hole pairs in the Lex region. This process is taken into account in (2) by the term n0/τ . Of course, the relaxation process in a long crystal is described by the exponential law (11) with a time constant τ (figure 6). In the case of 1062

structures with a short base the recovery process runs much faster (characteristic time ∼L2/D). For example, at L ∼ LD this time is 0.16τ . Thus, our results provide evidence of the possibility of implementing two situations in the base under exclusion. These situations differ from each other in physical nature. In the structure with a long base the transient process of exclusion and thermal generation of the carriers runs with the time constant τ , after the voltage switching both on and off. In the short-base structures the establishment process runs much faster. The process duration in this case may be controlled by applied voltage and it can be made equal to the duration of relaxation to equilibrium. To put this another way, the possibility exists to control the exclusion process by a pulse sequence with a 0.5 duty ratio. Let us now focus on some details of the discrepancy between theory and experiment. As follows from figures 3 and 4, there is a difference between experimental and theoretical values of the exclusion length (up to 30% by magnitude). Moreover, the front of the exclusion region in experiment is more sloping (figure 3). The slope of the n(x) front in experiment is usually lower by ∼3 times than in theory. These discrepancies, from our point of view, are caused mainly by the fact that one-dimensional theory eliminates, by definition, the impact of the surface carrier generation on the spatial distribution of carriers in the crystal. As stated in [15], with a finite-surface recombination velocity S the transverse profile of carrier concentration in the exclusion region takes a U shape. A decrease in concentration near the lateral sides becomes much less than near the central axis of the crystal. Indeed, as can be seen from experimental studies (figure 5), a deep concentration decrease (∼100%) arises in the central part of crystal (along A–A ). The decrease near the surface (along B–B ) is substantially less: ∼80% near the p+–p contact and 40–60% at the front end of the exclusion region. Thus, the strongest impact of surface generation is observed at the front end of exclusion. This is connected to the fact that the strength of the electric field decreases along the exclusion region when moving from the p+–p contact. With lower values of longitudinal Ex, the relative contribution of the transverse carrier diffusion induced by the surface generation is larger. This results in a larger recovery of concentration toward the equilibrium level and, respectively, in distortion of the exclusion front. It should be noted that a significant influence of surface generation takes place even at sufficiently small values of S. As following from measurements, this parameter under our experimental conditions did not exceed S = 200 cm s−1.

5. Conclusions The features of the transient exclusion process after switching on the voltage are caused by the formation of a high-field domain in the structure base. The duration of the current relaxation to a steady-state value in this process, t  τ , is, in fact, the duration of a high-field domain formation near the h–l contact. The duration of the establishment of a steady-state carrier distribution in the base is, in fact, the lifetime τ of the carriers in a long-base structure and the duration of a highfield domain expansion over all the structure length in the case of a short base. The first process determines the speed of the

The transient exclusion effect

structure as an active element of the electric circuit; the second one determines the speed of the structure as an optoelectronic device, for example, a modulator of IR radiation. The first process is controlled by the n0/Na ratio. The voltage applied and the base length can easily control the second process. The recovery of equilibrium of the charge-carrier distribution in the base, after switching off the voltage, is caused by the thermal generation of carriers. The sources of thermal generation are the structure bulk and ohmic contact. The transient-process parameters are described by a numerical solution of the non-stationary continuity equation. In the range of the base depletion lengths important for applications these parameters may be expressed by simple analytical formulae. A quantitative discrepancy is observed between experimental results and theory based on the one-dimensional approximation. The surface generation of electron–hole pairs significantly disturbs the shape of the depletion region, shortens the exclusion length (making this parameter spatially dependent in the transverse direction) and lowers the extent of base depletion with charge carriers. The results obtained may be used for the optimization of the parameters of IR radiation modulators and sources, with regard to the speed of operation and the spatial homogeneity of the exclusion region.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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