氮化物半導體光學與電學特性之研究(II)Studies on the Electrical and Optical Properties of Nitride Semiconductors(II)

行政院國家科學委員會專題研究計畫 成果報告

氮化物半導體光學與電學特性之研究 (II)

中 華 民 國 95 年 10 月 31 日

一、摘要 我們系統性的研究長在矽基板上氮化銦鎵/氮化鎵半導體量子井結構之光學 與應變等物理性質。我們研究以不同緩衝層在有機金屬氣相沉積所成長樣品的光 學特性。為了克服晶格不匹配與熱膨脹係數不等所造成之高缺陷密度問題，我們 採用了一種高低溫交錯生長緩衝層的方法。藉由拉曼光譜發現伸張應力可以因緩 衝層的適當配置得到舒緩。對復合時間的研究顯示發光機制是由侷限激子所主導 此外，我們發現室溫的內在量子效率也能夠得到提升，我們的結果指出了這樣的 光學性能的改善是由於適當配置此種複合緩衝層所造成非輻射復合中心大量減 少所致。 我們應用微波調制的技巧來研究雙子帶佔據的氮化鋁鎵/氮化鎵異質結構之 傳輸特性。微波調制的技術加強了低磁場的細微量子震盪，使我們能從實驗中直 接能夠比較不同子帶的載子移動率。此外，這個技術可以幫助我們決定子帶能階 的距離，特別是當第二子帶的載子密度遠低於第一子帶的載子密度時會特別的好 用。我們研究了不同空乏層厚度的樣品並且求得了子帶能階的能量差。我們的結 果證明了這是一種研究雙子帶佔據的氮化鋁鎵/氮化鎵異質結構之強有力的工 具。 我們研究了砷化鋁鎵/砷化鎵的二維電子系統來探討由半古典傳輸到量子霍 爾效應的過渡過程。藉由對量子磁震盪的實驗研究，我們以實驗證實了即便縱向 磁阻已經接近零值，SdH 理論的描述依然能夠成立，這表示了目前學界 Coleridge 等人的看法是不完備的。SdH 理論適用範圍的擴張，我們認為是和無序度與溫度 引發的阻耗有關。此外，伴隨著理論適用範圍的擴張，我們發現了如同平台-平 台躍遷的行為。我們的研究指出引入正磁阻項對於改進 SdH 理論的描述非常重 要。 關鍵詞：半導體、量子井、光激螢光、傳輸特性

Abstract

We have systematically studied the optical and strain properties of InGaN/GaN quantum wells grown on silicon substrate. We study on the influence of buffer layer on the optical properties of InGaN/GaN multiple quantum wells (MQWs) grown on silicon substrates by metalorganic vapor phase epitaxy. To overcome the large lattice mismatch and difference in thermal expansion coefficient by which a high dislocation density occurs, the MQWs are grown on an advanced buffer structure consisting of low-temperature (LT) AlN and high-temperature (HT) AlN/AlGaN/GaN stack. The Raman spectra confirm that the biaxial tensile strain is reduced by the insertion of the alternating LT and HT buffer layers. The spectral dependence of the recombination lifetime indicates that the emission mechanism is due to localized excitons. Moreover, we found the room-temperature internal quantum efficiency can be improved. Our results suggest that the enhanced optical performance comes from the reduced nonradiative recombination centers brought about by the LT and HT composite buffer layers.

We apply the microwave-modulated technique to study the transport properties of two-subband-populated AlGaN/GaN heterostructures. The microwave modulation enhances the small Shubnikov-de Haas oscillations at low magnetic fields, providing a direct way to compare the mobilities of different subbands from the experimental data. In addition, this technique can help us to determine the subband-energy separation, especially when the population of the second subband is much lower than that of the first one. Variation of subband-energy separation due to different spacer thickness is obtained. Therefore, we showed a powerful way to probe parameters of two-subband-populated AlGaN/GaN heterostructures.

The crossover from the semiclassical transport to quantum Hall effect is studied by examining a two-dimensional electron system in an AlGaAs/GaAs heterostructure. By probing the magneto-oscillations, it is shown that the semiclassical Shubnikov-de Haas (SdH) formulation can be valid even when the minima of the longitudinal resistivity approach zero. The extension of the applicable range of the SdH theory could be due to the damping effects resulting from disorder and temperature. Moreover, we observed plateau-plateau transition like behavior with such an extension. From our study, it is important to include the positive magnetoresistance to refine the SdH theory.

二、前言 寬能隙三族氮化物半導體是目前極重要的光電半導體材料，因此引發了廣大 的重視。儘管已經有商業化的產品問世，但是對於他們的光學機制與缺陷模式、 還是有許多不清楚的地方。如何能解決晶格不匹配與熱膨脹係數之差異所造成高 缺陷密度的問題，一直是人們追求的目標，但到目前為止還沒有產生一個主流的 方式能夠優於其他的選擇，因此這類型的問題還有許多研究的空間可供發揮。 氮化鋁鎵/氮化鎵異質結構在高功率高電子移動率電晶體的發展上佔有重要 的地位，而對其子帶電子特性的實驗研究，是分析及最佳化其量子結構的重要工 作。一般的寬能隙三族氮化物異質結構，因為沒有最佳之基板成長，品質無法大 幅提高，因此傳統傳輸實驗方法受到大幅度的限制。因此我們要尋求一個改善的 方法來解決這個問題。 經過了許多年的研究，人們對半導體磁傳輸現象漸漸有了一些理解。在高場 極限有藍道量子化效應的影響，在低場極限則有半古典的描述模型，然而對於中 介場範圍的電子物理行為，尚缺乏一個深切的認識。這樣的理解是不完備的，尤 其是考慮到氮化物半導體異質結構的電子研究中，這個範圍是實驗中經常要處理 的部份。因此我們要更深入的來探討這個問題。 三、研究目的、文獻探討、研究方法、結果與討論 以下將報告內容分由三段描述。

GaN-based nitride semiconductors have played important roles in optoelectronic devices such as light-emitting diodes and laser diodes [1-6]. Due to the lack of suited homosubstrates, these devices are usually grown on heterosubstrates such as sapphire and SiC. Recently, GaN-based materials grown on silicon substrates have drawn a lot of attention [7-13]. Silicon offers the advantages of high crystalline quality, large wafer size and low cost. Moreover, this approach is a promising candidate for integration of GaN-based optoelectronics with Si-based electronics in the future. One of the major challenges for this approach is to overcome the large lattice mismatch and difference in thermal expansion coefficient by which a high dislocation density occurs. From the device point of view, defects induced by lattice imperfection and residual strain influence their optical performance by a large extent because they can act as nonrecombination centers (NRCs). Considerable efforts have been made to reduce the dislocation density and residual strain to improve the crystalline quality deposited on silicon substrates. Silicon delta-doping, epitaxial lateral overgrowth, and substrate engineering have been proposed to improve the crystalline growth [14-19]. In this paper, we report an approach based on an advanced buffer structure grown prior to the active layers. The influence of such buffer layers on the optical properties of InGaN/GaN multiple quantum wells (MQWs) grown on silicon substrates is investigated. We show that this approach is able to improve the optical efficiency and the enhanced optical performance is related to the reduction of strain and NRCs brought about by the buffer structures.

The samples used for our study are InGaN/GaN MQWs grown on 2-inch Si (111) substrates by metalorganic vapor phase epitaxy. Trimethylgallium, trimethyindium, trimethylaluminum, ammonia and SiH4 were used for the growth. The MQWs are

grown on an advanced buffer structure consisting of low-temperature (LT) AlN and high-temperature (HT) AlN/AlGaN/GaN stack. The LT layer serves to reduce the large strain while the HT layer lessens the difference in thermal expansion. For sample A, a 32 nm LT AlN nucleation layer was grown on the substrate. This was followed by HT 50 nm AlN and 500 nm GaN deposition. Then a second thin LT AlN nucleation layer and 600-nm-thick HT GaN:Si (nSi ~ 1019 cm-3) was grown on top of

the first HT layers. On these buffer structures, the main MQW structure consisting of ten-period 2-nm-thick In0.25Ga0.75N and 4-nm-thick GaN:Si (nSi ~ 1020 cm-3) was

grown psudomorphically. Finally, a 50 nm GaN were grown as a cap layer. Refined buffer was used for sample B, in which two HT stacks consist of HT 100 nm AlN, 100 nm Al0.3Ga0.7N, and 60 nm Al0.2Ga0.8N layers were inserted before the start of the

growth of each thick GaN layer in the above-mentioned structures. The Raman spectra were measured under 514.5 nm excitation and recorded by a Jobin-Yvon T64000 micro-Raman system under the z(x_)z backscattering configuration with the scattering light parallel to the c axis of epilayers. The laser light was focused to a 1 μm spot size. For the photoluminescence (PL) measurements, the sample was excited by a chopped He-Cd laser beam. The emission was collected and sent to a Jobin-Yvon Triax 550 monochromator and detected by a Hamamatsu R928 photomultiplier tube. For the time-resolved photoluminescence (TRPL) measurements, a pulsed diode laser operating at 405 nm was used as an excitation light source. The signal was analyzed by a PicoQuant time-correlated single photon counting system. Samples were mounted onto the cold head of a closed cycle refrigerator and cooled down to desired temperatures for measurements.

Figure 1 shows the micro-Raman spectra of our samples at room temperature. Both the E2-high and A1-Lo modes of GaN are observed. The Raman modes for

sample A is 563.73 cm-1, while for sample B, it is blue shifted to 564.5 cm-1.

Compared with the standard value of 567.5 cm-1 for bulk GaN [15,20], the E2-high

peaks for both samples both show redshift with respect to that of bulk GaN. Because the lattice and thermal mismatch, the GaN epilayer is subjected to biaxial tensile stress when grown pseudomorphically on silicon (111) substrates. The E2-high mode

in the Raman measurements is often used to probe quantitatively the in-plane strain since it is found that this mode is sensitive to biaxial stress in GaN layers [21]. The biaxial stress can be estimated according to the relation [22]

σ

ω

=

C

Δ

where Δω is the phonon peak shift, C is the stress coefficient and σ is the biaxial stress. By using the stress coefficient 4.3 cm-1/GPa [8,19,23], the calculated stress in

the underlying GaN layer is 0.877 GPa for sample A, and 0.698 GPa for sample B. A reduction of tensile stress by an amount of 0.179 GPa is observed in our study. Such estimation indicates that the insertion of additional HT stacks give rise to a significant reduction of the in-plane tensile stress in the epilayers. It is beneficial because the crack formation and threading dislocation density can be reduced in this way [24,25]. Figure 2 (a) and (b) display the PL spectra measured at 10 K for both samples. They show similar band profile and the main peak wavelength is 456 and 446 nm for sample A and B. With similar top quantum well structure, the peak shift can be largely related to the reduced strain in sample B revealed by Raman observation. To further understand the optical emission of our samples, we performed TRPL measurements. We found the PL decay in general can be fitted with exponential decay. The emission energy dependence of PL lifetime τPL at 10 K is shown in Fig. 2 as closed circles for both samples. It is found that the τPL values increase with decreasing photon energy for both samples, which is characteristic of the localized system. In addition, temperature-dependent TRPL measurements were performed. The

temperature dependence of τPL is shown in Fig. 3 (a) and (b) as closed diamonds. The decay of excitons consists of both radiative recombination and the transfer process to localized states. As temperature rises to a certain extent, the excitons delocalize with the aid of enhanced thermal energy and recombine via nonradiative centers quickly. Therefore the lifetime τPL drops, as shown in Fig. 3 (a) and (b) for both samples.

The temperature dependence of the integrated PL intensity of the main PL band for our samples is shown in Fig. 4. The general trend is that the integrated PL intensity stays near unity as temperature lower than 100 K because the PL is expected to be dominated by localized excitons in this temperature range. The integrated PL intensity then decrease with the increase of temperature due to exciton delocalization. We found that the integrated PL intensity for sample A drops more rapidly than that for sample B. Since the nonradiative recombination process is generally frozen at low temperature, the internal quantum efficiency ηint can be set to be unity. As temperature increases, the internal quantum efficiency ηint(T) drops due to the enhanced nonradiative recombination. At room temperature, the internal quantum efficiency ηint(RT) for sample A is 17 %, while for sample B, it is raised to 25 %. This efficiency is quite good even compared with those grown on conventional sapphire substrates.

The thermal quenching of the integrated PL intensity was fitted by using [26]

)]

/

exp(

1

/[

)

(

T

I

D

E

kT

I

=

+

−

where I(T) is the integrated PL intensity at T K, I0 is a scaling factor, D is a process

rate parameter, Ea is the activation energy, and k is the Boltzmann constant. The

dashed lines in Fig. 4 are the least square fit of data with Eq. (2). The fitted values of

Ea are 20 and 24 meV for sample A and B, respectively.

)

(

1

)

(

1

)

(

1

T

T

T

τ

τ

τ

=

+

where τrad(T) and τnonrad(T) are radiative and nonradiative lifetimes at T K, respectively. On the other hand, The internal quantum efficiency ηint(T) is related to the radiative lifetime τrad(T) and nonradiative lifetime τnonrad(T) by the equation [27]

)

(

)

(

)

(

)

(

T

T

T

T

τ

τ

τ

η

+

=

From the above equations, we can estimate the temperature dependence of radiative and nonradiative lifetimes in terms of τPL and ηint(T):

)

(

)

(

)

(

T

T

T

η

τ

τ

=

)

(

1

)

(

)

(

T

T

T

η

τ

τ

−

=

The extracted radiative and nonradiative lifetimes for both samples are shown in Fig. 3 (a) and (b) as closed circles and open triangles, respectively.

At low T, the PL lifetime is dominated by τrad, above a certain critical temperature

Tc, the dominance is taken over by τnonrad. For both samples, it can be seen that τrad is nearly constant below 100 K and increases almost linearly with T above 150 K, suggesting that the density of states from which the emission originates change from zero dimension to two dimension with increasing T [28]. Above 100 K, more nearly-free-excitons are trapped by the activated nonradiative channels, resulting in shorter τnonrad. We found the decrease of τnonrad is more rapid for sample A than that for sample B. Therefore, Tc increases from ~ 140 K for sample A to ~ 180 K for

sample B. This is in reasonable agreement with the activation energy of 20 and 24 meV derived from the thermal quenching in Eq. (2). It indicates that the decrease in lifetime is mainly due to an increased influence of nonrecombination process.

At room temperature, the excitons involved in the radiative recombination are delocalized due to thermalization and then trapped by NRCs. The τnonrad is influenced

by the defect density which provides sources for NRCs, and the ηint(RT) is also determined by the NRCs. Our Raman results indicate the strain and dislocation density can be reduced by the proper insertion of HT stacks. The improved ηint(RT) for sample B can be attributed to the reduced defect density and NRCs originated from dislocations, which is consistent with a recent report [29].

In conclusion, the influence of an advanced buffer layer on the optical properties of InGaN/GaN MQWs grown on silicon substrates is studied. From the Raman measurements, it is found that the insertion of proper composite HT stacks effectively relieves the strain. Hence the strain and dislocation density can be reduced by this way. In addition, the room-temperature internal quantum efficiency can be improved, which is attributed to the reduced density of NRCs by the composite buffer structure. Our results suggest that to improve the optical performance of GaN-based devices on silicon, it is important to reduce the strain which brings about a high dislocation density and nonrecombination centers.

This work is supported by the National Science Council of the Republic of China under grant no: NSC 94-2112-M-110-009.

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Figure captions

Figure 1: Room temperature micro-Raman spectra of our samples. The Raman modes for sample B show blueshift with respect to that for sample A.

Figure 2: (a) PL spectra and PL lifetime (closed circle) measured at 10 K for sample A. (b) Similar for sample B.

Figure 3: The temperature dependence of PL lifetime τPL (closed diamond), radiative lifetime τrad (closed circle), and nonradiative lifetime τnonrad (open triangle) values for (a) sample A and (b) sample B.

Figure 4: The temperature dependence of the integrated PL intensity of the main emission band for both samples. The constant I(T) value in the low temperature range is normalized to unity. The dashed lines are the least-squares fit of data with Eq. (2).

Figure 1: Room temperature micro-Raman spectra of our samples. The Raman modes for sample B show blueshift with respect to that for sample A.

Figure 2: (a) PL spectra and PL lifetime (closed circle) measured at 10 K for sample A. (b) Similar for sample B.

Figure 3: The temperature dependence of PL lifetime τPL (closed diamond), radiative lifetime τrad (closed circle), and nonradiative lifetime τnonrad (open triangle) values for (a) sample A and (b) sample B.

Figure 4: The temperature dependence of the integrated PL intensity of the main emission band for both samples. The constant I(T) value in the low temperature range is normalized to unity. The dashed lines are the least-squares fit of data with Eq. (2).

Two-subband-populated AlGaN / GaN heterostructures probed

by electrically detected and microwave-modulated

magnetotransport measurements

Department of Materials Science and Optoelectronic Engineering, National Sun Yat-sen University, Kaohsiung 804, Taiwan, Republic of China and Center for Nanoscience and Nanotechnology, National Sun Yat-sen University, Kaohsiung 804, Taiwan, Republic of China

C. F. Huang

National Measurement Laboratory, Center for Measurement Standards, Industrial Technology Research Institute, Hsinchu 300, Taiwan, Republic of China

Y. F. Chenb兲

Department of Physics, National Taiwan University, Taipei 106, Taiwan, Republic of China

共Received 5 March 2006; accepted 7 July 2006; published online 31 August 2006兲

The authors apply the microwave-modulated technique to study the transport properties of two-subband-populated AlGaN / GaN heterostructures. The microwave modulation enhances the small Shubnikov–de Haas oscillations at low magnetic fields, providing a direct way to compare the mobilities of different subbands from the experimental data. In addition, this technique can help us to determine the subband-energy separation, especially when the population of the second subband is much lower than that of the first one. Variation of subband-energy separation due to different spacer thicknesses is obtained. Therefore, the authors showed a powerful way to probe parameters of two-subband-populated AlGaN / GaN heterostructures. © 2006 American Institute of Physics. 关DOI:10.1063/1.2339030兴

Recently fundamental properties of AlGaN / GaN hetero-structures have received much attention for the application of heterostructure-field-effect transistors and quantum cascade lasers using intersubband transitions.1–4The strong polariza-tion and large conducpolariza-tion-band offset bring about a better carrier confinement at AlGaN / GaN interface than that at AlGaAs/ GaAs interface.3,4 It naturally leads to a recent re-search interest in high-carrier-density AlGaN / GaN hetero-structures in which multiple subbands were occupied.5–7To optimize the performance of devices with high electron den-sities, it is essential to be able to probe efficiently the sub-band properties of two-dimensional electron gas共2DEG兲 of high densities at AlGaN / GaN heterointerface. Shubnikov–de Haas共SdH兲 measurements are often employed to character-ize the properties of 2DEG at semiconductor heterointer-faces. As carriers populate to the second subband, the SdH pattern has double periodicity and the detailed analysis counts on the effectiveness of numerical methods. In samples with low second-subband carrier population, it is especially difficult to resolve the second periodicity because it is easily obscured by the floating background in the longitudinal re-sistance ␳xx共B兲 during fast Fourier transformation. To

over-come this difficulty, we study high-carrier-density 2DEG confined at AlGaN / GaN heterostructures by microwave-modulated magnetotransport technique. We show that the modulated patterns of SdH oscillations with multifrequency can be drastically enhanced by employing the microwave modulation technique, allowing accurate determination of many useful parameters. Hence we demonstrate a powerful way to study two-subband-populated 2DEG systems.

The samples are two modulation-doped AlGaN / GaN heterostructures grown by atmospheric pressure metal-organic chemical vapor deposition on the 共0001兲 sapphire substrates. A nucleation GaN buffer layer was deposited, fol-lowed by an unintentionally doped GaN 共i-GaN兲 layer of 2␮m thickness. The barrier layer is a Si-doped Al0.22Ga0.78N

共n-AlGaN兲 layer of thickness of 25 nm for sample A and 30 nm for sample B. The doping concentration is 1.2 ⫻1018_cm−3_{. The one-side doping results in the triangular}

confinement of carriers in the heterojunctions. Between the n-AlGaN and i-GaN layer, a 5-nm-thick unintentionally doped Al_0.22Ga_0.78N spacer was inserted for sample A to re-duce remote impurity scattering. The samples were placed inside a 6 T Oxford superconducting magnet and cooled by liquid helium.

In the microwave modulation technique, the signal is detected electrically in phase with the microwave modula-tion. We have shown that the mechanism for the enhanced sensitivity and resolution is attributed to the hot carrier effect induced by microwave absorption and the suppression of the nonoscillatory background.8,9 Moreover, this technique has the distinctiveness of unchanging carrier concentration, which is very important for studying our high-carrier-density 2DEG systems. The microwave modulation was provided by a Gigatronics GT 9000 S microwave sweeper and was guided to the sample surface by a coaxial cable. The shape of the microwave pulses is a sine wave and the frequency range can be 2 – 20 GHz. It is also possible to use other wave forms such as a square wave. The estimated microwave power den-sity for a 10 dBm surface irradiation is 0.125 W / cm2_.

Figure 1 shows the conventional SdH oscillations taken at 3.6 K for 共a兲 sample A and 共b兲 sample B. The strong double periodicity of the SdH oscillations is easily

recog-a兲_{Electronic mail: [email protected]} b兲_{Electronic mail: [email protected]}

nized for sample A, as shown in Fig. 1共a兲. The modulation doping and the strong polarization field give rise to a large amount of confined 2DEG in the heterointerface. The elec-trons start to populate the second subband, resulting in double periodicity. In sample B shown in Fig. 1共b兲, however, features of double subband occupancy are not directly iden-tified as clearly as in sample A, which can be attributed to the low second-subband population. It is straightforward to ob-tain the carrier concentrations corresponding to the first sub-band from high-frequency oscillations. The successive oscil-lation numbers as a function of inverse magnetic fields of the SdH oscillation minima for both samples are plotted in the insets of Fig. 1. The data can be described by the simple linear equation10

1 BN

= N e

␲បn+ C, 共1兲

where BNrepresents each magnetic field at successive

oscil-lation minimum, N is an integer, C is a constant, and n is the carrier concentration. The solid lines in both insets in Fig. 1 show the fitting to Eq.共1兲, corresponding to carrier densities of 8.48⫻1012_cm−2 _{for sample A and 9.6⫻10}12_cm−2 _for

sample B, respectively.

Figures 2共a兲 and 2共b兲 display the microwave-modulated SdH patterns at the same temperature under the modulation of a 3.7 GHz microwave radiation for sample A and sample B, respectively. The SdH patterns are considerably enhanced for both samples. Comparing Fig. 2共a兲 with Fig. 1共a兲, we can see that the nonoscillatory background of␳xxis suppressed

While in Fig. 1共a兲 the nonoscillatory background still distorts the SdH pattern even as SdH oscillations get stronger at high magnetic fields, there remains virtually no distortion origi-nated from the nonoscillatory background in Fig. 2共a兲. As opposed to the SdH oscillations, the nonoscillatory back-ground is not altered under microwave illumination.8 Since the detected signal is proportional to the change of␳xxunder

microwave modulation, this underlying background signal is effectively suppressed, and the remaining oscillatory part is enhanced. Therefore the sensitivity of the measurement can be considerably improved. In addition, it is quite amazing that the visible signal noise due to the electrical circuit is almost washed out by the microwave modulation technique, as shown in the low-field region in Fig. 2共a兲. This is a strong evidence of greatly enhanced sensitivity, and hence the onset of SdH oscillations is noticeably lowered. Furthermore, it is significant to note that the short-period oscillations die away at magnetic fields lower than 2.5 T, while the long-period oscillations persist. Therefore we provide direct experimental evidence that the 2DEG in the second subband has a higher mobility than that in the first subband. On the other hand, the resolution of the conventional SdH measurements under similar condition is not capable of comparing these oscilla-tions directly.

Figure 3共a兲 shows the detailed plot of the microwave-modulated magnetoresistance for sample A at lower netic fields. There is only one series of oscillations at mag-netic fields lower than 2.5 T and the SdH oscillations due to the second-subband occupation can still be resolved at 1.5 T. This is among the lowest magnetic field possible to resolve the SdH oscillations in AlxGa1−xN / GaN systems placed in

He4 bath refrigerators,5–15 and thus manifests the power of the microwave modulation technique. This technique there-fore enables us to separate the contribution of the second-subband population from the composite SdH pattern at low magnetic fields. Since the long-period oscillations can be clearly observed in the low-field region, the carrier

concen-FIG. 1.共Color online兲 Magnetoresistance of 2DEG as a function of mag-netic field at 3.6 K in共a兲 sample A and 共b兲 sample B, respectively. The insets in each plot show successive oscillation numbers共open circles兲 as a function of inverse magnetic fields of the SdH minimum.

FIG. 2.共Color online兲 Electrically detected microwave-modulated SdH os-cillations at a temperature of 3.6 K for共a兲 sample A and 共b兲 sample B. The double periodicity is clearly resolved.

out further numerical analysis. Figure 3共b兲 shows the succes-sive oscillation numbers as a function of inverse magnetic fields in the low-field region. From the fit to Eq. 共1兲, the second-subband carrier density is determined to be 1.78 ⫻1012_cm−2_{. Because our technique has the advantage of}

unchanging carrier concentration, the total 2DEG concentra-tion in sample A can be determined unambiguously to be 1.02⫻1013_cm−2_.

Due to the long-period modulation, the double periodic-ity in sample B is not straightforwardly recognizable in con-ventional measurement. But the modulated SdH pattern can be readily identifiable in microwave-modulated SdH oscilla-tions, as shown in Fig. 2共b兲. Such comparisons show the advantage of our approach. The second-subband carrier den-sity can then be estimated to be 4.8⫻1011_cm−2_{, which is}

shown in Fig. 3共c兲. Thus, the total 2DEG concentration that resides in sample B is 1.01⫻1013_cm−2_{, which is very close}

to that for sample A. This example demonstrates that our technique is especially useful when there is a subband with low carrier concentration, which often brings about uncer-tainty and difficulty in numerical analysis. Once the carrier concentrations can be unambiguously determined, the sub-band separation can be inferred from the respective carrier concentration, and hence Fermi levels,

EF0= EF1+ E01=

␲ប2

m* n0=

␲ប2

m* n1+ E01, 共2兲

where m*_/␲ប2 _{is the two-dimensional density of state, and}

ond subbands, respectively. E01 is the energy separation

be-tween the first and second subbands, and EF0and EF1are the

energy differences between the Fermi levels and the minima of the first and second subbands, respectively. The effective mass for the two subbands is assumed to be about the same,4 m*_{= 0.23m}

0, and the energy separation between the two

sub-bands can be determined to be 69.6 and 94.6 meV for sample A and sample B, respectively. Our result shows that the subband-energy separation decreases as the spacer layer thickness increases from 0 to 5 nm. The subband-energy separation depends on the quantum confinement in the trian-gular potential well, which is determined by the Coulomb interaction. With the increase of the spacer layer thickness, the Coulomb interaction decreases, and the band bending also decreases. As a consequence, the quantum confinement reduces and the energy separation of the two subbands de-creases.

In conclusion, we demonstrated that the microwave-modulated measurement is a powerful technique to probe two-subband-populated 2DEG confined at AlGaN / GaN het-erostructures. The measured SdH spectra possess higher resolution and enhanced pattern, and merely requires direct data analysis without numerical artifacts, which provides an excellent opportunity to determine many important physical parameters accurately.

This work was supported by the National Science Coun-cil of Taiwan and the Ministry of Education of the Republic of China. One of the authors共D.R.H.兲 acknowledges finan-cial support from Aim for the Top University Plan, Taiwan.

1_{J. D. Heber, C. Gmachl, H. M. Ng, and A. Y. Cho, Appl. Phys. Lett. 81,}

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4_{D. R. Hang, C. H. Chen, Y. F. Chen, H. X. Jiang, and J. Y. Lin, J. Appl.}

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Zheng, S. L. Guo, Y. Shi, P. Han, Y. D. Zheng, T. Someya, and Y. Ar-akawa, Phys. Rev. B 62, R7739共2000兲.

6_{C. P. Jiang, S. L. Guo, Z. M. Huang, J. Yu, Y. S. Gui, G. Z. Zheng, J. H.}

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Chen, G.-H. Kim, J.-H. Lee, and J.-H. Lee, J. Appl. Phys. 93, 2055 共2003兲.

9_{D. R. Hang, J.-R. Juang, T.-Y. Huang, C.-T. Liang, W. K. Hung, Y. F.}

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P. Smorchkova, B. Heying, E. Haus, P. Fini, J. P. Ibbetson, S. Keller, P. M. Petroff, S. P. DenBaars, U. K. Mishra, and J. S. Speck, J. Appl. Phys. 87, 369共2000兲.

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FIG. 3.共Color online兲 共a兲 Microwave-modulated SdH oscillations at mag-netic fields lower than 2.5 T, in which only one oscillation frequency per-sists. The threshold magnetic field for SdH oscillations is greatly lowered. 共b兲 Successive oscillation numbers 共open circles兲 as a function of inverse magnetic fields of the SdH minimum for sample A. The solid line shows the fitting to Eq.共1兲. 共c兲 Similar for sample B.

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www.elsevier.com/locate/ssc

From semiclassical transport to quantum Hall effect under low-field

Landau quantization

D.R. Hanga,b,∗, C.F. Huangc, Y.W. Zhangd, H.D. Yehc, J.C. Hsiaoc, H.L. Pangc

a_{Department of Materials Science and Optoelectronic Engineering, National Sun Yat-sen University, Kaohsiung 804, Taiwan, ROC} b_{Center for Nanoscience and Nanotechnology, National Sun Yat-sen University, Kaohsiung 804, Taiwan, ROC}

c_{National Measurement Laboratory, Center for Measurement Standards, Industrial Technology Research Institute, Hsinchu 300, Taiwan, ROC} d_{Institute of Materials Science and Engineering, National Sun Yat-sen University, Kaohsiung 804, Taiwan, ROC}

Received 8 April 2006; received in revised form 8 August 2006; accepted 25 September 2006 by V. Pellegrini

Abstract

The crossover from the semiclassical transport to the quantum Hall effect is studied by examining a two-dimensional electron system in an AlGaAs/GaAs heterostructure. By probing the magneto-oscillations, it is shown that the semiclassical Shubnikov–de Haas (SdH) formulation can be valid even when the minima of the longitudinal resistivity approach zero. The extension of the applicable range of the SdH theory could be due to the damping effects resulting from disorder and temperature. Moreover, we observed plateau–plateau transition- like behavior with such an extension. From our study, it is important to include positive magnetoresistance to refine the SdH theory.

c

2006 Published by Elsevier Ltd PACS:73.40.-c; 73.43.-f

Keywords:A. Heterojunctions; D. Electronic transport; D. Quantum Hall effect; D. Phase transitions

Considerable efforts have been made to understand how

1

Landau quantization affects the magnetotransport properties of

2

two-dimensional electron systems (2DESs) under a

perpendic-3

ular magnetic field B. It is well known that Landau

quantiza-4

tion can modulate the density of states and induce

magneto-5

oscillations, which are periodic with respect to the inverse of

6

B. The two-dimensional Shubnikov–de Haas (SdH) theory,

de-7

rived from a semiclassical approach, acts as the conventional

8

tool to describe the low-field oscillations [1–5]. In practice, the

9

analysis of low-field oscillations provides a common way to

10

obtain three basic parameters of a 2DES, the carrier

concentra-11

tion, the quantum mobility, and the effective mass. In contrast,

12

to explain the integer quantum Hall effect (IQHE) appearing at

13

higher B, we shall consider quantum localization effects [1,6–

14

9]. In the IQHE regime, there are a series of quantum Hall states

15

characterized by quantized Hall plateaux and zero

longitudi-16

∗_{Corresponding author at: Department of Materials Science and} Optoelec-tronic Engineering, National Sun Yat-sen University, Kaohsiung 804, Taiwan, ROC. Tel.: +886 7 5252000x4066; fax: +886 7 5254099.

E-mail address:[email protected](D.R. Hang).

nal resistivity. The magnetic-field-induced phase transitions in 17

the IQHE provide good examples of continuous quantum phase 18

transitions [10,11]. Universal properties based on the scaling 19

theory and the modular symmetry have been investigated in the 20

phase transitions in the IQHE [12–15]. While the universalities 21

can be broken because of some unexpected factors, it has been 22

shown that features of the scaling theory and modular symme- 23

try can still be found after suitable analysis [16]. 24

It should be noted that the quantum localization effects are 25

also important even as B approaches zero in the standard theory 26

of the IQHE [7]. The low-field insulator induced by such effects 27

has been observed in 2DESs with large disorder [17–19]. For a 28

typical 2DES, in reality, the localization length becomes much 29

larger than the realistic sample size with decreasing B [7, 30

17]. In this way the semiclassical SdH theory, in which the 31

localization effects are ignored, remains valid at low B for most 32

2DESs while quantum localization effects are still important 33

to the appearance of the IQHE at high B [1]. Therefore, as 34

the magnetic field B is increased, the crossovers from classical 35

(semiclassical) transport to IQHE are expected in the low-field 36

Landau quantization for a wide range of disorders. 37 0038-1098/$ - see front matter c 2006 Published by Elsevier Ltd

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Fig. 1. The longitudinal and Hall resisitivity at the temperature 4.2 K. We can observe Shubnikov–de Haas oscillations at lower magnetic fields and quantum Hall plateaux at higher magnetic fields.

Although successful theories have been developed to

1

understand the magnetotransport properties of 2DESs, there

2

still exist many unresolved questions and ambiguities. At high

3

B, an experimental study inconsistent with the scaling theory

4

has been reported [20]. On the other hand, at low B there are

5

also debates on the validity of the Lifshitz–Kosevich formula,

6

originally derived for the three-dimensional Fermi liquid, to

7

be applied in the two-dimensional SdH theory [4,5,21]. It has

8

been suggested theoretically, in Ref. [4], that the damping

9

due to disorder and temperature is important in applying

10

such a formula to two-dimensional cases. Moreover, while

11

band parameter-like effective mass derived from

magneto-12

oscillations is taken conventionally as a constant, the field

13

dependence has been studied by several groups [22–24]. Our

14

group has also reported the enhancement of the effective mass

15

of a two-dimensional GaN electron system with increasing

16

B [22]. Therefore, more experimental investigations are

17

necessary to probe the SdH theory. To understand further

18

the exact behavior in the crossover from the semiclassical

19

to the IQHE regime, we look into the quantum

magneto-20

oscillations and low-field IQHE in a 2DES in an AlGaAs/GaAs

21

heterostructure. The effective mass m∗ is a well-established

22

constant in a two-dimensional GaAs electron system when

23

carrier concentration lies within the typical range. Therefore,

24

we can probe the applicable range of SdH formulation in such

25

a system by examining the value of m∗ while for some other

26

materials it is appropriate to investigate the meaning of band

27

parameters by using this formulation. We show that such a

28

semiclassical formulation can be valid even when the minima

29

of the longitudinal resistivity approach zero. In addition, the

30

positive magnetoresistance should be taken into account to

31

refine the SdH formulation.

32

The sample used for our study is an AlGaAs/GaAs

33

heterostructure, in which a 2DES resides in the GaAs side of

34

the heterojunction. The two-dimensional channel was followed

35

by a 15 nm spacer layer of AlGaAs, a 40 nm layer of

36

AlGaAs doped with Si at 1 × 1018 cm−3 and a 12 nm

37

GaAs cap layer doped at 1 × 1018 cm−3. A Hall pattern

38

was made by the standard lithography and etching process.

39

Fig. 2. The temperature dependence of longitudinal resistivity. The amplitudes of the oscillations are damped when the temperature is increased. Analysis of the amplitudes of quantum magneto-oscillations is done betweenν = 29 and ν = 9, as is indicated by the arrows. The dashed line in the inset shows the nonoscillatory background at T = 1.9 K obtained by averaging the magneto-oscillations.

Magnetotransport measurements were done with a 12 T 40

superconducting magnet and a He4 refrigerator.Fig. 1 shows 41

the four-terminal longitudinal and Hall magnetoresistivityρxx 42

andρxyat temperature T = 4.2 K. When a small perpendicular 43

magnetic field is applied, the 2DES is governed by the classical 44

transport theory, so we observeρxx∼constant andρxy=B/ne. 45

Here n is the carrier concentration and e is the electronic charge. 46

With gradually increasing magnetic fields, the 2DES enters the 47

semiclassical regime. A set of magneto-oscillations, commonly 48

known as SdH oscillations, whose periodicity is determined 49

by the two-dimensional carrier density, can be observed. The 50

two-dimensional carrier concentration can be obtained from the 51

low-field oscillations to be 3.17 × 1011 cm−2. The classical 52

mobilityµcis estimated to be 5.3×105cm2/V s from ρxx(B = 53

0) = 1/neµc. As shown inFig. 1, at higher magnetic fields, in 54

which the 2DES is in the strong localization regime, we can 55

observe well-developed quantum Hall states withρxx →0 and 56

ρxy=h/νe2and of filling factors down toν = 2. 57

The low-field oscillation amplitude1ρxx at finite tempera- 58

tures obtained from semiclassical SdH theory is given by [2] 59

1ρxx(B, T ) = 4ρ0D(m∗, T ) exp  − π µqB  , (1) 60

where ρ0 is a constant, µq is the quantum mobility, 61

the temperature factor D(m∗_{, T ) = χ/ sinh χ, χ =} 62

2π2kBm∗T/¯heB, ¯h is the reduced Planck constant, kB is the 63

Boltzmann constant, and m∗ is the electron effective mass. 64

This equation is expected to hold true for small magneto- 65

oscillations before well-developed quantum Hall states and zero 66

longitudinal resistivity appear with increasing B. In addition, 67

the constant ρ0 is expected to be the zero-field longitudinal 68

resistivity ρxx(B = 0) although there are reports on the 69

deviations [2]. 70

The detailed temperature dependence of low-field magneto- 71

oscillations in ρxx is shown in Fig. 2. With increasing 72

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Fig. 3. (a) According to the semiclassical theory, the effective mass can be extracted by plotting ln(1ρxx/T ) versus T ; (b) the obtained values are close to the expected value 0.067m0not only in the initial small SdH oscillations, but also in the low-field IQHE where fixed points in Hall resistivities are observed.

relatively high temperatures whereχ > 1, Eq.(1)can be further

1 simplified to yield: 2 ln 1ρ xx(B, T ) T  =C1− 2π2kBm∗ ¯ he B T, (2) 3

where C1 is a constant. This equation provides a powerful 4

way to obtain the carrier effective mass in magnetotransport

5

measurements according to the semiclassical SdH theory.

6

Asν < 9, as shown in the inset ofFig. 2, the spin-splitting

7

is resolved and the enhancement of exchange spin gaps should

8

be considered [25]. To focus on the range of the semiclassical

9

transport theory, we have analyzed the oscillating amplitudes

10

of filling factors from 29 to 9, where the spin enhancement

11

effect can be ignored.Fig. 3(a) shows the fitting for the effective

12

mass using Eq. (2) for two filling factors, 29 and 9, which

13

correspond to the boundary of the analyzed filling-factor region.

14

Atν = 29, where the oscillation amplitude is reasonably small

15

to follow the SdH theory, an effective mass value of 0.069m0 16

is found, which is in good agreement with the expected value

17

0.067m0. When the filling factor lowers as a consequence of 18

increasing the magnetic field, the minima ofρxxapproach zero 19

as T decreases forν < 14, where quantized plateaux can be

20

observed in ρxy. Therefore for ν < 14 the theory for IQHE 21

should be considered since it is expected to go beyond the scope

22

Fig. 4. The inset shows ln[1ρxx/ρxx(B = 0)D(m∗, T )] as a function of inverse magnetic field, from which the quantum mobility can be obtained. The plot of1ρxx/4ρ0exp(−π/µqB) with respect to T/B can be done accordingly for various fixed temperatures. The symbols squares, circles, up triangles, down triangles and diamonds are for the points at T = 1.9, 2.6, 3.2, 3.7 and 4.2 K, respectively. The solid symbols for each temperature stand for conditions as B > 1 T where minima of ρxxapproach zero. The numerical evaluation of D(m∗, T ) = χ/ sinh χ as a function of (T/B) is shown as the solid line.

of the SdH formulation. However, we can see inFig. 3(a) that 23

Eq. (2) is still valid at ν = 9 with m∗ = 0.0693m0, which 24

is close to 0.067m0. To see whether it is just a coincidence, 25

inFig. 3(b) we show the complete result of the effective mass 26

value based on Eq.(2). It is found that the obtained effective 27

masses throughout the region investigated possess a striking 28

consistency with an averaged value of 0.0688m0. The effective 29

mass value fluctuates within an extent of only 0.8% even when 30

the IQHE appears at 8 < ν < 14. The obtained result 31

indicates that the precision of effective mass obtained by the 32

semiclassical formula remains good even as minima ofρxx→0 33

andρxy=h/νe2. 34

From Eq.(1), we have 35

ln  1ρ xx ρxx(B = 0)D(m∗, T )  =C2− π µq 1 B, (3) 36 where C2is a constant. The inset ofFig. 4 shows the plot of 37

ln[1ρxx/ρxx(B = 0)D(m∗, T )] versus 1/B, obtained with the 38

averaged effective mass value. We obtain a quantum mobility 39

µq ∼5.38 × 104cm2/V s from the slope of the graph and the 40

constant C2 equals 1.55 which is close to the expected value 41

ln 4. 42

To examine further the SdH theory, we proceed to rearrange 43

Eq.(1)as 44

1ρxx(B)

4ρ0exp(−π/µqB)

=D(m∗, T ). (4) 45

In this form, the comparison can be checked easily by plotting 46

[1ρ_xx/4ρ₀exp(−π/µ_qB)] with respect to T/B, as shown in 47

Fig. 4. If Eq. (1) is a valid description, according to Eq.(4), 48

the amplitudes taken with different T/B should collapse on 49

the solid line given by calculated D(m∗_{, T ) with the obtained} 50

parameters. It is found that the entire data points coincide with 51

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minima ofρxx approach zero with decreasing T , as shown by 1

solid symbols, the temperature factor is still followed well.

2

The agreement with Eq.(4)throughout the region investigated

3

indicates that the semiclassical SdH formula remains a good

4

description in the low-field IQHE in our study.

5

In the conventional SdH theory, Eq.(1)is obtained for

low-6

field situations where the oscillations of the density of states1g

7

are much smaller than the zero-field density of states [6]. As

8

the minima ofρxxapproach zero,1g is no longer small and the 9

quantum diffusion model is considered for high-mobility

sam-10

ples at low temperatures where Eq.(1)is invalid [6,26].

How-11

ever, the applicable range of semiclassical SdH theory could

12

expand under the damping effects due to temperature and

dis-13

order [4]. It is shown that such effects play important roles in

14

transferring the three-dimensional Lifshitz–Kosevich formula

15

to the two-dimensional SdH theory [4]. In fact, different

mech-16

anisms are responsible under different types of disorder [27].

17

We note that the mobility of our sample is lower than that in

18

Ref. [26], and the experiments are performed at high

tempera-19

tures whereχ > 1. Our study reveals the extension of

semiclas-20

sical SdH theory to the IQHE under strong damping effects.

21

In the conventional SdH theory, we have

22

ρxx∼ρxx(B = 0) + 1ρxx (5) 23

as higher-order terms are ignored. Because ρxx ≥ 0, the 24

violation of SdH formula can be expected when the oscillation

25

term 1ρxx > ρxx(B = 0). However, in our study, Eq. (1) 26

holds true even when 1ρxx becomes larger than ρxx(B = 27

0) with decreasing T as ν < 16. At T = 1.9 K, as

28

indicated by the dashed line in the inset of Fig. 2, we have

29

an apparent positive magnetoresistance as the nonoscillatory

30

background after averaging the magneto-oscillations. Because

31

of such a nonoscillatory part, Eq. (1) remains true when

32

1ρxx > ρxx(B = 0) without inducing the negative 33

resistivity. Different mechanisms have been introduced for

34

the positive magnetoresistance [28–30], and our study shows

35

the importance of extending the applicable range of the SdH

36

formula.

37

To study further the crossover from the semiclassical

38

transport regime to IQHE, we also investigate the behaviors

39

of ρxy when the semiclassical SdH formula Eq. (1) is valid. 40

Fig. 5 shows the temperature dependence of ρxy at B ∼ 41

1 T, near which the positive magnetoresistance is important

42

for the extension of the SdH formula for1ρxx as mentioned 43

above. In addition to the Hall plateaux, there is a

temperature-44

independent point at B = 1.022 T between the plateaux of

45

the high filling factorν = 14 and 12. Similar

temperature-46

independent points exist between the plateauxν = 12 and 10,

47

andν = 10 and 8 as well. Temperature-independent points

48

are expected in both ρxx and ρxy at critical magnetic fields 49

of plateau transitions under high-field quantum localization,

50

which is important to the standard theories for IQHE [7,

51

31]. In our study, however, such points appear only in ρxy. 52

It has been reported that quantum localization leading to

53

IQHE is more robust in ρxy than in ρxx [1]. By inverting 54

the corresponding conductivities [13,32], the horizontal dash

55

line inFig. 5 indicates the expected T -independent position,

56

Fig. 5. Between the plateaux of the high filling factorν = 14 and 12, we observed the temperature-independent point inρxy, as shown by the arrow. Such a point is close to the expected universal value indicated by the horizontal dash line.

which deviates a little from the experimental one. Note that the 57

deviation may exist even under the high-field localization [7, 58

13]. Alternatively, the features of IQHE inρxycan be explained 59

by fixing the chemical potential without considering quantum 60

localization [33,34]. Therefore, more studies are necessary to 61

clarify the origins of T -independent points between adjacent 62

Hall plateaux when1ρxxfollows the SdH formula. 63

To conclude, we report magnetotransport measurement on 64

the 2DES in an AlGaAs/GaAs heterostructure to study the 65

crossover from semiclassical transport to strong localization. 66

Both the longitudinal and Hall resistivities are investigated 67

in such a crossover. While fixed points appear in ρxy with 68

increasing B, we found that semiclassical SdH formula is still 69

valid for the magneto-oscillations in ρxx. Such a formula, in 70

fact, survives even when the minima of ρxx approach zero at 71

low temperature. The extension of the applicable range of the 72

SdH theory could be due to the damping effects resulting from 73

disorder and temperature. It is clear that we should incorporate 74

the positive magnetoresistance to refine the SdH formula, and 75

the suggestion is that more studies are required to explain the 76

coexistence of plateau–plateau transition-like behavior and the 77

semiclassical SdH formula. 78

Acknowledgments 79

This work is supported by the National Science Council of 80

the Republic of China under grant no: NSC 94-2112-M-110- 81

009. D.R. Hang acknowledges financial support from ACORC 82

and Aim for the Top University Plan of National Sun Yat-sen 83

University, Taiwan. 84

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四、計畫成果自評 2006 年的成果良好，2006 至少在氮化物半導體光學與低維度電子物理等的 題目上已經發表了兩篇國際期刊論文1,2，兩篇審查中之國際期刊論文3,4以及八 篇會議論文5-12，我們也將持續整理研究成果陸續投稿。同時我們的研究生在半 導體光學與電學的研究學習上也得到充分而良好的訓練。 五、參考文獻

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