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Performance investigations of quantum dot infrared photodetectors
Infrared Physics & Technology 55 (2012) 320–325
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Infrared Physics & Technology journal homepage: www.elsevier.com/locate/infrared
Performance investigations of quantum dot infrared photodetectors Hongmei Liu a,b,⇑, Jianqi Zhang a a b
School of Technical Physics, XiDian University, Xi’an City, ShaanXi Province 710071, China School of Physical Science and Electronics, Shanxi Datong University, Datong City, ShanXi Province 037009, China
a r t i c l e
i n f o
Article history: Received 1 January 2012 Available online 14 March 2012 Keywords: Dark current Photocurrent Detectivity
a b s t r a c t Quantum dot infrared photodetectors (QDIPs) have already attracted more and more attention in recent years due to a high photoconductive gain, a low dark current and an increased operating temperature. In the paper, a device model for the QDIP is proposed. It is assumed that the total electron transport and the self-consistent potential distribution under the dark conditions determine the dark current calculation of QDIP devices in this model. The model can be used for calculating the dark current, the photocurrent and the detectivity of QDIP devices, and these calculated results show a good agreement with the published results, which illustrate the validity of the device model. Crown Copyright Ó 2012 Published by Elsevier B.V. All rights reserved.
1. Introduction In the past decade, QDIPs (quantum dots infrared photodetectors) have already been proved to possess more efficiencies than other types of semiconductor photodetectors and have become a very hot field of research due to a broader infrared response, a higher photoconductive gain, a lower dark current and an increased operating temperature [1–5]. With the expanding and deepening of the detecting application, the needs for the high performance and the high quality QDIP may be continuously increasing in the future. The device model of QDIPs plays an important theoretical role in optimizing the device design, reducing the dark current and improving the device performance, and thus more and more attention has been paid to the study of the device model in the process of pursuing the high performance QDIP [6–9]. In the paper, a device model is proposed to theoretically evaluate the performance of the QDIP. This device model accounts for the features of the total electron transport including the microscale transport and the nanoscale transport and the self-consistent potential distribution under the dark conditions, and it can be used to calculate the main performance parameters of QDIP devices such as the dark current, the photocurrent and the detectivity as functions of the temperature and the electric field. The calculated results show a good agreement with the published results, which provide with a good evidence for the validity of the proposed model. As a result, based on its contribution in the theoretical prediction of QDIP performance, the model can not only play an important theoretical role in further optimizing the device design and improving the device performance, but can also provide device ⇑ Corresponding author at: School of Technical Physics, XiDian University, Xi’an City, ShaanXi Province 710071, China. E-mail address: [email protected] (H. Liu).
designers with the theoretical guidance and the experimental verification to pursue the high performance QDIP devices, which may be widely applied to the commercial, defense and space fields in the future.
2. Model QDIP devices detect an infrared light by electrons translation between subband and subband or subband and continuum in quantum dots, and generally consists of barrier layers of excited regions and repetitive quantum dots layers [10]. The schematic view of the QDIP layers structure is shown in Fig. 1, where the top and bottom contacts are usually used as the emitter and collector, respectively. The sandwiched layers structure between the emitter and the collector is a stack of quantum dots layers separated by barrier layers. In this paper, each quantum dots layer mainly consists of many periodically distributed identical quantum dots with P the density QD , and the sheet density of doping donors equals P D . The lateral size of quantum dots aQD is supposed as large enough, so each quantum dot has a large number of bound states to accept more electrons. However, the transverse size of quantum dots is smaller than the distance between the quantum dots layers L to provide with the single energy level related to the quantization in the direction. The lateral distance between quantum dots is LQD qffiffiffiffiffiffiffiffiffiffi P which is equal to QD , hN k i is the average number of electrons in a quantum dot belonging to the k-th quantum dot layer, where k is within 1 K (K is the total number of quantum dots layer). Based on these assumptions, by considering the influence of the total electron transport on the current under dark conditions, we propose a device model of QDIP for estimating the performance of QDIP such as the dark current, the photocurrent and the detectivity.
1350-4495/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.infrared.2012.03.001
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Fig. 1. Schematic view of the QDIP layers structure, where the spheres represent quantum dots.
2.1. Dark current The dark current density can be obtained by counting the mobile carrier density in the barrier [11,12], which is shown as:
hjdark i ¼ ev n3D
ð1Þ
where e is the electron charge, v the drift velocity of electrons, and n3D the three-dimensional electron density which can be obtained by the following equation [11].
n3D ¼ 2
3=2 mb kT 2
2ph
Ea exp kT
ð2Þ
where mb is the effective mass of electron, k the Boltzmann constant, T the temperature, h the reduced Planck constant, and Ea the activation energy depending on the total electron transport. Since the present investigation indicates the simultaneous existence of the nanoscale transport and the microscale transport in the total electron transport process of the QDIP devices [13,14], the activation energy under the total electron transport process including the microscale and nanoscale electron transport Ea can be calculated as [13]:
Ea ¼ E0;micro expðE=E0 Þ þ E0;nano bE
ð3Þ
where E is an applied electric field intensity, E0,micro and E0,nano respectively represent the activation energies under the microscale and the nanoscale transport at zero bias (E = 0 kV/cm), E0 and b are the experimental fitting parameters, which represent the change rate of the activation energy under the microscale and the nanoscale electron transport mechanism with the electric field, respectively. Furthermore, it is very clear that Eq. (3) can be divided into two parts, E0,micro exp (E/E0) corresponds to the activation energy under the microscale transport mechanism which is thermal emission over the effective potential barrier [15], and E0,nano bE indicates the activation energy under the nanoscale transport mechanism which means the escaping from quantum dots related to tunnelling [16,17]. Substituting Eq. (3) into Eq. (2), then we can get the dark current density which can be shown by the following equation [14].
hjdark i ¼ 2ev
ered in the dark current calculation. On the other hand, since the dark current is also considered as the current, which is formed from the currents flow the punctures in the planar potential barriers formed by the charged quantum dots [18], it can be obtained with the consideration of the influence of electron transport on each current through the punctures. This dark current calculation and above proposed model not only essentially put emphasis on the electron transport on dark current, but both are the algorithms which mainly depend on the structure parameters of QDIP devices. Hence, in the same photodetector, it is via the current balance relation in dark conditions between the two dark current models that we can obtain the average number of electrons in a quantum dot and further estimate the photocurrent and the detectivity. On the basis of the theory that the dark current should depend on the electron transport through the punctures in the planar potential barriers formed by the charged quantum dots, the dark current density [18–20] of QDIP can also be calculated as:
hjdark i ¼ jmax
Z
QD
1
2
dr exp
0
euðhNk iÞ kT
ð5Þ
P where QD is the quantum dot density in a layer, Nk the average number of electrons in a quantum dot belonging to the k-th quantum dot layer, which is actually the same for all indexes (i.e. hN k i ¼ hNi), uðhN k iÞ is the potential distribution in the quantum dot layer as a function of the average number of electrons in each quantum dot, and jmax is the maximum current density, and can be estimated from Richardson-Dushman relation [19,21]:
jmax ¼ A T 2
ð6Þ
⁄
where A is the Richardson constant. After integration, an expression for the average the dark current density [19,22] can be obtained as:
hjdark i ¼ jdark
V þ V D ðhNi=NQD ÞV QD exp e hNi ðK þ 1ÞkT
H
ð7Þ
where
2
H¼
3=2 mb kT
P
p4 4
0 erf @0:47LQD
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P3=2 132 hNi QD A5
e0 er
e0 er kB T e2
qffiffiffiffiffiffiffiffiffiffi P
ð8Þ
QD
2
2ph E0;micro expðE=E0 Þ þ E0;nano bE exp kT
and these characteristic voltages can be calculated as:
ð4Þ
From the analysis above, in this dark current model, the influence of the total electron transport (including thermal emission over the effective barrier and the electron escaping from the quantum dots related to tunnelling) on the activation energy is consid-
V QD ¼
VD ¼
P e KðK þ 1Þ Lð1 #ÞNQD 2e0 er QD
P e KðK þ 1Þ L 2e0 er D
ð9Þ
ð10Þ
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pffiffiffi 0:72 2 ffiffiffiffiffiffiffiffiffiffi q #¼ P pKL QD
ð11Þ
In the above equations, K is the total number of the quantum P dot layers, L the distance between the quantum dot layers, D the sheet density of doping donors, NQD the maximum in the number of electrons which can occupy each quantum dot, LQD the lateral distance between quantum dots, V the applied voltage, and er the dielectric constant of the material used to fabricate quantum dots. Let Eq. (4) equal Eq. (7), we can get the current balance relationship under the dark conditions which can be shown as:
3=2 mb kB T
Ea;micro þ Ea;nano 2ev exp 2 kT 2ph H V þ V D ðhNi=NQD ÞV QD ¼ jdark exp e hNi ðK þ 1ÞkT
ð12Þ
By solving Eq. (12), we can get the average electrons numbers in a quantum dot hNi, further obtain new models for the photocurrent and detectivity of QDIP. 2.2. Photocurrent As well known to us, the photocurrent defined as the current of the detector under an illumination [23] can be determined by the following equation.
hjPhoto i ¼ eUs gg p
ð13Þ
where Us is the incident photo flux density on a detector, g the quantum efficiency, and gp the photoconductive gain which can be calculated by the ratio of the lifetime of carriers to the transit time of the carriers through the device. In QDIP devices, since the electron–electron scattering is main relaxation processing, the carriers lifetime becomes very long [24] and furthermore leads to a high photoconductive gain, the typical value of which is within 1– 106 [19]. Based on the relation between the quantum efficiency and the average electron number in a quantum dot, which can be shown P as: g ¼ dhNiK QD , the expression of the photocurrent can be written as:
hjPhoto i ¼ deg p hNi
P
Us K
ð14Þ
QD
where d is the electron capture cross section coefficient and it is adjusted to meet experimental comparison, the value of the incident photo flux density Us is supposed as 8 1017 photons/cm2 s [19]. 2.3. Detectivity According to the photoconductive detection mechanism, with a thermal noise ignored, the detectivity of QDIP can be determined as:
pffiffiffi Ri A D ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4eIdark g n
ð15Þ
where Ri is the current responsivity of QDIP, A the area of QDIP device, gn the noise gain of photodetectors, which approximately equals photoconductive gain gp in the quantum dot infrared photodetector, Idark is the dark current. Here, the current responsivity, which is the ratio of the detector photocurrent to the incident photopower, can be given as:
P deg p hNi QD K Ri ¼ hv
ð16Þ
where v is the frequency of an incident infrared light. Substituting Eqs. (4) and (16) into Eq. (15), we can ultimately obtain the detectivity which is shown as:
D ¼ hv
P deg p hNi K QD rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8g n e2 v
mb kT 2ph2
3=2
exp
ð17Þ
E0;micro expðE=E0 ÞþE0;nano bE kT
3. Simulation and data The dark current, photocurrent and detectivity of QDIP, which are shown as functions of the applied electric field and temperature, are discussed and analyzed in details in the section. These obtained results are compared with experimental results, which are transferred from the voltage coordinates into the electric field intensity coordinates, reported in literatures to show the validity of the proposed model. In our calculations, the drift velocity of electrons within 105 1.8 106 m/s [25] is supposed as the constant 8 105 m/s, the parameter mb is chosen as 0.34 me [26], the other structure parameters from the GaAs or InGaAs QDIP devices [19,20,27,28] are displayed in Table 1 and are adjusted to the same values as those of the QDIP devices used for the experimental verification. Fig. 2 presents the dark current density as functions of the applied electrical field and temperature. It is clear that these curves have the same characteristics in changing of the dark current density with electrical field (Fig. 2a) and temperature (Fig. 2b). The increase trend of the dark current density with the electric field is shown in Fig. 2a and it clearly proved that total electron transport including the nanoscale and microscale transport determines the dark current density of QDIP devices. As what is shown at the beginning of these curves in Fig. 2a, the dark current density changes rapidly with the increase of the applied electric field, which can be mainly ascribed to the contribution of the microscale electron transport on the dark current density, whereas the subsequent slow increase of the dark current density reveals the significant contribution of the nanoscale electron transport. More importantly, our theoretical results obviously show a good agreement with the previous published experimental results [15] plotted with black squares at 77 K in Fig. 2a, which well directly examines the validity of our dark current density model. But it can also be found that there is a slight deviation between the theoretical results and the experimental results. The reasons of the deviation are not hitherto clear, and it possibly is ascribed to the operation without consideration of the influence of the applied voltage on the parameters such as v, E0, b and g in our model. In addition, just as shown in Fig. 2a, the temperature also makes a great contribution to the dark current density in a small temperature range (23 K). For example, under the same electrical field of 5 kV/cm, the dark current density is only 1.03 107 A/cm2 at 77 K, and then rapidly reaches to 2.40 104 A/cm2 when the temperate is of 100 K, which is 3 orders of magnitude larger than that at 77 K. In order to make clear of this phenomenon, the influence of temperature on the dark current density at the electric field of 5 kV/cm is displayed in details in Fig. 2b. Obviously, the dark current density which exhibits exponential increase can be greatly
Table 1 Parameters in QDIP calculations. E0,micro
E0
L
K
34.6, 90 meV E0,nano 210, 224.7 meV
1.62, 3 kV/cm b 2, 2.79 meV cm/kV
30–40 nm P
10 P
QD
4 1010 cm2
NQD 8
D
0:3
P
QD
er 12
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Fig. 2. Dark current density as functions of the electric field and temperature with E0,micro = 34.6, 90 meV, E0 = 1.62, 3 kV/cm, E0,nano = 210, 224.7 meV, b = 2, 2.79 meV cm/kV. (a) Dark current density as a function of an applied electric field at 77 K and 100 K. (b) Dark current density as a function of temperature with different applied electric fields.
Fig. 3. Average number of electrons in a quantum dot as functions of electric field and temperature with E0,micro = 34.6 meV, E0 = 1.62 kV/cm, E0,nano = 224.7 meV, P b = 2.79 meV cm/kV, L = 31.5 nm, QD ¼ 0:4 1015 cm2 , NQD = 8. (a) Average number of electrons in a quantum dot as a function of the applied electric field at different temperatures. (b) Average number of electrons in a quantum dot as a function of temperature at the different applied electric fields.
affected by the temperature under the certain electric field. In addition, at the electrical field of 8 and 11 kV/cm, the curves of dark current density vs temperature are plotted with green and blue lines, respectively. It is clear that these curves have the same increase trend with the increase of the temperature, which can be ascribed to the following reasons: the increasing of temperature results in more and more electrons escaping from the quantum dot by the thermionic emission and further generates the increase of the dark current density. Fig. 2b also describes the dependence of the dark current density on the electric field. At the certain temperature, the higher the electrical field is, the larger the dark current density will be. In fact, when the applied field is increased, a greater band bending will result in reduction of the potential barrier, and thus more and more electrons will escapes from the quantum dot across the reduced potential barrier. As a result, the larger dark current density can be generated by the increase of electric field. From the analysis above, a conclusion can be drawn that the electrical field and the temperature both have a very great contribution to dark current density due to the reducing the height of potential barrier and enhancing thermionic emission, respectively. Fig. 2a also shows the dependence of the dark current density on the parameters E0,micro, E0,nano, E0 and b. The dark current density values at 100 K (corresponding to the curve 100 K) are used as the values of reference, the dark current density values will correspondingly change when the parameters E0,micro, E0,nano, E0 and b are changed, respectively. These changed dark current density val-
ues form into the curves E0,micro, E0,nano, E0 and b, respectively. From the difference between the curve E0,nano and the curve 100 K, it can be found that the dark current density values with E0,nano = 210 meV are very larger than those with E0,nano = 224.7 meV. This corresponding relationship between the small E0,nano and the large dark current density is mainly due to the inverse exponential relationship between the dark current density between E0,micro. For the similar reason, the dark current density values from the curve E0,micro with E0,micro = 90 meV also show the excepted decrease compared with the data in the curve 100 K. Moreover, Fig. 2a also indicates that the change of the dark current density with the parameters E0 and b, which are the change rate of the activation energy under the microscale and the nanoscale electron transport with the electric field. For instance, at the same electric field of 2 kV/cm, the dark current density is 3.40 105 A/cm2 with E0 = 1.62 kV/cm (in the curve 100 K), however, it becomes 1.39 105 A/cm2 (in the curve E0) when the parameters E0 is 3 kV/cm. The decrease of the dark current density with the increase of E0 demonstrates the influence of the microscale electron transport on the dark current density. Similarly, the increase of dark current density with the decrease of b with 5–12 kV/cm of the electric field intensity also shows the influence of the nanoscale electron transport on the dark current density. As a result, the parameters E0,micro, E0,nano, E0 and b, which are the same as the electric field and temperature, have a great effect on the dark current density calculation.
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H. Liu, J. Zhang / Infrared Physics & Technology 55 (2012) 320–325
Fig. 4. Photocurrent as functions of electric field and temperature with E0,micro = 34.6 meV, E0 = 1.62 kV/cm, E0,nano = 224.7 meV, b=2.79 meV cm/kV, P 15 cm2 , L = 31.5 nm, NQD = 8. (a) Photocurrent as a function of the QD ¼ 0:4 10 applied electric field at 60 K and 100 K. (b) Photocurrent as a function of temperature at the different applied electric fields.
With the consideration of the current balance relation under the dark condition, the average number of electrons in a quantum dot, the photocurrent and the detectivity under the different electrical field and temperature can be calculated out and simulated in Figs. 3–5. Here, Fig. 3a reveals the average number of electrons in a quantum dot increases with the increase of the electric field at the different temperature, which is similar to the increases of the dark current density in Fig. 2a. The reasons for the similar increase existence are as follows, when the electric field is very low, a large number of electrons are limited to the previous location because of high potential barriers, but with increase of electrical field, the potential barriers will be greatly reduced as a result of a greater band bending, and thus more and more electrons will be easily captured by quantum dots across the reduced potential barrier. In the end, the number of electrons in a quantum dot will be increased. In addition, Fig. 3a also presents the dependence of the average electron number in a quantum dot on the temperature. It can be found that, at the electrical field of 5 kV/cm, the average electron number in a quantum dot is 2.68 at 100 K and 2.83 at 60 K in Fig. 3a. At the same electrical field, the average electron number in a quantum dot at temperature within 60–110 K in Fig. 3b shows the similar decrease trend with the temperature. The decrease of the average electrons number in a quantum dot mainly results from the increase of electrons escaping out of quantum dots by thermionic emission with the enhanced temperature. Of course, it can be also noted from Fig. 3b that the higher the electrical field, the more the average electron number in a quantum dot at the certain temperature. The finding is also consistent with what is shown in Fig. 3a.
Fig. 5. Detectivity as functions of electric field and temperature with E0,micro = 34.6 P 15 meV, E0 = 1.62 kV/cm, E0,nano = 224.7 meV, b = 2.79 meV cm/kV, QD ¼ 0:4 10 2 cm , L = 36.6 nm, NQD = 8. (a) Detectivity as a function of applied electric field at 80 K and 100 K. (b) Detectivity as a function of temperature with different applied electric fields.
Fig. 4a presents the photocurrent values vs the applied electrical filed in the range of 5–12 kV/cm. Due to the interdependence of the photocurrent and the average number of electrons in a quantum dot from Eq. (14), the changes of the photocurrent with the electric field in Fig. 4a are similar to those results in Fig. 3a. Therefore, for the same reason, the change trends of the curves in Fig. 4b are similar to those in Fig. 3b. From the physical mechanism point of view, the increased electric field brings about the enhanced reduction of the potential barrier. More electrons possibly escape from the quantum dots over this reduced barrier, thus this behavior makes the photocurrent increase in the end. However, for the photocurrent declining with the temperature at a constant bias voltage, since the electrons in a quantum dot becomes lower in number with the temperature, and thus, fewer electrons in quantum dots can be excited from the quantum dots to form into the photocurrent when the incident infrared light irradiates on the activation region. As a result, the photocurrent will be decreased with the temperature. In addition, it is obvious in Fig. 4a that the experimental values at 100 K from the Ref. [27] agree with our calculated results, which demonstrated the correctness of our model again. From the above analysis, it is evident that these curves in Figs. 3 and 4 not only show the dependence of these parameters including the average number of electrons in a quantum dot and photocurrent on the electric field and the temperature, but also demonstrate the nature of correlation among these parameters. Fig. 5 illustrates the influence of electric field and temperature on the detectivity of QDIP devices. As observed in Fig. 5a, a theoretical evidence for the declined detectivity can support the fact that
H. Liu, J. Zhang / Infrared Physics & Technology 55 (2012) 320–325
the increase of the electric field intensity can greatly influence the performances of QDIP devices. The decrease trends of these detectivity curves in Fig. 5a can be ascribed to the increase of the dark current with enhanced electron emission as a result of reducing the height of potential barrier as the above analysis. Moreover, as exhibited in Fig. 5a, the detectivity values are demonstrated to show a good agreement with the experimental results at 80 K from the published literature [28] and the consistency gives a direct evidence to the validity of our detectivity model. Fig. 5b not only reveals the decreased detectivity with the increase of temperature, but also indicates the influence of the electric field on the detectivity. For example, at the same temperature of 80 K, the detectivity under the electrical field of 5 kV/cm is 8.60 109 cm Hz1/2/W, which is about 4 times as large as that at the electrical field of 11 kV/cm. More details of the changing trends of the detectivity with the electric field can be found in Fig. 5a. Fundamentally speaking, the increase of the dark current with the increase of the applied electric field intensity and the temperature ultimately results in the decrease of the detectivity for QDIP devices.
4. Conclusions In the paper, the device model of QDIP is proposed, which accounts for the influence of the total electron transport. Based on this model, main performance parameters such as the dark current, the photocurrent, the detectivity are calculated and simulated. The obtained results can well explain the strong dependence of the dark current and the other performances on the applied electric field and the temperature in the QDIP devices, and they show a good consistency with the published experimental results as expected, which further test and verify the accuracy of the proposed models. Although these models can be well used for estimating performances of QDIP and directing the design of QDIP devices, it can be found that they cannot apply to the conditions at too low voltage, the reasons leading to the phenomenon are not clear so far. Therefore, further research is required to better understand and make clear the dependence of our developed device model including the total electron transport on the very low electric field, and the corresponding experiments will be carried out in our lab. Acknowledgments This work was supported by the Fundamental Research Funds for the Central Universities (Grant Nos. K50510050001 and K50510050004); the National Science Foundation of China (NSFC, Grant No. 60777042); Zhejiang Provincial Science foundation (Grant No. Y4110529); and China Postdoctoral Science Foundation (Grant No. 20100481063). References [1] S.K. Das, T.D. Das, S. Dhar, M. de la Mare, A. Krier, Near infrared photoluminescence observed in dilute GaSbBi alloys grown by liquid phase epitaxy, Infrared Phys. Technol. 55 (2012) 156–160.
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Contents lists available at SciVerse ScienceDirect
Infrared Physics & Technology journal homepage: www.elsevier.com/locate/infrared
Performance investigations of quantum dot infrared photodetectors Hongmei Liu a,b,⇑, Jianqi Zhang a a b
School of Technical Physics, XiDian University, Xi’an City, ShaanXi Province 710071, China School of Physical Science and Electronics, Shanxi Datong University, Datong City, ShanXi Province 037009, China
a r t i c l e
i n f o
Article history: Received 1 January 2012 Available online 14 March 2012 Keywords: Dark current Photocurrent Detectivity
a b s t r a c t Quantum dot infrared photodetectors (QDIPs) have already attracted more and more attention in recent years due to a high photoconductive gain, a low dark current and an increased operating temperature. In the paper, a device model for the QDIP is proposed. It is assumed that the total electron transport and the self-consistent potential distribution under the dark conditions determine the dark current calculation of QDIP devices in this model. The model can be used for calculating the dark current, the photocurrent and the detectivity of QDIP devices, and these calculated results show a good agreement with the published results, which illustrate the validity of the device model. Crown Copyright Ó 2012 Published by Elsevier B.V. All rights reserved.
1. Introduction In the past decade, QDIPs (quantum dots infrared photodetectors) have already been proved to possess more efficiencies than other types of semiconductor photodetectors and have become a very hot field of research due to a broader infrared response, a higher photoconductive gain, a lower dark current and an increased operating temperature [1–5]. With the expanding and deepening of the detecting application, the needs for the high performance and the high quality QDIP may be continuously increasing in the future. The device model of QDIPs plays an important theoretical role in optimizing the device design, reducing the dark current and improving the device performance, and thus more and more attention has been paid to the study of the device model in the process of pursuing the high performance QDIP [6–9]. In the paper, a device model is proposed to theoretically evaluate the performance of the QDIP. This device model accounts for the features of the total electron transport including the microscale transport and the nanoscale transport and the self-consistent potential distribution under the dark conditions, and it can be used to calculate the main performance parameters of QDIP devices such as the dark current, the photocurrent and the detectivity as functions of the temperature and the electric field. The calculated results show a good agreement with the published results, which provide with a good evidence for the validity of the proposed model. As a result, based on its contribution in the theoretical prediction of QDIP performance, the model can not only play an important theoretical role in further optimizing the device design and improving the device performance, but can also provide device ⇑ Corresponding author at: School of Technical Physics, XiDian University, Xi’an City, ShaanXi Province 710071, China. E-mail address: [email protected] (H. Liu).
designers with the theoretical guidance and the experimental verification to pursue the high performance QDIP devices, which may be widely applied to the commercial, defense and space fields in the future.
2. Model QDIP devices detect an infrared light by electrons translation between subband and subband or subband and continuum in quantum dots, and generally consists of barrier layers of excited regions and repetitive quantum dots layers [10]. The schematic view of the QDIP layers structure is shown in Fig. 1, where the top and bottom contacts are usually used as the emitter and collector, respectively. The sandwiched layers structure between the emitter and the collector is a stack of quantum dots layers separated by barrier layers. In this paper, each quantum dots layer mainly consists of many periodically distributed identical quantum dots with P the density QD , and the sheet density of doping donors equals P D . The lateral size of quantum dots aQD is supposed as large enough, so each quantum dot has a large number of bound states to accept more electrons. However, the transverse size of quantum dots is smaller than the distance between the quantum dots layers L to provide with the single energy level related to the quantization in the direction. The lateral distance between quantum dots is LQD qffiffiffiffiffiffiffiffiffiffi P which is equal to QD , hN k i is the average number of electrons in a quantum dot belonging to the k-th quantum dot layer, where k is within 1 K (K is the total number of quantum dots layer). Based on these assumptions, by considering the influence of the total electron transport on the current under dark conditions, we propose a device model of QDIP for estimating the performance of QDIP such as the dark current, the photocurrent and the detectivity.
1350-4495/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.infrared.2012.03.001
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Fig. 1. Schematic view of the QDIP layers structure, where the spheres represent quantum dots.
2.1. Dark current The dark current density can be obtained by counting the mobile carrier density in the barrier [11,12], which is shown as:
hjdark i ¼ ev n3D
ð1Þ
where e is the electron charge, v the drift velocity of electrons, and n3D the three-dimensional electron density which can be obtained by the following equation [11].
n3D ¼ 2
3=2 mb kT 2
2ph
Ea exp kT
ð2Þ
where mb is the effective mass of electron, k the Boltzmann constant, T the temperature, h the reduced Planck constant, and Ea the activation energy depending on the total electron transport. Since the present investigation indicates the simultaneous existence of the nanoscale transport and the microscale transport in the total electron transport process of the QDIP devices [13,14], the activation energy under the total electron transport process including the microscale and nanoscale electron transport Ea can be calculated as [13]:
Ea ¼ E0;micro expðE=E0 Þ þ E0;nano bE
ð3Þ
where E is an applied electric field intensity, E0,micro and E0,nano respectively represent the activation energies under the microscale and the nanoscale transport at zero bias (E = 0 kV/cm), E0 and b are the experimental fitting parameters, which represent the change rate of the activation energy under the microscale and the nanoscale electron transport mechanism with the electric field, respectively. Furthermore, it is very clear that Eq. (3) can be divided into two parts, E0,micro exp (E/E0) corresponds to the activation energy under the microscale transport mechanism which is thermal emission over the effective potential barrier [15], and E0,nano bE indicates the activation energy under the nanoscale transport mechanism which means the escaping from quantum dots related to tunnelling [16,17]. Substituting Eq. (3) into Eq. (2), then we can get the dark current density which can be shown by the following equation [14].
hjdark i ¼ 2ev
ered in the dark current calculation. On the other hand, since the dark current is also considered as the current, which is formed from the currents flow the punctures in the planar potential barriers formed by the charged quantum dots [18], it can be obtained with the consideration of the influence of electron transport on each current through the punctures. This dark current calculation and above proposed model not only essentially put emphasis on the electron transport on dark current, but both are the algorithms which mainly depend on the structure parameters of QDIP devices. Hence, in the same photodetector, it is via the current balance relation in dark conditions between the two dark current models that we can obtain the average number of electrons in a quantum dot and further estimate the photocurrent and the detectivity. On the basis of the theory that the dark current should depend on the electron transport through the punctures in the planar potential barriers formed by the charged quantum dots, the dark current density [18–20] of QDIP can also be calculated as:
hjdark i ¼ jmax
Z
QD
1
2
dr exp
0
euðhNk iÞ kT
ð5Þ
P where QD is the quantum dot density in a layer, Nk the average number of electrons in a quantum dot belonging to the k-th quantum dot layer, which is actually the same for all indexes (i.e. hN k i ¼ hNi), uðhN k iÞ is the potential distribution in the quantum dot layer as a function of the average number of electrons in each quantum dot, and jmax is the maximum current density, and can be estimated from Richardson-Dushman relation [19,21]:
jmax ¼ A T 2
ð6Þ
⁄
where A is the Richardson constant. After integration, an expression for the average the dark current density [19,22] can be obtained as:
hjdark i ¼ jdark
V þ V D ðhNi=NQD ÞV QD exp e hNi ðK þ 1ÞkT
H
ð7Þ
where
2
H¼
3=2 mb kT
P
p4 4
0 erf @0:47LQD
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P3=2 132 hNi QD A5
e0 er
e0 er kB T e2
qffiffiffiffiffiffiffiffiffiffi P
ð8Þ
QD
2
2ph E0;micro expðE=E0 Þ þ E0;nano bE exp kT
and these characteristic voltages can be calculated as:
ð4Þ
From the analysis above, in this dark current model, the influence of the total electron transport (including thermal emission over the effective barrier and the electron escaping from the quantum dots related to tunnelling) on the activation energy is consid-
V QD ¼
VD ¼
P e KðK þ 1Þ Lð1 #ÞNQD 2e0 er QD
P e KðK þ 1Þ L 2e0 er D
ð9Þ
ð10Þ
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pffiffiffi 0:72 2 ffiffiffiffiffiffiffiffiffiffi q #¼ P pKL QD
ð11Þ
In the above equations, K is the total number of the quantum P dot layers, L the distance between the quantum dot layers, D the sheet density of doping donors, NQD the maximum in the number of electrons which can occupy each quantum dot, LQD the lateral distance between quantum dots, V the applied voltage, and er the dielectric constant of the material used to fabricate quantum dots. Let Eq. (4) equal Eq. (7), we can get the current balance relationship under the dark conditions which can be shown as:
3=2 mb kB T
Ea;micro þ Ea;nano 2ev exp 2 kT 2ph H V þ V D ðhNi=NQD ÞV QD ¼ jdark exp e hNi ðK þ 1ÞkT
ð12Þ
By solving Eq. (12), we can get the average electrons numbers in a quantum dot hNi, further obtain new models for the photocurrent and detectivity of QDIP. 2.2. Photocurrent As well known to us, the photocurrent defined as the current of the detector under an illumination [23] can be determined by the following equation.
hjPhoto i ¼ eUs gg p
ð13Þ
where Us is the incident photo flux density on a detector, g the quantum efficiency, and gp the photoconductive gain which can be calculated by the ratio of the lifetime of carriers to the transit time of the carriers through the device. In QDIP devices, since the electron–electron scattering is main relaxation processing, the carriers lifetime becomes very long [24] and furthermore leads to a high photoconductive gain, the typical value of which is within 1– 106 [19]. Based on the relation between the quantum efficiency and the average electron number in a quantum dot, which can be shown P as: g ¼ dhNiK QD , the expression of the photocurrent can be written as:
hjPhoto i ¼ deg p hNi
P
Us K
ð14Þ
QD
where d is the electron capture cross section coefficient and it is adjusted to meet experimental comparison, the value of the incident photo flux density Us is supposed as 8 1017 photons/cm2 s [19]. 2.3. Detectivity According to the photoconductive detection mechanism, with a thermal noise ignored, the detectivity of QDIP can be determined as:
pffiffiffi Ri A D ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4eIdark g n
ð15Þ
where Ri is the current responsivity of QDIP, A the area of QDIP device, gn the noise gain of photodetectors, which approximately equals photoconductive gain gp in the quantum dot infrared photodetector, Idark is the dark current. Here, the current responsivity, which is the ratio of the detector photocurrent to the incident photopower, can be given as:
P deg p hNi QD K Ri ¼ hv
ð16Þ
where v is the frequency of an incident infrared light. Substituting Eqs. (4) and (16) into Eq. (15), we can ultimately obtain the detectivity which is shown as:
D ¼ hv
P deg p hNi K QD rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8g n e2 v
mb kT 2ph2
3=2
exp
ð17Þ
E0;micro expðE=E0 ÞþE0;nano bE kT
3. Simulation and data The dark current, photocurrent and detectivity of QDIP, which are shown as functions of the applied electric field and temperature, are discussed and analyzed in details in the section. These obtained results are compared with experimental results, which are transferred from the voltage coordinates into the electric field intensity coordinates, reported in literatures to show the validity of the proposed model. In our calculations, the drift velocity of electrons within 105 1.8 106 m/s [25] is supposed as the constant 8 105 m/s, the parameter mb is chosen as 0.34 me [26], the other structure parameters from the GaAs or InGaAs QDIP devices [19,20,27,28] are displayed in Table 1 and are adjusted to the same values as those of the QDIP devices used for the experimental verification. Fig. 2 presents the dark current density as functions of the applied electrical field and temperature. It is clear that these curves have the same characteristics in changing of the dark current density with electrical field (Fig. 2a) and temperature (Fig. 2b). The increase trend of the dark current density with the electric field is shown in Fig. 2a and it clearly proved that total electron transport including the nanoscale and microscale transport determines the dark current density of QDIP devices. As what is shown at the beginning of these curves in Fig. 2a, the dark current density changes rapidly with the increase of the applied electric field, which can be mainly ascribed to the contribution of the microscale electron transport on the dark current density, whereas the subsequent slow increase of the dark current density reveals the significant contribution of the nanoscale electron transport. More importantly, our theoretical results obviously show a good agreement with the previous published experimental results [15] plotted with black squares at 77 K in Fig. 2a, which well directly examines the validity of our dark current density model. But it can also be found that there is a slight deviation between the theoretical results and the experimental results. The reasons of the deviation are not hitherto clear, and it possibly is ascribed to the operation without consideration of the influence of the applied voltage on the parameters such as v, E0, b and g in our model. In addition, just as shown in Fig. 2a, the temperature also makes a great contribution to the dark current density in a small temperature range (23 K). For example, under the same electrical field of 5 kV/cm, the dark current density is only 1.03 107 A/cm2 at 77 K, and then rapidly reaches to 2.40 104 A/cm2 when the temperate is of 100 K, which is 3 orders of magnitude larger than that at 77 K. In order to make clear of this phenomenon, the influence of temperature on the dark current density at the electric field of 5 kV/cm is displayed in details in Fig. 2b. Obviously, the dark current density which exhibits exponential increase can be greatly
Table 1 Parameters in QDIP calculations. E0,micro
E0
L
K
34.6, 90 meV E0,nano 210, 224.7 meV
1.62, 3 kV/cm b 2, 2.79 meV cm/kV
30–40 nm P
10 P
QD
4 1010 cm2
NQD 8
D
0:3
P
QD
er 12
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323
Fig. 2. Dark current density as functions of the electric field and temperature with E0,micro = 34.6, 90 meV, E0 = 1.62, 3 kV/cm, E0,nano = 210, 224.7 meV, b = 2, 2.79 meV cm/kV. (a) Dark current density as a function of an applied electric field at 77 K and 100 K. (b) Dark current density as a function of temperature with different applied electric fields.
Fig. 3. Average number of electrons in a quantum dot as functions of electric field and temperature with E0,micro = 34.6 meV, E0 = 1.62 kV/cm, E0,nano = 224.7 meV, P b = 2.79 meV cm/kV, L = 31.5 nm, QD ¼ 0:4 1015 cm2 , NQD = 8. (a) Average number of electrons in a quantum dot as a function of the applied electric field at different temperatures. (b) Average number of electrons in a quantum dot as a function of temperature at the different applied electric fields.
affected by the temperature under the certain electric field. In addition, at the electrical field of 8 and 11 kV/cm, the curves of dark current density vs temperature are plotted with green and blue lines, respectively. It is clear that these curves have the same increase trend with the increase of the temperature, which can be ascribed to the following reasons: the increasing of temperature results in more and more electrons escaping from the quantum dot by the thermionic emission and further generates the increase of the dark current density. Fig. 2b also describes the dependence of the dark current density on the electric field. At the certain temperature, the higher the electrical field is, the larger the dark current density will be. In fact, when the applied field is increased, a greater band bending will result in reduction of the potential barrier, and thus more and more electrons will escapes from the quantum dot across the reduced potential barrier. As a result, the larger dark current density can be generated by the increase of electric field. From the analysis above, a conclusion can be drawn that the electrical field and the temperature both have a very great contribution to dark current density due to the reducing the height of potential barrier and enhancing thermionic emission, respectively. Fig. 2a also shows the dependence of the dark current density on the parameters E0,micro, E0,nano, E0 and b. The dark current density values at 100 K (corresponding to the curve 100 K) are used as the values of reference, the dark current density values will correspondingly change when the parameters E0,micro, E0,nano, E0 and b are changed, respectively. These changed dark current density val-
ues form into the curves E0,micro, E0,nano, E0 and b, respectively. From the difference between the curve E0,nano and the curve 100 K, it can be found that the dark current density values with E0,nano = 210 meV are very larger than those with E0,nano = 224.7 meV. This corresponding relationship between the small E0,nano and the large dark current density is mainly due to the inverse exponential relationship between the dark current density between E0,micro. For the similar reason, the dark current density values from the curve E0,micro with E0,micro = 90 meV also show the excepted decrease compared with the data in the curve 100 K. Moreover, Fig. 2a also indicates that the change of the dark current density with the parameters E0 and b, which are the change rate of the activation energy under the microscale and the nanoscale electron transport with the electric field. For instance, at the same electric field of 2 kV/cm, the dark current density is 3.40 105 A/cm2 with E0 = 1.62 kV/cm (in the curve 100 K), however, it becomes 1.39 105 A/cm2 (in the curve E0) when the parameters E0 is 3 kV/cm. The decrease of the dark current density with the increase of E0 demonstrates the influence of the microscale electron transport on the dark current density. Similarly, the increase of dark current density with the decrease of b with 5–12 kV/cm of the electric field intensity also shows the influence of the nanoscale electron transport on the dark current density. As a result, the parameters E0,micro, E0,nano, E0 and b, which are the same as the electric field and temperature, have a great effect on the dark current density calculation.
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Fig. 4. Photocurrent as functions of electric field and temperature with E0,micro = 34.6 meV, E0 = 1.62 kV/cm, E0,nano = 224.7 meV, b=2.79 meV cm/kV, P 15 cm2 , L = 31.5 nm, NQD = 8. (a) Photocurrent as a function of the QD ¼ 0:4 10 applied electric field at 60 K and 100 K. (b) Photocurrent as a function of temperature at the different applied electric fields.
With the consideration of the current balance relation under the dark condition, the average number of electrons in a quantum dot, the photocurrent and the detectivity under the different electrical field and temperature can be calculated out and simulated in Figs. 3–5. Here, Fig. 3a reveals the average number of electrons in a quantum dot increases with the increase of the electric field at the different temperature, which is similar to the increases of the dark current density in Fig. 2a. The reasons for the similar increase existence are as follows, when the electric field is very low, a large number of electrons are limited to the previous location because of high potential barriers, but with increase of electrical field, the potential barriers will be greatly reduced as a result of a greater band bending, and thus more and more electrons will be easily captured by quantum dots across the reduced potential barrier. In the end, the number of electrons in a quantum dot will be increased. In addition, Fig. 3a also presents the dependence of the average electron number in a quantum dot on the temperature. It can be found that, at the electrical field of 5 kV/cm, the average electron number in a quantum dot is 2.68 at 100 K and 2.83 at 60 K in Fig. 3a. At the same electrical field, the average electron number in a quantum dot at temperature within 60–110 K in Fig. 3b shows the similar decrease trend with the temperature. The decrease of the average electrons number in a quantum dot mainly results from the increase of electrons escaping out of quantum dots by thermionic emission with the enhanced temperature. Of course, it can be also noted from Fig. 3b that the higher the electrical field, the more the average electron number in a quantum dot at the certain temperature. The finding is also consistent with what is shown in Fig. 3a.
Fig. 5. Detectivity as functions of electric field and temperature with E0,micro = 34.6 P 15 meV, E0 = 1.62 kV/cm, E0,nano = 224.7 meV, b = 2.79 meV cm/kV, QD ¼ 0:4 10 2 cm , L = 36.6 nm, NQD = 8. (a) Detectivity as a function of applied electric field at 80 K and 100 K. (b) Detectivity as a function of temperature with different applied electric fields.
Fig. 4a presents the photocurrent values vs the applied electrical filed in the range of 5–12 kV/cm. Due to the interdependence of the photocurrent and the average number of electrons in a quantum dot from Eq. (14), the changes of the photocurrent with the electric field in Fig. 4a are similar to those results in Fig. 3a. Therefore, for the same reason, the change trends of the curves in Fig. 4b are similar to those in Fig. 3b. From the physical mechanism point of view, the increased electric field brings about the enhanced reduction of the potential barrier. More electrons possibly escape from the quantum dots over this reduced barrier, thus this behavior makes the photocurrent increase in the end. However, for the photocurrent declining with the temperature at a constant bias voltage, since the electrons in a quantum dot becomes lower in number with the temperature, and thus, fewer electrons in quantum dots can be excited from the quantum dots to form into the photocurrent when the incident infrared light irradiates on the activation region. As a result, the photocurrent will be decreased with the temperature. In addition, it is obvious in Fig. 4a that the experimental values at 100 K from the Ref. [27] agree with our calculated results, which demonstrated the correctness of our model again. From the above analysis, it is evident that these curves in Figs. 3 and 4 not only show the dependence of these parameters including the average number of electrons in a quantum dot and photocurrent on the electric field and the temperature, but also demonstrate the nature of correlation among these parameters. Fig. 5 illustrates the influence of electric field and temperature on the detectivity of QDIP devices. As observed in Fig. 5a, a theoretical evidence for the declined detectivity can support the fact that
H. Liu, J. Zhang / Infrared Physics & Technology 55 (2012) 320–325
the increase of the electric field intensity can greatly influence the performances of QDIP devices. The decrease trends of these detectivity curves in Fig. 5a can be ascribed to the increase of the dark current with enhanced electron emission as a result of reducing the height of potential barrier as the above analysis. Moreover, as exhibited in Fig. 5a, the detectivity values are demonstrated to show a good agreement with the experimental results at 80 K from the published literature [28] and the consistency gives a direct evidence to the validity of our detectivity model. Fig. 5b not only reveals the decreased detectivity with the increase of temperature, but also indicates the influence of the electric field on the detectivity. For example, at the same temperature of 80 K, the detectivity under the electrical field of 5 kV/cm is 8.60 109 cm Hz1/2/W, which is about 4 times as large as that at the electrical field of 11 kV/cm. More details of the changing trends of the detectivity with the electric field can be found in Fig. 5a. Fundamentally speaking, the increase of the dark current with the increase of the applied electric field intensity and the temperature ultimately results in the decrease of the detectivity for QDIP devices.
4. Conclusions In the paper, the device model of QDIP is proposed, which accounts for the influence of the total electron transport. Based on this model, main performance parameters such as the dark current, the photocurrent, the detectivity are calculated and simulated. The obtained results can well explain the strong dependence of the dark current and the other performances on the applied electric field and the temperature in the QDIP devices, and they show a good consistency with the published experimental results as expected, which further test and verify the accuracy of the proposed models. Although these models can be well used for estimating performances of QDIP and directing the design of QDIP devices, it can be found that they cannot apply to the conditions at too low voltage, the reasons leading to the phenomenon are not clear so far. Therefore, further research is required to better understand and make clear the dependence of our developed device model including the total electron transport on the very low electric field, and the corresponding experiments will be carried out in our lab. Acknowledgments This work was supported by the Fundamental Research Funds for the Central Universities (Grant Nos. K50510050001 and K50510050004); the National Science Foundation of China (NSFC, Grant No. 60777042); Zhejiang Provincial Science foundation (Grant No. Y4110529); and China Postdoctoral Science Foundation (Grant No. 20100481063). References [1] S.K. Das, T.D. Das, S. Dhar, M. de la Mare, A. Krier, Near infrared photoluminescence observed in dilute GaSbBi alloys grown by liquid phase epitaxy, Infrared Phys. Technol. 55 (2012) 156–160.
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