Terahertz-frequency electronic noise in In0.53Ga0.47As n+nn+ nanostructure

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Terahertz-frequency electronic noise in In0.53Ga0.47As n+nn+ nanostructure

Fatima Zohra Mahi, Luca Varani

To cite this version:

Fatima Zohra Mahi, Luca Varani. Terahertz-frequency electronic noise in In0.53Ga0.47As n+nn+

nanostructure. Results in Physics, Elsevier, 2013, 3, pp.209-213. �10.1016/j.rinp.2013.09.010�. �hal-

02450421�

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Terahertz-frequency electronic noise in In

_0.53

Ga

_0.47

As n

⁺

nn

⁺

nanostructure

Fatima Zohra Mahi

^a,^⇑

, Luca Varani

^b

aScience and Technology Institute, University of Bechar, Algeria

bInstitut d’Electronique du Sud (CNRS UMR 5214), Université Montpellier II, 34095 Montpellier, France

a r t i c l e i n f o

Article history:

Received 1 May 2013 Accepted 25 September 2013 Available online 3 October 2013

Keywords:

High frequency Electronic noise Nanostructure Noise resonance Current spectral density

a b s t r a c t

In this article we present an analytical model for the calculation of high-frequency electronic noise in n⁺nn⁺structure based on In0.53Ga0.47As material by using the analytical approach of Heterostructure Barrier Varactor (HBV) proposed in Ref.[1]. The model enables to interpret the different resonances appearing in the current and voltage spectral densities. In particular, we discuss the effect of geometrical parameters such as the total length of device and the free carriers concentration on the noise resonance.

The results can be useful in optimizing the device parameters for the generation of high frequency resonances noise.

Ó 2013 The Authors. Published by Elsevier B.V.

1. Introduction

In recent years significant attention has been paid to the physical processes responsible for electronic noise in n⁺nn⁺diodes. The final goal is to use them for various terahertz applications[2]. A considerable effort has been devoted to the theoretical investiga- tion of noise spectra by using Monte Carlo simulations of n⁺nn⁺ diodes[3,4]. In particular, the fluctuations spectra in submicron n⁺nn⁺structures for different semiconductors (Si, GaAs) have been evaluated by Monte Carlo simulations in Ref.[5]. In the terahertz frequency, some basic questions concerning the analysis and the discussion of the noise spectra in n⁺nn⁺structure have not been ex- plained yet. Therefore, the development of an analytical model able to reproduce the noise spectra of these diodes represents an inter- esting physical subject to be pursued.

In this contribution we present an analytical model enabling the calculation of current and voltage spectra in n⁺nn⁺diodes for In0.53Ga0.47As material and providing an accurate description of the resonances appearing in the high-frequency region. Moreover, our analytical model takes into account the ﬂuctuations of the free carriers, the returning carriers and the plasma resonance at the n⁺n homojunctions for the spectrum noise calculation. Under these

considerations the conduction current may be neglected and we consider the drift-current ﬂuctuations in our calculation.

The results of current noise are investigated at different spatial and spectral points in one dimensional inhomogeneous structures.

This can determine, on one hand, the contribution of the n⁺and n regions and, on the other hand, can describe the appearance of the resonances in high-frequency region. The discussion of the appearance of the resonances noise is useful to optimize the geometrical parameter of device for the extraction of the high-order harmonics.

To extract the high-frequency signal, the diode is embedded into a parallel output resonant circuit when a large-amplitude microwave voltage is applied.

The effectiveness of this model, for the calculation of ﬂuctua- tions spectrum, has been veriﬁed for the nanometric diodes such as Schottky barrier diodes (SBD) in article[6]and Heterostructure Barrier diodes (HBV) in article[1].

2. Analytical model

We are interested in the first step to the evolution of the spectral current density in inhomogeneous n⁺nn⁺In0.53Ga0.47As structure. For this case we evaluate the fluctuations of the drift current di in terahertz frequency region and the thermal current fluctuations in gigahertz frequency. In the second step, we calculate the spectrum of voltage fluctuations between the terminals of the n⁺nn⁺diode embedded into an external RLC resonant circuit.

For the ﬁrst step, we consider an n⁺nn⁺structure as shown in Fig. 1.

2211-3797 Ó 2013 The Authors. Published by Elsevier B.V.

http://dx.doi.org/10.1016/j.rinp.2013.09.010

⇑ Corresponding author. Tel.: +213 6 62 70 34 53.

E-mail address:[email protected](F.Z. Mahi).

Contents lists available atScienceDirect

Results in Physics

j o u r n a l h o m e p a g e : w w w . j o u r n a l s . e l s e v i e r . c o m / r e s u l t s - i n - p h y s i c s

Open access under CC BY-NC-ND license.

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Here liis the length and Niis the doping of the i-region. We assume that the dopings are equal to the free carrier concentrations in each region of the diode at room temperature.

2.1. Low frequency current noise model in n⁺nn⁺diodes

In the low frequency (f < 1 THz), the dominant contribution to the total noise comes from the thermal noise when the applied voltage is sufﬁciently low (the diode exhibits ohmic characteristics in this case). In addition, we assume that the contributions of generation-recombination and 1/f noise are not taken into account in our theoretical model. The thermal current noise spectrum has a Lorenzian form given by:

Slowð

x

Þ ¼ 4kT 1 þ ð

xs

Þ²

X³

i¼1

1

Ri ð1Þ

where Riis the resistance of i-regions and

s

is the average collision (relaxation) time deﬁned as

s

¼^lⁿ_e^m (where

l

n is the electron mobility and m^⁄is the effective masse of materials).

The resistance of i-regions can be written as:

Ri¼

q

_i^lⁱ

Ai

ð2Þ

where

q

iis the electrical resistivity and Aiis the cross section of i-regions. The electrical resistivity calculated at moderate temperature by the expression:

q

i¼ 1

eNi

l

_n ^ð3Þ

According to the Eq.(1), the current spectral density exhibits the high value^4kT_R

i whenx?0.

2.2. High-frequency current noise model in n⁺nn⁺diodes

It should be note that the current noise spectrum in terahertz frequency region takes into account only the ﬂuctuations of free carriers when the drift component of the current is considered.

The analytical expressions, which describe the current noise characteristics of n⁺nn⁺diodes, are derived to the analytical model of Heterostructure barrier varactor [1]. Moreover, for the high- frequency region, the current noise behavior to a n⁺nn⁺diode is evaluated at room temperature under a constant voltage applied between its terminals.

Siið

x

Þ ¼

x

^eA

L

2X³

j¼1

X³

i¼1

Nilibij

2

S^j_ff ð4Þ

where Sⁱ_ff¼ 4kT_Am¹ mi

N_il_iis the spectral density of the thermal Lange- vin force normalized to the free carrier number in i-regions, k is the Boltzmann constant and T the lattice temperature,

m

ithe momen- tum relaxation, L¼P3

i¼1liis the total length of diode and the bi,jele- ments of the inverse matrix developed in[1] and expressed as:

a11¼

x

²þ i

xm

1þ

x

²₁ð1 r1Þ, a32¼ a12¼ r2

x

²₂, a22¼

x

²þ i

xm

2þ

x

²₂ð1 r2Þ, a31¼ a21¼ r1

x

²₁; a23¼ a13¼ r3

x

²₃, a33¼

x

²þ i

xm

1þ

x

₃²ð1 r3Þ. Here ri¼^l_Lⁱ are the relative lengths and

x

i¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðe²=

0mÞN_i

p is the plasma frequency of the i-regions.

In the results, the plasma frequency (the hybrid plasma resonance) corresponding to the frequency position of the resonance peak. The appearance of this resonance peak is due to the plasma oscillations when the presence of different free carrier concentrations N_iin the structure (presence of homojunction) is considered (seeFigs. 4, Fig. 6 and Fig. 9).

The Eq.(4)constitutes the basis for our analysis of the current noise spectra in the high-frequency region. According to previous investigations (see Ref.[1,6]) this model is expected to describe correctly the current fluctuations in the THz domain. In addition, the spectrum of current fluctuations Siiin Eq.(4)can also be used to obtain a spatial map of current fluctuations inside the device.

For this purpose the lengths lican be replaced by the variation zL with 0 6 z 6 1 and i = 1,2,3 so that we can compute the position- dependent current spectral density Sii(x,li). This approach, on one hand allows us to describe the frequency dependent noise through the term

x

^eA_L and, on the other hand, it allows us to determine the contribution of n⁺and n regions when the dependence with position is included in li.

3. Voltage noise spectrum at n⁺nn⁺terminals

Here we shall consider the n⁺nn⁺diode embedded into a particular parallel resonant circuit when a microwave voltages U(t) = U0+ U1sin(x0t) is applied to the whole circuit. The corresponding equivalent scheme is presented inFig. 2.

The diode is simplified to the resistance Rd(Ud), the capacitance Cv(U^d) which depend on the voltage drop Udbetween the diode terminals and the drift-current fluctuations described by the spectral density Sii(x). The resonant circuit is characterized by the inductance L, the capacitance Cl and the noiseless resistance R. The voltage fluctuations drop between the diode terminals, dU_d, is obtained by the analytical model detailed in article [7].

The ﬁrst iteration of spectrum voltage ﬂuctuations obtains as:

S¹_U_d_U_dð

x

Þ ¼ Zðj

x

Þj²Siið

x

Þ þ S^v_iið

x

Þ

ð5Þ where Z(x) is the linear impedance of the diode + circuit and S^v_ii is the spectrum of extra current ﬂuctuations given as:

S^v_iið

x

Þ ¼ ð

x

C⁰_vÞ²X¹

n–0

c

_n

j j²jZð

x

nÞj²Siið

x

nÞ ð6Þ

Fig. 1. Schematic view of the n⁺nn⁺diode with: l1and l3the lengths of the n⁺ regions, l2the length of the n regions, N1,N2and N3the doping concentrations of the three regions.

Fig. 2. The series connection of the n⁺nn⁺diode with a RLC parallel resonant circuit.

210 F.Z. Mahi, L. Varani / Results in Physics 3 (2013) 209–213

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The terms of Eq.(6)is detailed in articles [7,8], where C⁰_v is the constant component capacity of diode and

c

n is the relative contribution of the varying part of the n⁺nn⁺diode capacitance at the frequency of the nth harmonic of the pumping voltagexn= nx0.

4. Results and discussion

The spectral current density in In0.53Ga0.47As n⁺nn⁺structure is calculated by using Eqs.(4) and (1). The evaluation of the current spectrum takes into account the mobility, the electrical resistivity and the collision time constant, whose values are given in Ref.[9].

TheFig. 3presents the total spectral current density, the high and low frequency current noise, in In0.53Ga0.47As n⁺nn⁺structure.

In Fig. 3, we remark the appearance of resonance peaks near 3.2 THz and 2 THz corresponding to the presence of n⁺n and nn⁺ homojunctions. The resonance peaks associated with plasma frequencyx1and the intermediate frequency between the plasma frequenciesx1andx2.

Near 2 THz the contribution of the anode region, corresponding to the position zli> 0.7

l

m (i = 2,3), to the current noise is found to be of great importance.

TheFig. 4presents the separate spectrum contributions coming from the low frequency current noise and from n⁺, n regions obtained by the analytical model(the low frequency spectrum noise of Eq.(1)and the three terms of Eq.(4), respectively).

The spectral densities in Fig. 4exhibits three resonant peaks near 0.2 THz, 2 THz and 3.2 THz when the resonances of 0.2 and 3.2 THz ﬁt well with the plasma frequenciesx2and x1 respectively. Indeed, The position of the peak at 2 THz can be ﬁtted by the empirical expression of reference[10](the expression is for a symetrical structure):

x

²_res¼

x

²₁ ^l²

l1þ l2þ

x

²₂ ^l¹

l1þ l2 ð7Þ

According to Eq.(7), the appearance of the intermediate resonance depends on the plasma frequencies (x2 andx1) and the lengths values (l1and l2). In the case considered here, using of Eq.(4)for spectrum noise evaluation, the presence of the second n⁺ region with plasma frequencyx3 gives a supplementary contribution to the resonance noise near 2 THz (seeFig. 4).

Moreover, we can increase the frequency position of resonances, for the extraction of harmonics in In_0.53Ga_0.47As n⁺nn⁺ diode, by the discussion of the lengths effect on the current spectral density (seeFig. 5).

In Fig. 5, we remark that the frequency position of the resonance peak increases when the total length of diode decreases according to Eqs.(7) and (4). In addition, the decreasing of the total

length introduces an increasing of the relative lengths of i regions (ri= li/L). This leads to the appearance of high-frequency resonances at 3 THz and 3.7 THz for the length value L = 0.7

l

m. There- fore, the n⁺nn⁺nanostructure is more favorable for obtaining the highest frequency resonances.

TheFig. 6presents the effect of carrier concentrations on the current spectrum by using the Eq.(4). We observe that the resonance of plasma frequencyx2appears in 1 THz for 10¹⁶cm³compared to free carrier concentration 10¹⁴cm³when this resonance appears near 0.2 THz (see Fig. 6). Therefore, the ﬁrst resonance

0 0.5

0 1 2 3 4 5 1

0 2 4 6

x 10⁻⁹

S ii/A (A2 m−2 s)

Frequency (THz) Position (10

−6 m)

Fig. 3. Spectral density of current ﬂuctuations per unit surface in In0.53Ga0.47As n⁺nn⁺structure with: l1= l3= 0.5lm, l2= 0.25lm, N1,3= 10¹⁷cm³and N2= 10¹⁴ cm³.

10⁻²² 10⁻²⁰ 10⁻¹⁸ 10⁻¹⁶ 10⁻¹⁴ 10⁻¹² 10⁻¹⁰ 10⁻⁸

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Sii/A (A2 m−2 s)

Frequency (THz) contribution : n⁺(1)

n n⁺(2) S_low total

Fig. 4. Spectral density of current ﬂuctuations per unit surface in each region of a In0.53Ga0.47As n⁺nn⁺structure. The structure is the same as that ofFig. 3.

10⁻¹³ 10⁻¹² 10⁻¹¹ 10⁻¹⁰ 10⁻⁹ 10⁻⁸

0 0.5 1 1.5 2 2.5 3 3.5 4 Sii/A (A2 m−2 s)

Frequency (THz) L(µm) 1.6

1.10.7

Fig. 5. Spectral density of current ﬂuctuations in In0.53Ga0.47As n⁺nn⁺structure as a function of the total length of device. With: l2= 0.25lm, N1,3= 1 10¹⁷cm³and N2= 10¹⁴cm³.

10⁻¹³ 10⁻¹² 10⁻¹¹ 10⁻¹⁰ 10⁻⁹

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Sii/A (A2 m−2 s)

Frequency (THz) N₂ (cm⁻³) 10¹⁴

10¹⁶

Fig. 6. Spectral density of current ﬂuctuations in In0.53Ga0.47As n⁺nn⁺structure as a function of free carrier concentrations of n region. With: the total length L = 1.6lm, l2= 0.25lm, N1,3= 10¹⁷cm³.

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peak increase with the increasing of the dopings according to

x

²₂¼ ðe²=

0mÞN2.

The ﬁrst iteration of voltage noise spectrum between the diode terminals is evaluated by using Eq.(5).

Here S^vis the extra part of displacement - current ﬂuctuations given by equation(6)and included in the voltage noise spectrum (in Eq.(5)), where the resonance circuit condition (diode + circuit RLC) is tuned to the frequency of the third harmonicsxres= 3x0, the pumping signal at frequency

m

0= 200 GHz and the amplitude is U₀= 0.45 V.

The dependence of voltage noise spectrum to the length of device is calculated inFig. 7by using Eq.(5)when the length values are 1.6

l

m, 1.1

l

m and 0.7

l

m. In addition, the concentrations of n and n⁺regions are N2= 10¹⁴cm³and N1,3= 10¹⁷cm³respectivelly.

We observe inFig. 7that the high-frequency resonance peak generated by the n⁺nn⁺ diode appeared near 1 THz for different lengths values associated to circuit frequency resonance. More- over, we remark the presence of an additional resonances peaks at 1.5 THz and 3 THz corresponding to the total length 1.6

l

m when the higher resonances peaks appear near 2.5 THz and 3.5 THz for the length 0.7

l

m. Therefore, the frequencies positions of resonances series increase for the n⁺nn⁺nanostructure.

By using Eq.(5)we can extract n harmonics generated between the resonances peaks appearing at frequencies 1 THz and 3.5 THz (seeFig. 7). The results ofFig. 8are calculated in accordance with Eq. (5) and by changing the parameters of the resonant circuit compared to the condition of Fig. 7 (changing the resistance R, inductance L and the capacitance C values).

InFig. 8, we observe the appearance of resonances peaks in fre- quenciesxres± nx0when the most important resonance noise is at the frequencies 1 THz and at 4 THz for the total length 0.7

l

m.

Therefore, the level of the noise appearance can be considerably determined by a proper choice of the diode parameters and its operation point (i.e. U0).

Finally, inFig. 9we report the comparison of the spectra of current ﬂuctuations for a GaAs n⁺nn⁺ diode with N_1,3= 10¹⁷ cm³, N2= 10¹⁶cm³and the total length L = 0.6

l

m calculated at 300 K with the MCP simulations in Ref.[10] and with the analytical model described above. The results refer to the two different cases:

the homogeneous (l2= 0.6

l

m) and the homojunction (l2= 0.2

l

m and 0.4

l

m) structures corresponding, respectively, to absence and presence of n or/and n⁺regions. For the case of a homogeneous structure in the absence of the homojunction n⁺n (solid and dashed dot curves) the spectral density of current ﬂuctuations exhibits the usual Lorentzian form proportional to ¹

1þðxsÞ²according to Eq.(1).

Moreover, the presence of a homojunction leads to the appearance

of a resonance peak due to plasma oscillations (the so called hybrid plasma resonance) associated with the ﬂuctuations of the free carrier concentration.

It should be emphasized that the good agreement found between analytical (Fig. 9(a)) and MCP (Fig. 9(b)) results in the frequency range of interest fully validates the analytical model proposed here.

10⁻³⁶ 10⁻³⁵ 10⁻³⁴ 10⁻³³ 10⁻³² 10⁻³¹ 10⁻³⁰ 10⁻²⁹ 10⁻²⁸

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 SUdUd (V2 m2 s)

Frequency (THz) L(µm): 1.6

1.10.7

Fig. 7. Spectral density of voltage ﬂuctuations between the In0.53Ga0.47As n⁺nn⁺ diode terminals for different total lengths. With R = 9.5 10¹⁰Xm², L = 1.02 10²³ Hm², Cl= 2.7 10⁴Fm², R⁰d= 3.7 10⁵Xm², C⁰_v= 2.2 10³ Fm².

10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 10⁰ 10²

0 1 2 3 4 5 6

SUd (10−29 V2 m2 s)

Frequency (THz) L (µm): 0.7

1.6

Fig. 8. Spectral density of voltage ﬂuctuations between the In0.53Ga0.47As n⁺nn⁺ terminals calculated for 1.6lm and 0.7lm (dashed and solid lines, respectively).

With R = 10⁸Xm²,L = 1.02 10²³Hm², Cl= 0.8 10³Fm², R⁰_d= 3.7 10⁵Xm², C⁰_v= 2.2 10³Fm².

0.001 0.01 0.1 1

0.5 1 1.5 2 2.5 3 3.5 4 Sii/A ( 10−9 A2 s m−2 )

Frequency (THz)

(a)

(b)

l₂ (µm) : 0.2 0.40.6

Fig. 9. Spectral density of current ﬂuctuations per unit surface in GaAs n⁺nn⁺diode at room temperature calculated with (a) analytical model and (b) Monte Carlo simulations[10].

212 F.Z. Mahi, L. Varani / Results in Physics 3 (2013) 209–213

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5. Conclusions

The analytical model proposed in this article describes the high frequency part of noise spectral densities in In0.53Ga0.47As n⁺nn⁺ diodes.

The evaluation of the current noise can determine the appearance of the intrinsic resonance noise in terahertz frequency and thier dependance on total length and dopings of the diode. The discussion of the diode parameters dependence on the current spectrum is useful for optimizing the level noise appearance.

The results of the current noise can be useful to calculate the voltage noise between the n⁺nn⁺diode terminals. For the voltage noise evaluation the diode is embedded into an external resonant circuit for the extraction of the high-order harmonics.

The noise peaks of the voltage spectrum appears in frequencies xres± nx0 from 1 THz to 4 THz for the device parameters L = 0.7

l

m and N2= 10¹⁷cm³. These parameters of diode can pro- duce the high frequency order of harmonic extraction.

Acknowledgement

This work is partly supported by the Algerian ministry of higher education and research through contract N. D03820100006 and by TeraLab-Montpellier.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the online version, athttp://dx.doi.org/10.1016/j.rinp.2013.09.010.

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