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多量子阱红外探测器的工作原理是基于子能级跃迁的,或者说带内跃迁,以此区别带间跃迁。要想在量子阱中形成子能级,量子阱的厚度必须要小于电子的波尔半径,电子此时受到量子限制作用,从而形成分立的子能级。这里,我们采用最简单的单量子阱和单电子模型来描述多量子阱红外探测器的工作原理。根据Fermi黄金定则,电子从能态i到能态f的跃迁几率可由以下公式表述:
$${W_{fi}} = \frac{{2\pi }}{\hbar }{\left| {\left\langle {{\psi _f}\left| {{H^\prime }} \right|{\psi _i}} \right\rangle } \right|^2}\delta \left( {{E_f} - {E_i} - \hbar \omega } \right)$$ (1) 式中:
${H^\prime }$ 是相互作用的哈密顿量;$\hbar \omega $ 是辐射能;$\psi $ 和$E$ 是初态(i)和终态(f)的波函数和本征能量。由于在带间跃迁时辐射波长远大于晶格周期,在子带跃迁时辐射波长大于势阱宽度,因此可以采用偶极近似:$${H^\prime } = \frac{{{q^2}F_0^2}}{{4m{ * ^2}{\omega ^2}}}(\vec \varepsilon \cdot \vec p)$$ (2) 式中:
$q$ 是元电荷;$F_0^{}$ 是电场振幅;$m * $ 是有效质量;$\vec \varepsilon $ 和$\vec p$ 分别是极化矢量和动量算符。将公式(2)代入公式(1),可以得到:$${W_{fi}} = \frac{{2\pi }}{\hbar }\frac{{{q^2}F_0^2}}{{4m{ * ^2}{\omega ^2}}}{\left| {\left\langle {{\psi _f}\left| {\vec \varepsilon \cdot \vec p} \right|{\psi _i}} \right\rangle } \right|^2}\delta \left( {{E_f} - {E_i} - \hbar \omega } \right)$$ (3) 应用包络函数形式,电子波函数
${\psi _i}(\vec r)$ 可以表示为周期性Bloch函数${u_\nu }(\vec r)$ 和缓慢变化的包络函数${f_n}(\vec r)$ 的乘积,其中$n$ 表示所研究体系的量子数。在假设所有组成材料的晶格周期函数都相同的条件下,可以得出仅用于包络函数的薛定谔方程:$$\frac{{ - \hbar }}{{2m * }}{\nabla ^2}{f_n}(\vec r) + V(\vec r){f_n}(\vec r) = {E_n}{f_n}(\vec r)$$ (4) 式中:
$V(\vec r)$ 是势能曲线;${E_n}$ 是能量本征值。在单量子阱中引入包络函数表达式:$${f_n}(\vec r) = \frac{1}{{\sqrt S }}\exp \left( {i{{\vec k}_ \bot } \cdot {{\vec r}_ \bot }} \right)\chi (z)$$ (5) 式中:
$S$ 是样本面积;$\chi ({\textit{z}})$ 是沿生长轴的包络函数分量;${\vec k_ \bot }$ 和${\vec r_ \bot }$ 表示二维(2D)矢量(kx,ky)和(x,y)。公式(4)的解推出以下形式的能量本征值:$${E_n}\left( {{k_ \bot }} \right) = {E_{n0}} + \frac{{{\hbar ^2}k_ \bot ^2}}{{2{m^ * }}}$$ (6) 其中子带能量
${E_{n0}}$ 取决于电势曲线$V({\textit{z}})$ 。对于具有无限深势垒的对称量子阱,可以得到以下本征值:$${E_n}\left( {{k_ \bot }} \right) = \frac{{{n^2}{\hbar ^2}k_ \bot ^2}}{{2m * {L^2}}} + \frac{{{\hbar ^2}k_ \bot ^2}}{{2{m^*}}}$$ (7) 其中L是势阱厚度。回到公式(3),矩阵元素
$\left\langle {{\psi _f}|\vec \varepsilon \cdot \vec p|{\psi _i}} \right\rangle $ 分解为:$$\left\langle {{\psi _f}|\vec \varepsilon \cdot \vec p|{\psi _i}} \right\rangle = \vec \varepsilon \cdot \left\langle {{u_\nu }|\vec p|{u_{{v^\prime }}}} \right\rangle \left\langle {{f_n}\mid {f_{{n^\prime }}}} \right\rangle + \vec \varepsilon \cdot \left\langle {{u_\nu }\mid {u_{{v^\prime }}}} \right\rangle \left\langle {{f_n}|\vec p|{f_{{n^\prime }}}} \right\rangle $$ (8) 式中:
$\nu $ 和$\nu '$ 以及$n$ 和$n'$ 分别是带间和子带终态和初始态的指数。第一项表示带间跃迁,在带内跃迁情况下此项为零。它由Bloch函数的偶极矩阵元(决定带间偏振选择规则)和包络函数的重叠积分组成,并决定了电子子能级和空穴子能级的跃迁选择定则。第二项表示子带跃迁过程,它由Bloch函数的重叠积分(当两个包络状态从同一能带中获取时,该积分不为零)以及包络函数的偶极矩阵元组成,该偶极矩阵元决定了子带跃迁的偏振选择定则:$$\left\langle {{f_n}|\vec \varepsilon \cdot \vec p|{f_{{n^\prime }}}} \right\rangle = \frac{1}{S}{\varepsilon _{\textit{z}}}\delta \left( {k_ \bot ^i - k_ \bot ^f} \right)\int {\rm{d}} {\textit{z}}\chi _n^*({\textit{z}}){p_{\textit{z}}}{\chi _{{n^\prime }}}({\textit{z}})$$ (9) 因此,只有电场的z分量能耦合到子带跃迁中,即量子阱结构只对横磁(TM)偏振光有响应,这就需要使用表面光栅或波导装置来将光耦合到有源区域中。
将公式(9)代入公式(3),并考虑到动量可以描述为
${\vec p_{nn'}} = im * \omega {\vec r_{nn'}}$ ,则子带跃迁几率可以改写为:$$\begin{split}{W_{fi}} = & \frac{{2\pi }}{\hbar }\frac{{{q^2}F_0^2}}{4}\varepsilon _{\textit{z}}^2{\left| {\left\langle {{\chi _f}({\textit{z}})|{\textit{z}}|{\chi _i}({\textit{z}})} \right\rangle } \right|^2} \times\\& \delta \left( {k_ \bot ^i - k_ \bot ^f} \right)\delta \left( {{E_f} - {E_i} - \hbar \omega } \right)\end{split}$$ (10) 态i到f之间的子带吸收系数通常由单位时间和体积所吸收的电磁能量
$\hbar \omega {W_{fi}}/V$ 与入射辐射强度的比值$I = \dfrac{1}{2}nc{\varepsilon _0}F_0^2$ 来定义,其中ε0是真空介电常数,n是材料的折射率,c是光速。$$\begin{split} \alpha (\omega ) = &\frac{{\pi {q^2}\left( {{E_f} - {E_i}} \right)}}{{nc{\varepsilon _0}\hbar V}} \times \varepsilon _{\textit{z}}^2\sum\limits_{i,f} 2 {\left| {\left\langle {{\chi _f}({\textit{z}})|{\textit{z}}|{\chi _i}({\textit{z}})} \right\rangle } \right|^2}\\& \left[ {f\left( {{E_i}} \right) - f\left( {{E_f}} \right)} \right]\delta \left( {{E_f} - {E_i} - \hbar \omega } \right) \end{split}$$ (11) 式中:
$f\left( E \right)$ 是费米函数。该求和可以表示为两个子带的载流子面密度之差$n_i^S - n_f^S$ :$$\sum\limits_{i,f} 2 \left[ {f\left( {{E_i}} \right) - f\left( {{E_f}} \right)} \right] = S\left( {n_i^S - n_f^S} \right)$$ (12) 式中:S是晶体表面。因此,子带跃迁吸收系数可以写为:
$$\alpha (\omega ) = \frac{{\pi \left( {{E_f} - {E_i}} \right)}}{{nc{\varepsilon _0}\hbar L}}\left( {n_i^S - n_f^S} \right)\varepsilon _z^2\mu _{if}^2\delta \left( {{E_f} - {E_i} - \hbar \omega } \right)$$ (13) 其中,
${u_{if}}{\rm{ = }}q\left\langle {{\chi _f}({\textit{z}})|{\textit{z}}|{\chi _i}({\textit{z}})} \right\rangle$ 是子带跃迁偶极矩阵元。偶极矩阵元仅涉及初始和最终子带的包络波函数。因此,由于z为奇数,在对称势阱中仅允许包络波函数具有相反奇偶性的子带之间的跃迁,但此选择规则不适用于非对称势能分布。
当给定势能
$V(\vec r)$ 分布,可以根据公式(4)求解得到量子阱中电子的包络波函数${f_n}(\vec r)$ 及其能量本征值${E_n}$ 。图1(a)为Kandaswamy等人计算得到的GaN/AlN多量子阱中电子和空穴的波函数及能级[15]。由于强的量子阱限制作用,电子和空穴在量子阱中形成了分立能级,能级间距可由势阱的厚度等参数自由调控。利用分立能级上电子或空穴的跃迁,可以实现对红外光的探测。图 1 (a)计算得到的GaN/AlN多量子阱能带图及其电子和空穴的波函数和能级,(b)计算得到的e2-e1和e3-e1跃迁能量与势阱厚度的关系,其中三角形为实验数据
Figure 1. The calculated wavefunction and energy level of electrons and holes iin GaN/AlN MQWs, (b) the calculated relationship of e2-e1 and e3-e1 transition energy and quantum well width, the triangle symbol is the experimental data
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摘要: 多量子阱红外探测器是一种新型的利用子带跃迁机制的探测器件,具有非常高的设计自由度。GaN/Al(Ga)N量子阱由于大的导带带阶,超快的电子驰豫时间,超宽的红外透明区域以及高的声子能量,使得其成为继GaAs量子阱红外探测器之后又一潜在的探测材料结构。文中详细综述了国内外关于GaN基量子阱红外子带吸收及其探测器件的研究进展。首先介绍了量子阱红外探测器的工作原理及其选择定则,接着从极性GaN基多量子阱、非极性或半极性GaN基多量子阱以及纳米线结构GaN基多量子阱三个方面回顾当前GaN基多量子阱红外吸收的一些重要研究进展,包括了从近红外到远红外甚至太赫兹波段范围的各种突破。最后回顾了GaN基多量子阱红外探测器件的研究进展,包括其光电响应特性和高频响应特性,并对其未来的发展进行总结和展望。Abstract: Quantum well infrared photodetector (QWIP) is a new device utilizing the intersubband transition in conduction band or valance band, which has a very high free degree of device design. Due to the large conduction band-offset, the ultrafast electron relax time, the ultra-wide infrared transparency and the high energy LO-phonon, the GaN/Al(Ga)N multi-quantum wells (MQWs) has become a potential candidate for the infrared detector since the GaAs based MQWs. In this paper, the research progresses of intersubband transition absorption (ISBT) and corresponding photoresponse of GaN based MQWs were systematically reviewed. First, the operation principle and the selection rule of the quantum well infrared photodetector was explained. Then, the main research work was introduced including the ISBT absorption of polar, nonpolar and nanowire GaN based MQWs, from the near infrared to far infrared, even the THz range. Finally, the progress of GaN based QWIP and quantum cascade detectors (QCD) was reviewed including the photofresponse and the frequency response of the device. A conclusion and perspective was presented for the future research in GaN based QWIP and QCDs.
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Key words:
- GaN /
- quantum well /
- infrared photodetector /
- intersubband transition absorption
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图 1 (a)计算得到的GaN/AlN多量子阱能带图及其电子和空穴的波函数和能级,(b)计算得到的e2-e1和e3-e1跃迁能量与势阱厚度的关系,其中三角形为实验数据
Figure 1. The calculated wavefunction and energy level of electrons and holes iin GaN/AlN MQWs, (b) the calculated relationship of e2-e1 and e3-e1 transition energy and quantum well width, the triangle symbol is the experimental data
图 2 (a) GaN/AlGaN单量子阱能带示意图,图中包含电子的第一能级e1和第二能级e2,(b)计算得到的子带吸收系数谱,case1、case2、case3、case4四个量子阱结构的极化电场分别为0、1.36、1和2.4 MV/cm
Figure 2. (a) Band energy diagram of GaN/AlGaN single quantum well, including the first energy level e1 and the second energy level e2, (b) the calculated intersubband absorption spectra of four different quantum wells, the polarization field in case1, case2, case3 and case4 quantum wells are 0, 1.36, 1 and 2.4 MV/cm, respectively
图 4 (a) 蓝宝石衬底上不同势阱厚度的GaN/AlN多量子阱中的子带跃迁吸收谱,(b)蓝宝石和硅衬底上不同势阱厚度的GaN/AlGaN多量子阱中的子带跃迁吸收谱
Figure 4. (a) The intersubband absorption spectra of GaN/AlN MQWs grown on sapphire with different quantum well widths, (b) intersubband absorption spectra of GaN/AlGaN MQWs grown on sapphire and Si substrates with different quantum well widths
图 7 (a)不同掺杂浓度的GaN/Al0.2Ga0.8N (3 nm/3 nm)多量子阱的子带吸收谱,(b)不同掺杂浓度的GaN/Al0.1Ga0.9N (7 nm/4 nm)多量子阱的子带吸收谱,(c)不同掺杂位置的GaN/AlN (1.5 nm/3 nm)多量子阱的子带吸收谱
Figure 7. (a) The intersubband absorption spectra of GaN/Al0.2Ga0.8N (3 nm/3 nm) MQWs with different doping densities, (b) intersubband absorption spectra of GaN/Al0.1Ga0.9N (7 nm/4 nm) MQWs with different doping densities, (c) intersubband absorption spectra of GaN/AlN (1.5 nm/3 nm) MQWs with different doping locations
图 10 (a)不同势阱厚度的c面、m面和a面GaN/AlN多量子阱中的短波红外子带吸收谱,(b)吸收峰与势阱厚度的关系,(c) m面GaN/AlGaN多量子阱中的中波红外子带吸收谱,(d)对应的c面GaN/AlGaN多量子阱中的中波红外子带吸收谱
Figure 10. (a) The intersubband short wavelength infrared absorption spectra of c, m and a plane GaN/AlN MQWs with different quantum well widths, (b) the relationship of QW with and absorption peak, (c) intersubband Mid-infrared absorption spectra of m plane GaN/AlGaN MQWs, (d) intersubband Mid-infrared absorption spectra of c plane GaN/AlGaN MQWs
图 11 (a)不同势阱厚度非极性面GaN/AlGaN量子阱中的太赫兹波段子带吸收谱,(b)不同Si掺杂浓度非极性面GaN/AlGaN量子阱中的太赫兹波段子带吸收谱,(c)不同Ge掺杂浓度非极性面GaN/AlGaN量子阱中的太赫兹波段子带吸收谱
Figure 11. (a) The intersubband THz absorption spectra of nonpolar GaN/AlGaN MQWs with different quantum well widths, (b) the intersubband THz absorption spectra of nonpolar GaN/AlGaN MQWs with different Si doping concentrations, (c) the intersubband THz absorption spectra of nonpolar GaN/AlGaN MQWs with different Ge doping concentrations
图 12 (a)和(b) GaN/AlN纳米线量子阱的TEM图,(c)不同Ge掺杂浓度的GaN/AlN纳米线量子阱的子带吸收谱,(d)不同势阱厚度的GaN/AlN纳米线量子阱的子带吸收谱
Figure 12. (a) and (b) The TEM image of GaN/AlN nanowire quantum wells, (c) the intersubband absorption spectra of GaN/AlN nanowire quantum wells with different Ge doping concentrations, (d) the intersubband absorption spectra of GaN/AlN nanowire quantum wells with different QW widths.
图 16 (a) GaN/AlGaN/AlN量子级联探测器的能带示意图,(b) GaN/AlGaN/AlN量子级联探测器的透射谱和光伏响应谱,(c) GaN/AlGaN/AlN量子级联探测器的频率响应谱,(d) GaN/AlGaN量子级联探测器的频率响应谱
Figure 16. (a) The energy band diagram of GaN/AlGaN/AlN quantum cascade detectors, (b) the transimission and photovoltaic response spectra of GaN/AlGaN/AlN quantum cascade detectors, (c) the frequency response spectra of GaN/AlGaN/AlN quantum cascade detectors, (d) the frequency response spectra of GaN/AlGaN quantum cascade detectors
图 17 (a)双阶梯型GaN/AlGaN量子阱太赫兹探测器的光电流响应谱,(b) m面InGaN/AlGaN量子阱长波红外探测器的光电流响应谱,(c) GaN/AlN纳米线量子阱近红外探测器的器件示意图及其量子阱能带示意图,(d) GaN/AlN纳米线量子阱近红外探测器的光电流响应谱
Figure 17. (a) The photocurrent response spectra of double step quantum well GaN/AlGaN MQWs THz detector, (b) the photocurrent response spectra of m-plane InGaN/AlGaN MQWs long wavelength infrared detector, (c) the schematic and energy band diagram of GaN/AlN nanowire quantum well near infrared detector, (d) photocurrent response spectra of GaN/AlN nanowire quantum well near infrared detector
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