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中国科学院物理研究所 武建劳
2016年11月23日
瞄准世界范围高端作者群开展工作，
提高刊物国际影响力
中科院物理所主办４个刊物
第一主办单位: 中国科学院物理研究所
第二主办单位: 中国物理学会
主管: 中国科学院
发表文章和全文下载数量
2015年发表论文:
《物理学报》来稿3884篇 发表1325篇
《中国物理B》来稿2972篇 发表1161篇
《中国物理快报》来稿1498篇 发表540篇
《物理》来稿293篇发表185篇 (组稿率97%)
总计 来稿8647篇 发表3211篇
2015年编辑部网站全文下载:
《物理学报》504万篇次
《中国物理B》 228万篇次
《中国物理快报》121万篇次
《物理》219万篇次
总计全文下载1072万篇次
我国物理论文统计分析以及我们刊物的困难
影响档 很弱 弱 中 强 很强 极强 超强 论文总数
总被引次数0-2 3-9 10-29 30-99 100-299 300-999 >=1000
2006 14032 10274 5739 1856 247 20 3 14032
2007 16329 11790 6343 1946 213 19 0 16329
2008 18202 12881 6615 1829 195 29 2 18202
2009 18941 13153 6350 1500 131 16 1 18941
2010 19921 13284 6002 1375 147 26 1 19921
2011 22038 14315 6000 1246 162 19 2 22038
2012 24026 14531 5488 997 121 19 1 24026
2013 27629 15200 4931 868 102 4 0 27629
2014 29308 13196 3303 510 47 0 0 29308
2015 30596 7450 1256 120 7 1 0 30596
Company name
www.themegallery.com
被引次数门槛值Phys 0.01% 0.10% 1.00% 10.00%20.00%50.00%
2006 2546 552 161 41 23 8
2007 1360 506 153 39 23 8
2008 2245 559 152 37 22 7
2009 1996 439 133 34 20 7
2010 998 407 121 31 18 7
2011 725 305 98 26 16 6
2012 816 276 81 22 13 5
2013 375 182 59 17 10 4
2014 328 111 37 11 7 3
2015 151 43 16 5 3 1
Company name
www.themegallery.com
中国作者（包括港澳作者）独立发表的物理论文在各个影响力区间的分布
P CN 0.01% 0.10% 1.00% 10.00%20.00%50.00%
2006 0 8 83 1082 2379 5946
2007 0 1 81 1165 2441 6484
2008 0 7 89 1193 2528 7493
2009 0 5 78 1032 2442 7285
2010 1 15 98 1108 2543 6970
2011 2 19 163 1398 2875 8034
2012 1 21 169 1547 3388 9037
2013 2 28 250 2017 4207 10516
2014 0 39 319 2561 4747 10929
2015 3 40 482 3441 6372 14638
Company name
www.themegallery.com
全世界物理类论文分布
PHYS
ALL 0.01% 0.10% 1.00% 10.00%20.00%50.00%
2006 11 119 1186 11862 23724 59309
2007 12 121 1207 12067 24134 60336
2008 12 125 1251 12510 25021 62552
2009 12 124 1240 12401 24801 62004
2010 12 124 1236 12360 24719 61798
2011 13 131 1312 13123 26245 65614
2012 13 132 1322 13222 26445 66112
2013 13 138 1384 13840 27680 69200
2014 13 139 1394 13942 27884 69711
2015 13 139 1393 13929 27858 69646
Company name
www.themegallery.com
《中国物理快报》发表的文章区间分布
CPL 0.01% 0.10% 1.00% 10.00%20.00%50.00%
2006 0 0 0 8 21 144
2007 0 0 0 6 16 154
2008 0 1 1 9 25 259
2009 0 0 0 3 15 171
2010 0 0 0 1 11 131
2011 0 0 2 11 22 142
2012 0 0 1 10 33 143
2013 0 0 0 8 20 148
2014 0 0 1 2 8 92
2015 0 0 1 10 23 208
Company name
www.themegallery.com
《中国物理快报》
有19篇引用超过100次，
有77篇引用超过50次
最近10年，在全世界物理类论文中，
有3篇文章超过或接近top 0.1%的引用水平，即千分之一的水平；
有6篇文章超过或接近top 1%的引用水平，
有68篇文章超过或接近top 10%的引用水平，
有1592篇文章超过或接近top 50%的引用水平
(根据Web of Science数据库, 2016年11月18日数据)
Company name
www.themegallery.com
2016 Nobel 物理奖
Thouless、Haldane、Kosterlitz
获得了Nobel 物理奖
他们从理论上提出了二维物理体系中的拓扑
相和拓扑量子态，解释了某种薄膜层物质的导电率会以整倍数发生变化
Thouless－Kosterlitz 1976-1978
Duncan M. Haldane 1980
Company name
www.themegallery.com
《中国物理快报》两个原创性工作
１. 赵忠贤院士发表在《中国物理快报》关于铁基超导的原创新工作截止目前被引1177次，获得2012年中国物理学会期刊“最有影响力论文”特等奖。引用水平达到该年度全世界物理类论文(123104篇)的Top 0.05%,即万分之五的水平。
题目： Superconductivity at 55K in iron-based F-doped layered quaternary compound Sm[O(1-x)F(x)]FeAs
作者：REN Zhi-An(任治安)，LU Wei(陆伟)，YANG Jie(杨杰)，YI Wei(衣玮)，SHEN Xiao-Li(慎晓丽)，LI Zheng-Cai(李正才)，CHE Guang-Can(车广灿)，DONG Xiao-Li(董晓莉)，SUN Li-Ling(孙力玲)， ZHOU Fang(周放)， ZHAO Zhong-Xian(赵忠贤)
刊期：Chinese Physics Letters, 2008, 25(6): 2215
被引频次: 1177 (来自Web of Science, 2016年11月17日数据)
Company name
www.themegallery.com
《中国物理快报》两个原创性工作
２. 清华大学薛其坤院士发表在《中国物理快报》的文章被美国著名学术期刊《Science》专文评论，是薛其坤院士2016年获得“未来科学奖”（100万美元）的两项主要研究成果之一。
题目： Interface-Induced High-Temperature Superconductivity in Single Unit-Cell FeSe Films on SrTiO3
作者：WANG Qing-Yan(王庆艳)1,2, LI Zhi(李志) 2, ZHANG Wen-Hao(张文号)1, ZHANG Zuo-Cheng(张祚成)1, ZHANG Jin-Song(张金松)1, LI Wei(李渭) 1, DING Hao(丁浩)1, OU Yun-Bo(欧云波)2, DENG Peng(邓鹏)1, CHANG Kai(常凯)1, WEN Jing(文竞)1, SONG Can-Li(宋灿立)1, HE Ke(何珂)2, JIA Jin-Feng(贾金锋)1, JI Shuai-Hua(季帅华)1, WANG Ya-Yu(王亚愚)1, WANG Li-Li(王立莉)2, CHEN Xi(陈曦)1, MA Xu-Cun(马旭村)2, XUE Qi-Kun(薛其坤)1
刊期：Chinese Physics Letters, 2012, 29(3): 037402
被引频次: 260 (来自Web of Science数据库, 2016年11月16日Company name
www.themegallery.com
被Science 专文进行评论报道
Company name
www.themegallery.com
未来科学大奖
2016年9月19日，成立于2016年初的未来科学大奖首次揭晓两位获奖者，
生命科学奖得主是香港中文大学卢煜明教授，
物质科学奖得主是清华大学薛其坤院士。
每人获得100万美元奖金。
薛其坤院士获奖理由：他在利用分子束外延技术发现单层铁硒超导和反常量子霍尔效应等新奇量子效应方面做出了开拓性工作。
发表在《中国物理快报》的上述文章是获奖理由中两项主要研究成果之一，即单层铁硒超导新现象和新机理方面的研究。
Company name
www.themegallery.com
EXPRESS LETTERS栏目(CPL)
从2012年下半年开始，我们创建了EXPRESS LETTERS栏目，吸引一些对本研领域有重要推动意义又具有时效性的工作在我刊快速发表。承诺两周内给出文章是否录用的决定，一个月内发表。
著名学者
薛其坤 (清华大学/2014);
文小刚 (MIT/2013);
封东来 (复旦大学/2015);
周兴江 (中科院物理所/2013);
廖劲峰 (Brookhaven Natl Lab/2015);
吴从军 (UC San Diego/2016); ……
等均在本栏目发表了具有重要创新性的文章
目前，已经有18篇文章在EXPRESS LETTERS栏目发表。Company name
www.themegallery.com
（1）题目：A New Bipolar Type Transistor Created Based on Interface Effects of Integrated All Perovskite Oxides
作者：XIA Feng-Jin, WU Hao, FU Yue-Ju, XU Bo, YUAN Jie, ZHU Bei-Yi, QIU Xiang-Gang, CAO Li-Xin, LI Jun-Jie, JIN Ai-Zi, WANG Yu-Mei, LI Fang-Hua, LIU Bao-Ting, XIE Zhong, ZHAO Bai-Ru
卷期：Chin. Phys. Lett.. 2012, 29(10): 107402.
（至2016年7月8日下载698次, 被引1次）
（2）题目：Nonlocal Imaging by Conditional Averaging of Random Reference Measurements
作者：LUO Kai-Hong, HUANG Bo-Qiang, ZHENG Wei-Mou, WU Ling-An
卷期：Chin. Phys. Lett.. 2012, 29(7): 074216.
（至2016年7月8日下载711次, 被引24次）
Company name
www.themegallery.com
（3）题目：Breakdown of Energy Equipartition in Vibro-Fluidized Granular Media in Micro-Gravity
作者：CHEN Yan-Pei, Pierre Evesque, HOU Mei-Ying
卷期：Chin. Phys. Lett.. 2012, 29(7): 074501.
（至2016年7月8日下载481次, 被引10次）
（4）题目：Finding a Way to Determine the Pion Distribution Amplitude from the Experimental Data
作者：HUANG Tao, WU Xing-Gang, ZHONG Tao
卷期：Chin. Phys. Lett. 2013, 30(4): 041201
（至2016年7月8日下载452次, 被引6次）
Company name
www.themegallery.com
（5）题目：A Lattice Non-Perturbative Definition of an SO(10) Chiral Gauge Theory and Its Induced Standard Model
作者：WEN Xiao-Gang
卷期：Chin. Phys. Lett. 2013, 30(11): 11101
（至2016年7月8日下载525次, 被引8次）
（6）题目：Fermi Surface and Band Structure of (Ca,La)FeAs2 Superconductor from Angle-Resolved Photoemission Spectroscopy
作者：LIU Xu, LIU De-Fa, ZHAO Lin, GUO Qi, MU Qing-Ge, CHEN Dong-Yun, SHEN Bing, YI He-Mian, HUANG Jian-Wei, HE Jun-Feng, PENG Ying-Ying, LIU Yan, HE Shao-Long, LIU Guo-Dong, DONG Xiao-Li, ZHANG Jun, CHEN Chuang-Tian, XU Zu-Yan, REN Zhi-An, ZHOU Xing-Jiang
卷期：Chin. Phys. Lett. 2013, 30(12): 127402
（至2016年7月8日下载448次, 被引6次）
Company name
www.themegallery.com
（7）题目：Direct Observation of High-Temperature Superconductivity in One-Unit-Cell FeSe Films
作者：ZHANG Wen-Hao, SUN Yi, ZHANG Jin-Song, LI Fang-Sen, GUO Ming-Hua, ZHAO Yan-Fei, ZHANG Hui-Min, PENG Jun-Ping, XING Ying, WANG Hui-Chao, FUJITA Takeshi, HIRATA Akihiko, LI Zhi, DING Hao, TANG Chen-Jia, WANG Meng, WANG Qing-Yan, HE Ke, JI Shuai-Hua, CHEN Xi, WANG Jun-Feng, XIA Zheng-Cai, LI Liang, WANG Ya-Yu, WANG Jian, WANG Li-Li, CHEN Ming-Wei, XUE Qi-Kun, MA Xu-Cun
卷期：Chin. Phys. Lett. 2014, 31(1): 017401
（至2016年7月8日下载1819次，被引59次）
Company name
www.themegallery.com
（8）题目： Magnetism in Quasi-One-Dimensional A2Cr3As3 (A=K,Rb) Superconductors
作者：吴贤新 勒聪聪 袁静 范桁 胡江平
卷期：Chin. Phys. Lett. 2015, 32(5): 057401
（至2016年7月8日下载268次，被引17次）
（9）题目： Possible p-Wave Superconductivity in Epitaxial Bi/Ni Bilayers
作者：龚欣欣 周和心 徐鹏超 岳迪 朱凯 金晓峰 田鹤 赵格剑 陈庭勇
卷期：Chin. Phys. Lett. 2015, 32(06): 067402
（至2016年7月8日下载439次）
Company name
www.themegallery.com
（10）题目：First Evaluation and Frequency Measurement of the Strontium Optical Lattice Clock at NIM
作者：林弋戈 王强 李烨 孟飞 林百科 臧二军 孙震 房芳 李天初 方占军
卷期：Chin. Phys. Lett. 2015, 32(09): 090601
（至2016年7月8日下载421次）
（11）题目：Consistency of Perfect Fluidity and Jet Quenching in Semi-Quark-Gluon Monopole Plasmas
作者：徐杰谌，廖劲峰，Miklos Gyulassy
卷期：Chin. Phys. Lett. 2015, 32(09): 092501
（至2016年7月8日下载240次，被引7次）
Company name
www.themegallery.com
（12）题目：Observation of Fermi Arcs in Non-Centrosymmetric Weyl Semi-Metal Candidate NbP
作者：徐迪飞 杜永平 王震 李宇鹏 牛晓海 姚岐 Dudin Pavel 许祝安万贤纲 封东来
卷期：Chin. Phys. Lett. 2015, 32(10): 107101
（至2016年7月8日下载256次，被引7次）
（13）题目：New Superconductivity Dome in LaFeAsO1−xFx Accompanied by Structural Transition
作者：杨杰 周睿 尉琳琳 杨槐馨 李建奇 赵忠贤 郑国庆
卷期：Chin. Phys. Lett. 2015, 32(10): 107401
（至2016年7月8日下载284次，被引3次）
Company name
www.themegallery.com
（14）题目：Electron-Elastic-Wave Interaction in a Two-Dimensional Topological Insulator
作者：吴晓光
卷期：Chin. Phys. Lett. 2016, 33(02): 027303
（至2016年7月8日下载158次）
（15）题目：Microscopic Theory of the Thermodynamic Properties of Sr3Ru2O7
作者：李伟正, 吴从军
卷期：Chin. Phys. Lett. 2016, 33(03): 037201
（至2016年7月8日下载139次）
（16）题目： Relativistic Brueckner--Hartree--Fock Theory for Finite Nuclei
作者： 申时行, 胡金牛, 梁豪兆, 孟杰, Peter Ring, 张双全Company name
www.themegallery.com
（17）题目：Possible Nodeless Superconducting Gaps in Bi_2Sr_2CaCu_2O_8+\delta and YBa_2Cu_3O_7-x Revealed by Cross-Sectional Scanning Tunneling Spectroscopy
作者：Ming-Qiang Ren(任明强)1,2, Ya-Jun Yan(闫亚军)1,2, Tong Zhang(张童)1,2**, Dong-Lai Feng(封东来)1,2**
卷期：Chin. Phys. Lett. 2016, 33(12): 127402
（18）题目：Topological Phase in Non-Centrosymmetric Material NaSnBi
作者：Xia Dai(代霞)1, Cong-Cong Le(勒聪聪)1, Xian-Xin Wu(吴贤新)1, Sheng-Shan Qin(秦盛山)1, Zhi-Ping Lin(林志萍)1, Jiang-Ping Hu(胡江平)
卷期：Chin. Phys. Lett. 2016, 33(12): 127301
Company name
www.themegallery.com
EXPRESS LETTERS栏目
对应于2015 JCR根据Web of Science数据得出
Express Letters影响因子：9.75;
即时影响因子：2.83
篇均被引次数：9.87
h-index: 7 （15篇文章中有7篇文章至少被引次数在7次以上）
施引文献中
7%是 Nature子刊；
37% 是 Physical Review 系列刊 （30% PR A-X +7% PRL）
Company name
www.themegallery.com
“最有影响力”论文奖励
从2012年开始，已经进行了四届
“最有影响力论文”奖励活动
CPL评选特等奖论文4篇，一等奖论文10篇
Company name
www.themegallery.com
《中国物理快报》获奖论文
特等奖论文(1) Superconductivity at 55K in iron-based F-doped layered quaternary compound Sm[O(1-x)F(x)]FeAs
作者：REN Zhi-An(任治安)，LU Wei(陆伟)，YANG Jie(杨杰)，YI Wei(衣玮)，SHEN Xiao-Li(慎晓丽)，LI Zheng-Cai(李正才)，CHE Guang-Can(车广灿)，DONG Xiao-Li(董晓莉)，SUN Li-Ling(孙力玲)，ZHOU Fang(周放)， ZHAO Zhong-Xian(赵忠贤)
CPL 2008年25卷2215页
被引频次: 1177
《中国物理快报》获奖论文
特等奖论文(2) 中国人民大学物理系 鲍威 等人 发表在《中国物理快报》2011年第8期的文章
题目：铁基超导体（K0.8Fe16Se2）的一种新奇大磁矩反铁磁序A Novel Large Moment Antiferromagnetic Order in K0.8Fe16Se2 Superconductor
作者：BAO Wei(鲍威), HUANG Qing-Zhen(黄清镇), CHEN Gen-Fu(陈根富), M. A. Green, WANG Du-Ming(王笃明), HE Jun-Bao(何俊宝), QIU Yi-Ming(邱义铭)
Chinese Physics Letters, 2011, 28:086104
被引频次: 264
《中国物理快报》获奖论文
特等奖论文（3）美国加利福尼亚大学圣地亚哥分校物理系 吴从军
等人发表在《中国物理快报》2011年第9期的文章
题目：自旋轨道耦合导致的非传统玻色-爱因斯坦凝聚Unconventional Bose-Einstein Condensations from Spin-Orbit Coupling
作者：WU Cong-Jun(吴从军), Ian Mondragon-Shem, ZHOU Xiang-Fa(周祥发)
Chinese Physics Letters, 2011, 28:097102
被引频次: 165
《中国物理快报》获奖论文
(4) 2016年CPL特等奖论文
Interface-Induced High-Temperature Superconductivity in Single Unit-Cell FeSe Films on SrTiO3
Chinese Physics Letters, 2012, 29(3): 037402
作者：WANG Qing-Yan(王庆艳)1,2†, LI Zhi(李志)2†, ZHANG Wen-Hao(张文号)1†, ZHANG Zuo-Cheng(张祚成)1†, ZHANG Jin-Song(张金松)1, LI Wei(李渭)1, DING Hao(丁浩)1, OU Yun-Bo(欧云波)2, DENG Peng(邓鹏)1, CHANG Kai(常凯)1, WEN Jing(文竞)1, SONG Can-Li(宋灿立)1, HE Ke(何珂)2, JIA Jin-Feng(贾金锋)1, JI Shuai-Hua(季帅华)1, WANG Ya-Yu(王亚愚)1, WANG Li-Li(王立莉)2, CHEN Xi(陈曦)1, MA Xu-Cun(马旭村)2**, XUE Qi-Kun(薛其坤)1**
Chinese Physics Letters, 2012, 29(3): 037402
被引频次: 231
(来自Web of Science数据库, 2016年8月24日数据)Company name
www.themegallery.com
一等奖10篇：
(1) Superconductivity in hole-doped (Sr1-xKx)Fe2As2
作者：CHEN Gen-Fu(陈根富)，LI Zheng(李政)，LI Gang(李岗)，HU Wan-Zheng(胡婉铮)，DONG Jing(董靖)，ZHOU Jun(周军)，ZHANG Xiao-Dong(张晓冬)，ZHENG Ping(郑萍)，WANG Nan-Lin(王楠林)，LUO Jian-Lin(雒建林)
CPL 2008年25卷3403页
被引频次: 196
(2) Deterministic secure communication without using entanglement
作者：CAI Qing-Yu(蔡庆宇)，LI Bai-Wen(李白文)
CPL 2004年21卷601页
被引频次: 197
Company name
www.themegallery.com
(3) Binary Cu-Zr Bulk Metallic Glasses
作者：TANG Mei-Bo(汤美波)，ZHAO De-Qian(赵德乾)，PAN Ming-Xiang(潘明祥)，WANG Wei-Hua(汪卫华)
CPL 2004年21卷901页
被引频次: 200
(4) Multiple Nodeless Superconducting Gaps in (Ba0.6K0.4)Fe2As2 Superconductor from Angle-Resolved Photoemission
作者：ZHAO Lin(赵林)，LIU Hai-Yun(刘海云)，ZHANG Wen-Tao(张文涛)，MENG Jian-Qiao(孟建桥)，JIA Xiao-Wen(贾小文)，LIU Guo-Dong(刘国东)，DONG Xiao-Li(董晓莉)，CHEN Gen-Fu(陈根富)，LUO Jian-Lin(雒建林)，WANG Nan-Lin(王楠林)，LU Wei(陆伟)，WANG Gui-Ling(王桂玲)，ZHOU Yong(周永)，ZHU Yong(朱镛)，WANG Xiao-Yang(王晓洋)，XU Zu-Yan(许祖彦)，CHEN Chuang-Tian(陈创天)，ZHOU Xing-Jiang(周兴江)
被引频次: 140
Company name
www.themegallery.com
(5) Nodal gap in Fe-based layered superconductor LaO(0.9)F(0.1-delta)FeAs probed by specific heat measurements
Chinese Physics Letters, 2008, 25:2221
作者：MU Gang(牟刚), ZHU Xi-Yu(祝熙宇), FANG Lei(方磊), SHAN Lei(单磊), REN Cong(任聪),WEN Hai-Hu(闻海虎)
被引频次: 105
(6) Pseudospin symmetry in relativistic framework with harmonic oscillator potential and Woods-Saxon potential
Chinese Physics Letters, 2003, 20:358
作者：CHEN Ti-Sheng(陈惕生),LÜ Hong-Feng(吕洪凤),MENG Jie(孟杰),ZHANG Shuang-Quan(张双全), ZHOU Shan-Gui(周善贵)
被引频次: 88
Company name
www.themegallery.com
(7) Interface-Induced High-Temperature Superconductivity in Single Unit-Cell FeSe Films on SrTiO3
Chinese Physics Letters, 2012, 29(3): 037402
作者：WANG Qing-Yan(王庆艳)1,2†, LI Zhi(李志)2†, ZHANG Wen-Hao(张文号)1†, ZHANG Zuo-Cheng(张祚成)1†, ZHANG Jin-Song(张金松)1, LI Wei(李渭)1, DING Hao(丁浩)1, OU Yun-Bo(欧云波)2, DENG Peng(邓鹏)1, CHANG Kai(常凯)1, WEN Jing(文竞)1, SONG Can-Li(宋灿立)1, HE Ke(何珂)2, JIA Jin-Feng(贾金锋)1, JI Shuai-Hua(季帅华)1, WANG Ya-Yu(王亚愚)1, WANG Li-Li(王立莉)2, CHEN Xi(陈曦)1, MA Xu-Cun(马旭村)2**, XUE Qi-Kun(薛其坤)1**
被引频次: 219 (8) Viscous Flow over an Unsteady Shrinking Sheet with Mass
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Chinese Physics Letters, 2009, 26(1): 014703
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作者: GUO Bo-Ling(郭柏灵)1, LING Li-Ming(凌黎明)1,2
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CPL Express Letter 论文推送Microscopic Theory of the Thermodynamic Properties of Sr$_3$Ru$_2$O$_7$
Wei-Cheng Lee(李伟正)1,2, Congjun Wu(吴从军)1**
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A Series Journal of the Chinese Physical SocietyDistributed by IOP Publishing
Online: http://iopscience.iop.org/0256-307Xhttp://cpl.iphy.ac.cn
C H I N E S E P H Y S I C A L S O C I E T Y
ISSN: 0256-307X
中国物理快报
ChinesePhysicsLettersVolume 33 Number 3 March 2016
CHIN.PHYS. LETT. Vol. 33, No. 3 (2016) 037201 Express Letter
Microscopic Theory of the Thermodynamic Properties of Sr3Ru2O7∗
Wei-Cheng Lee(李伟正)1,2, Congjun Wu(吴从军)1**1Department of Physics, University of California, San Diego, California, 92093, USA
2Department of Physics, Applied Physics, and Astronomy, Binghamton University-State University of New York,Binghamton, USA
(Received 12 February 2016)The thermodynamic properties of the bilayer ruthenate compound Sr3Ru2O7 at very low temperatures are inves-tigated by using a tight-binding model yielding the realistic band structure combined with the on-site interactionstreated at the mean-field level. We find that both the total density of states at the Fermi energy and the entropyexhibit a sudden increase near the critical magnetic field for the nematic phase, echoing the experimental find-ings. A new mechanism to explain the anisotropic transport properties is proposed based on scatterings at theanisotropic domain boundaries. Our results suggest that extra cares are necessary to isolate the contributionsdue to the quantum criticality from the band structure singularity in Sr3Ru2O7.
PACS: 72.80.Ga, 73.20.−r, 71.10.Fd DOI: 10.1088/0256-307X/33/3/037201
The bilayer ruthenate compound Sr3Ru2O7 hasaroused considerable attentions with various inter-esting properties. It was first considered exhibit-ing a field-tuned quantum criticality of the metam-agnetic transition.[1−3] Later, in the ultra-pure sin-gle crystal it has been found that the metamagneticquantum critical point is intervened by the emer-gence of an unconventional anisotropic (nematic) elec-tronic state,[4,5] stimulating considerable theoreticalefforts.[6−18] Sr3Ru2O7 is a metallic itinerant systemwith the active 𝑡2𝑔-orbitals of the Ru sites in the bi-layer RuO2 (𝑎𝑏) planes. At very low temperatures(∼ 1 K), it starts as a paramagnet at small mag-netic fields. Further increasing field strength leads totwo consecutive metamagnetic transitions at 7.8 and8.1 Tesla if the field is perpendicular to the 𝑎𝑏-plane.The nematic phase is observed between these two tran-sitions, identified by the observation of anisotropic re-sistivity without noticeable lattice distortions.
This nematic phase in Sr3Ru2O7 can be under-stood as the consequence of a Fermi surface Pomer-anchuk instability.[3] It is a mixture in both den-sity and spin channels with the 𝑑-wave symmetry,[19]though its microscopic origin remains controversial.Different microscopic theories have been proposedbased on the quasi-1D bands of 𝑑𝑥𝑧 and 𝑑𝑦𝑧,[14,15,18]and based on the 2d-band of 𝑑𝑥𝑦.[8,9,16,17] In thetheories of ours[14,18] and Raghu et al.,[15] the un-conventional (nematic) magnetic ordering was inter-preted as orbital ordering between the 𝑑𝑥𝑧 and 𝑑𝑦𝑧-orbitals. In particular, in Ref. [18] a realistic tight-binding model is constructed taking into account themulti-orbital features, which reproduces accuratelythe results of the angle-resolved photon emission spec-
troscopy (ARPES)[20] and the quasiparticle interfer-ence in the spectroscopic imaging scanning tunnelingmicroscopy (STM).[21]
The influence of quantum critical fluctuations inSr3Ru2O7 seems to be novel as well. Rost et al.[22,23]measured the entropy and specific heat in ultra-puresamples and found divergences near the metamagnetictransitions in both quantities. Although it is a com-mon feature in a quantum critical state that the spe-cific heat diverges as 𝐶/𝑇 ∼ [(𝐵 − 𝐵𝑐)/𝐵𝑐]
−𝛼 dueto quantum fluctuations, the exponent of 𝛼 is fittedto be 1 instead of 1/3 as predicted by the celebratedHertz–Millis theory.[24,25] The total density of states(DOS) measured by Iwaya et al.[26] using the STMshowed that the DOS at the Fermi energy (𝒟(𝜖
F)) in-
creases significantly under the magnetic field, but theDOS at higher and lower energy does not change ac-cordingly. This indicates that the Zeeman energy doesnot simply cause a relative chemical potential shift toelectrons with opposite spins (the DOS evolution withthe external magnetic field will be discussed in Sup-plemental Material III B, i.e., SM III B). These findingshave posted a challenge to understand the critical be-havior in this material.
In this Letter, we show that the realistic band fea-tures of Sr3Ru2O7 make this material very sensitiveto small energy scales. Parts of the Fermi surface areclose to the van Hove singularities, and Fermi surfacereconstructions in the external magnetic fields lead toa singular behavior in 𝒟(𝜖F). This results in the diver-gences observed in the experiments mentioned above.Because of the strong spin-orbit coupling and the un-quenched orbital moments, the Zeeman energy tendsto reconstruct the Fermi surfaces rather than just pro-
∗Supported by the NSF DMR-1410375 and AFOSR FA9550-14-1-0168, the President’s Research Catalyst Award (No CA-15-327861) from the University of California Office of the President, and the CAS/SAFEA International Partnership Program forCreative Research Teams.
**Corresponding author. Email: [email protected]© 2016 Chinese Physical Society and IOP Publishing Ltd
037201-1
CHIN.PHYS. LETT. Vol. 33, No. 3 (2016) 037201 Express Letter
vides a relative chemical potential shift. Our resultssuggest that the influence of quantum critical fluctua-tions will be masked if 𝒟(𝜖
F) of the system exhibits a
non-monotonic behavior, implying that a more carefulanalysis is required in order to distinguish the role ofthe quantum criticality in the bilayer Sr3Ru2O7.
We employ the tight-binding model 𝐻0 for theband structure of Sr3Ru2O7 derived by two of usand Arovas in a previous work[18] as elaborated inSM I. It is featured by the 𝑡2𝑔-orbital structure (e.g.𝑑𝑥𝑧, 𝑑𝑦𝑧, 𝑑𝑥𝑦), the bilayer splitting, the staggered dis-tortion of the RuO octahedral, and spin-orbit couplingdescribed by a few parameters as follows. The intra-layer hoppings include the longitudinal (𝑡1) and trans-verse (𝑡2) hoppings for the 𝑑𝑥𝑧 and 𝑑𝑦𝑧-orbitals, thenearest neighbor (𝑡3), the next nearest neighbor (𝑡4),the next next nearest neighbor (𝑡5) hoppings for the𝑑𝑥𝑦-orbital, and the next nearest neighbor (𝑡6) hop-ping between 𝑑𝑥𝑧 and 𝑑𝑦𝑧-orbitals. Here 𝑡⊥ is thelongitudinal inter-layer hopping for the 𝑑𝑥𝑧 and 𝑑𝑦𝑧-orbitals. The rotations of the RuO octahedra induceadditional inra-(𝑡
INT) and inter-layer (𝑡⊥
INT) hoppings
between 𝑑𝑥𝑧 and 𝑑𝑦𝑧 orbitals. The onsite terms in-clude the spin-orbit coupling 𝜆𝜎 ·𝐿, the energy split-ting 𝑉𝑥𝑦 of the 𝑑𝑥𝑧 and 𝑑𝑦𝑧 orbitals relative to the𝑑𝑥𝑦-orbital, and the chemical potential 𝜇. A typicalFermi surface configuration is plotted in Fig. 1 withthe parameter values in the caption.
kx/π
ky/π
1.0
1.0
0.5
0.5
0.0
0.0
-0.5
-0.5-1.0-1.0
Fig. 1. The Fermi surfaces using the bilayer tight-bindingmodel Eq. (1) in SM I with the parameters in units of 𝑡1 as:𝑡2 = 0.1𝑡1, 𝑡3 = 𝑡1, 𝑡4 = 0.2𝑡1, 𝑡5 = −0.06𝑡1, 𝑡6 = 0.1𝑡1,𝑡⊥ = 0.6𝑡1, 𝑡INT = 𝑡⊥
INT= 0.1𝑡1, 𝜆 = 0.2𝑡1, 𝑉𝑥𝑦 = 0.3𝑡1,
and 𝜇 = 0.94𝑡1. The thick dashed lines mark the bound-ary of the half Brillouin zone due to the unit cell doublinginduced by the rotation of RuO octahedra. The Fermi sur-faces of the bonding (𝑘𝑧 = 0) and the anti-bonding bands(𝑘𝑧 = 𝜋) are denoted by black solid and red dashed lines,respectively.
The Hubbard model contains the on-site intra andinter orbital interactions as
𝐻int = 𝑈∑𝑖,𝑎,𝛼
��𝛼𝑖𝑎↑��
𝛼𝑖𝑎↓ +
𝑉
2
∑𝑖,𝑎,𝛼=𝛽
��𝛼𝑖𝑎 ��
𝛽𝑖𝑎, (1)
where the Greek index 𝛼 refers to the orbitals 𝑥𝑧, 𝑦𝑧
and 𝑥𝑦; the Latin index 𝑎 refers to the upper andlower layers. The other two possible terms in themulti-band Hubbard interaction are the Hund rulecoupling and pairing hopping terms, which do notchange the qualitative physics and are neglected. Weassume the external 𝐵-field lying in the 𝑥𝑧-planewith an angle 𝜃 tilted from the 𝑧-axis. The occu-pation and spin in each orbital and layer are de-fined as follows: 𝑛𝛼
𝑎 ≡∑
𝑠⟨𝑑𝛼 †𝑠,𝑎(𝑖)𝑑
𝛼𝑠,𝑎(𝑖)⟩, 𝑆𝛼
𝑧 𝑎 ≡12
∑𝑠 𝑠 ⟨𝑑𝛼 †
𝑠,𝑎(𝑖)𝑑𝛼𝑠,𝑎(𝑖)⟩, 𝑆𝛼
𝑥𝑎 ≡ 12
∑𝑠⟨𝑑𝛼 †
𝑠,𝑎(𝑖)𝑑𝛼𝑠,𝑎(𝑖)⟩,
where 𝑠 refers to spin index. The detailed mean-fieldtheory solution for the Hamiltonian 𝐻0 +𝐻int is pre-sented in SM II, and order parameters are computedself-consistently. It was pointed out in Refs. [14,15]that the nematic phase can be identified as the or-bital ordering between the 𝑑𝑥𝑧 and 𝑑𝑦𝑧-orbitals. Thenematic (𝒩 ) and the magnetization (ℳ) order pa-rameters are defined as
𝒩 =∑𝑎
(𝑛𝑦𝑧𝑎 − 𝑛𝑥𝑧
𝑎 ), ℳ =∑𝑎𝛼
𝑆𝛼𝑎 . (2)
Throughout this study, the interaction parameter val-ues are taken as 𝑈/𝑡1 = 𝑉/𝑡1 = 3.6, such that nospontaneous magnetization occurs in the absence ofthe external magnetic field.
0.25
0.20
0.15
0.10
0.05
0.000.001 0.002 0.003 0.004 0.005 0.006
Ord
er
para
mete
rs
µBB/t1
M
N
Fig. 2. The order parameters as a function of 𝜇B𝐵 for𝜃 = 0. The nematic phase (in the green area) is boundedby two magnetization increases which correspond to meta-magnetic transitions.
We present the mean-field results of low tempera-ture thermodynamic properties at 𝐵 ‖ 𝑐, i.e., 𝜃 = 0.Many experimentally observed results can be repro-duced and understood by the singular behavior of𝒟(𝜖
F) under the magnetic field. The order param-
eters |ℳ| and 𝒩 as functions of 𝜇B𝐵 are shown in
Fig. 2. There are three rapid increases in the mag-netization, consistent with the experiment measure-ments of the real part of the very low frequencyAC magnetic susceptibility at 7.5 T, 7.8 T and 8.1T,respectively.[4] Experimentally, only the last two ex-hibit dissipative peaks in the imaginary part of theAC susceptibility, which characterize the first ordermetamagnetic transitions. The first jump measured isconsidered as a crossover. The nematic ordering devel-ops in the area bounded by the last two magnetizationjumps, reproducing the well-known phase diagram ofthe Sr3Ru2O7.[4] In particular, if we adopt the resultsfrom LDA calculations[27,28] that 𝑡1 ≈ 300meV, we
037201-2
CHIN.PHYS. LETT. Vol. 33, No. 3 (2016) 037201 Express Letter
can find that three jumps in the magnetization appearat 𝐵 ≈ 0.0032𝑡1/𝜇B , 0.0053𝑡1/𝜇B , and 0.0059𝑡1/𝜇B ∼15.7T, 26T, and 29 T, which are within the sameorder to the experimental values. This is an im-provement compared to the results in previous theorycalculations.[8−11,15,16] in which the nematic orderingdevelops at much higher field strength 𝜇B𝐵/𝑡1 ≈ 0.02.
The sensitivity of the nematic phase to the smallenergy scale like the Zeeman energy is because a partof the Fermi surfaces, mostly composed of quasi-1Dbands, is approaching the van Hove singularities at(𝜋, 0) and (0, 𝜋). The evolution of the Fermi sur-face structures as increasing the 𝐵-field across thenematic phase boundaries is presented in detail inSM IIIA. When the system is in the nematic phase,the Fermi surfaces only have 2-fold symmetry as ex-pected. Particularly, the nematic distortion is mostprominent near (±𝜋, 0) and (0,±𝜋) whose Fermi sur-faces are dominated by the quasi-1D bands, support-ing the mechanism of orbital ordering in quasi-1Dbands driven by the van Hove singularities.
4.4
4.0
3.6
3.20.005 0.006 0.007
µBB/t1
S↼B↽/T
Fig. 3. (a) The entropy landscape represented by thequantity 𝑆(𝐵)/𝑇 in units of 𝑘2𝐵/𝑡1 within the range of0.0045 ≤ 𝜇B𝐵/𝑡1 ≤ 0.007. A sudden increase near thenematic region (green area) is clearly seen.
One of the intriguing puzzles experimentally ob-served is the critical exponent of the divergence inentropy (as well as specific heat) when approachingthe nematic region, or the quantum critical point. Asmentioned in the introduction, the exponent does notmatch the Hertz-Millis theory based on the assump-tion of a constant DOS 𝒟(𝜖
F). However, Sr2Ru3O7
has a complicated Fermi surface evolutions under the𝐵-field, and is in the Fermi liquid state at low temper-atures inferred from the temperature dependences ofthe resistivity. It is then worthy of studying first thecontribution from the band structure to the entropybefore considering the quantum fluctuations. The en-tropy per Ru atom can be evaluated by
𝑆(𝐵) = − 𝑘B
𝑁
′∑𝑘
∑𝑗
[𝑓(𝐸𝑗(𝑘)) ln 𝑓(𝐸𝑗(𝑘))
+ (1− 𝑓(𝐸𝑗(𝑘))) ln(1− 𝑓(𝐸𝑗(𝑘)))], (3)
where 𝑓 is the Fermi distribution function, and 𝐸𝑗(𝑘)is the mean-field energy spectra of the 𝑗th band.In Fig. 3, we plot 𝑆(𝐵)/𝑇 at a low temperature of1/(𝛽𝑡1) = 0.002 for the 𝐵-fields in the vicinity of
the nematic region. 𝑆(𝐵)/𝑇 increases first, beingsuppressed, and then decreases, which is consistentwith the experiment.[22] Since the nematic transitionis driven by the sudden increase of 𝒟(𝜖
F), the entropy
should also be enhanced from outside towards the ne-matic region.
While it is generally expected that the quantumfluctuations near the critical point contribute addi-tional entropy, our results demonstrate the singularbehavior of 𝒟(𝜖
F) already produces diverging behavior
in entropy under magnetic field at constant temper-ature, although the critical exponent 𝛼 is difficult beextracted from the current theory. Similar argumenthas been proposed in a previous study,[23] in whichthe effect of a rigid band shift away from van Hovesingularities in a perfect 1D band is discussed.
ρ
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5-0.050 0.000 0.050
ω/t
AllQuasi-1Dbands
µBB/t1
kBT/t 1
0.0051
0.009
0.006
0.003
0.0057 0.0063
0.0000.0050.0100.0150.020
(a) (b)
Fig. 4. (a) The 𝑇–𝐵 phase diagram. Magnitudes of 𝒩 arerepresented by the color scales. The areas with light colorshave large 𝒩 , defining the region for the nematic order.The re-entry of the nematic order at higher temperatureis seen at fields between 0.0058 < 𝜇B𝐵/𝑡1 < 0.0063. (b)The DOSs of the all bands (solid line) and quasi-1D bands(dashed line) at 𝜇B𝐵/𝑡1 = 0.006. The yellow areas referto the energy window bounded by ±𝑘B𝑇/𝑡1 with temper-ature 𝑘B𝑇/𝑡1 = 0.003. It can be seen that this thermalenergy window covers a region in which the DOS increasesabruptly, driving the nematic phase at finite temperature.
An intriguing experimental observation is the“muffin”-shaped phase boundary of the nematic phasein the 𝑇–𝐵 phase diagram.[22] At field strengthsslightly below 7.8T and above 8.1T the nematic phaseappears at finite temperature but vanishes at zerotemperature, to which we term as the “re-entry” be-havior. It means that the entropy is actually higherinside the nematic phase than the adjacent normalphases. By inspecting Fig. 3 closer, it can be seen thatthe entropy drops as entering the nematic phase fromthe lower-field boundary but raises as entering fromthe upper-field boundary, thus the present theory hasthe “re-entry” at the upper-field boundary but not atthe lower-field boundary, which is further confirmedby the temperature dependence of 𝒩 as a function of𝐵-field shown in Fig. 4(a).
The re-entry behavior can be understood as fol-lows. The mechanism for nematic ordering inSr3Ru2O7 based on van Hove singularities is all aboutincreasing 𝒟(𝜖
F) abruptly by driving the system closer
to the van Hove singularities with the magnetic field.At the field strength slightly above the upper critical
037201-3
CHIN.PHYS. LETT. Vol. 33, No. 3 (2016) 037201 Express Letter
field, 𝒟(𝜖F) is large but still not enough for the occur-rence of the nematic ordering. If the thermal energyis large enough to cover enough DOS within the ther-mal energy window 𝜖
F−𝑘
B𝑇 < 𝜖
F< 𝜖
F+𝑘
B𝑇 but still
low enough so that the thermal fluctuations are small,the re-entry of the nematic phase at higher temper-ature is possible. As an illustration, Fig. 4(b) plotsthe DOS for 𝜇B𝐵/𝑡1 = 0.006 at which the nematicordering first occurs at 𝑘
B𝑇/𝑡1 = 0.003 in our calcula-
tion. It can be seen that the thermal energy windowfor 𝑘
B𝑇/𝑡1 = 0.003 (yellow areas) covers a region in
which the DOS increases abruptly, consistent with themechanism for the re-entry behavior discussed above.
Now let us consider the case of titled magneticfield. Figure 5 presents the nematic order parame-ter 𝒩 as functions of 𝜇
B𝐵 and 𝜃. Since the in-plane
component of the orbital Zeeman energy explicitlybreaks the 𝐶4 symmetry down to the 𝐶2 symmetry,𝒩 is non-zero as long as 𝜃 = 0. Nevertheless, theexperimentally observable nematic phase can still beidentified by the jumps in 𝒩 . Our results show thatthe nematicity is strongly enhanced with increasing𝜃, which is due to the orbital Zeeman energy. Theanisotropic in-plane component of the orbital Zeemanenergy term −𝜇B𝐵
∑𝑖,𝑎 𝐿𝑥,𝑖𝑎 sin 𝜃 is clearly propor-
tional to 𝐵 and largest at 𝜃 = 90∘ (i.e. �� ‖ ��), whichinduces the anisotropy of Fermi surface as explicitlyshown in Fig. 3 in SM IV. Although such anisotropy inthe band structure is not important at low field, it canbe amplified by the effect of the interactions, drivingthe system more susceptible to the nematic phase asthe critical points are approached. As a result, theportion of the nematic phase in the phase diagram isenlarged as 𝜃 increases from 0∘ to 90∘ as seen in ourcalculations.
θ (
deg)
µBB/t1
90
60
30
00.002 0.003 0.004 0.005 0.006 0.007
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
Fig. 5. The nematic order parameter 𝒩 as functions of𝜇B𝐵 and 𝜃. The magnitudes of 𝒩 are represented by thecolor scales.
However, experimentally the resistive anisotropydisappears quickly as the magnetic field is tilted awayfrom the 𝑐-axis,[3,5,15] suggesting that the nematic or-dering vanishes with the increase of the field angle𝜃. This observation is seemingly in contradiction withour theory, but this disagreement can be reconciled
as follows. It has been argued that the resistivitymeasurement may not be a good indicator for the ne-matic phase in Ref. [15]: The nematic phase is mostlyassociated with states near the van Hove singular-ity where Fermi velocities are too small to contributesignificantly to transport properties. The observedanisotropic resistivity is mostly likely due to the scat-terings on nematic domains. The tilt of the magneticfield aligns domains, which makes the anisotropy ex-plicit. However, if the domains are fully aligned, theresistivity measurement will become insensitive to thenematic phase due to the diminished scatterings be-tween nematic domains even though the nematic ordercould be larger.
Our results have posted a possibility that the ne-matic order could occur in a larger range of the mag-netic field for 𝐵 ‖ �� than for 𝐵 ‖ 𝑧. Detection meth-ods other than resistivity would be desirable. One fea-sible way is to measure the quasiparticle interference(QPI) in the spectroscopic imaging STM, which hasbeen examined in detail in our previous work.[18] It hasbeen predicted by us that if there is a nematic order,QPI spectra will manifest patterns breaking rotationalsymmetry. Another possible experiment is the nuclearquadruple resonance (NQR) measurement, which hasbeen widely used to reveal ordered states in high-𝑇𝑐 cuprates[29−31] and recently the iron-pnictides.[32]This technique utilizes the feature that a nucleus witha nuclear spin 𝐼 ≥ 1 has a non-zero electric quadruplemoment. Because the electric quadruple moment cre-ates energy splittings in the nuclear states as a electricfield gradient is present, a phase transition could be in-ferred if substantial changes in the resonance peak areobserved in the NQR measurement. In addition, sincethis is a local probe at the atomic level, it is highlysensitive to the local electronic change. Given that Ruatom has a nuclear spin of 𝐼 = 5/2[33] and the orbitalordering in the quasi-1D bands significantly changesthe charge distribution around the nuclei, a system-atic NQR measurement as functions of magnetic fieldand field angle will reveal more conclusive informationabout nematicity.
One remaining puzzle on the transport anisotropyin the nematic phase is why the easy axis for the cur-rent flow is perpendicular to the in-plane componentof the 𝐵-field.[5] In the following, we provide a nat-ural explanation based on the anisotropic spatial ex-tension of domain boundaries. Assuming the 𝐵-fieldlying in the 𝑥𝑧-plane, the in-plane (𝑥𝑦) orbital Zee-man energy reads 𝐻in−plane = −𝜇
B𝐵 sin 𝜃
∑𝑖,𝑎 𝐿𝑥,𝑖𝑎,
which couples the 𝑑𝑥𝑦 and 𝑑𝑥𝑧-orbitals and breaks thedegeneracy between the 𝑑𝑥𝑧 and 𝑑𝑦𝑧-orbitals. Sincethe 𝑑𝑥𝑦-orbital has lower on-site energy due to thecrystal field splitting than that of 𝑑𝑥𝑧, the 𝑑𝑥𝑧-orbitalbands are pushed to higher energy than 𝑑𝑦𝑧-orbitalbands by this extra coupling. As a result, the ne-matic state with preferred 𝑑𝑦𝑧-orbitals (i.e., 𝒩 > 0)has lower energy in the homogeneous system. At small
037201-4
CHIN.PHYS. LETT. Vol. 33, No. 3 (2016) 037201 Express Letter
angles of 𝜃, domains with preferred 𝑑𝑥𝑧-orbitals (i.e.𝒩 ≤ 0) could form as depicted in Fig. 6(a) as meta-stable states, which occupy less volume than the ma-jority domain of 𝒩 > 0.
(b)
y
x
(a)
Fig. 6. Illustration of the energetically favored domainstructures as the in-plane magnetic field is along the �� axis.Orbital ordered phase with 𝒩 > 0 are dominant (shadedareas) and high-energy domains with 𝒩 ≤ 0 (green ovals)coexist. (b) The domain walls extend longer in the 𝑦 di-rection because it costs less energies if less 𝑦-than-�� bondsare broken. The white oval represents the wavefunction of𝑑𝑦𝑧 orbital on each site.
Let us consider the shape of the domain bound-aries. Because of the quasi-1D features of the 𝑑𝑥𝑧and 𝑑𝑦𝑧-orbitals, the horizontal (vertical) domain wallbreaks the bonds of the 𝑑𝑦𝑧(𝑑𝑥𝑧)-orbital as depictedin Fig. 6(b), respectively. Since the 𝑑𝑦𝑧-orbital is pre-ferred by 𝐻in−plane, the horizontal domain wall costsmore energy. Consequently, the domain structure il-lustrated in Fig. 6(a) with longer vertical walls thanthe horizontal walls is energetically favored. Since theelectrons suffer less domain scatterings hopping alongthe 𝑦-axis in this domain structure, it becomes theeasy axis for the current flow. At large values of 𝜃,higher energy domains are suppressed and eventuallyvanish, and thus the resistivity measurement becomesinsensitive to the nematic phase because of vanishingof the domain scatterings.
In summary, we have shown that many impor-tant properties observed in the Sr3Ru2O7 could bequalitatively consistent with a realistic tight-bindingmodel together with on-site interactions treated atmean-field level. The band structure is complicatedby multibands, bilayer splitting, rotations of RuO oc-tahedra, and the spin-orbit coupling, collectively lead-ing to the high sensitivity to the small energy scales,which is the main cause of the singular behavior in theevolution of the Fermi surfaces under magnetic field.For the case of magnetic field parallel to the 𝑐-axis, thenematic order, which is interpreted as the orbital or-dering in quasi-1D 𝑑𝑦𝑧 and 𝑑𝑥𝑧 bands, appears. Thesingular behavior in 𝒟(𝜖
F) also results in the diver-
gences in the entropy and specific heat landscapes.As the magnetic field is tilted away from the 𝑐 axis(𝜃 = 0), we find that the nematic region expands in-stead of shrinking as the resistivity measurement hasindicated. To explain this discrepancy, we adopt thedomain scattering argument.[15] Furthermore, we havegiven an explanation for another experimental puzzlethat the easy axis for the current flow is always per-pendicular to the in-plane magnetic field. Measure-ments like quasiparticle interference in the spectro-scopic imaging STM and NQR which could detect theorbital ordering directly have been proposed.
We thank J. E. Hirsch and A. Mackenzie for valu-able discussions.
References[1] Grigera S A et al 2001 Science 294 329[2] Perry R S et al 2001 Phys. Rev. Lett. 86 2661[3] Grigera S A et al 2003 Phys. Rev. B 67 214427[4] Grigera S A et al 2004 Science 306 1154[5] Borzi R A et al 2007 Science 315 214[6] Millis A J, Schofield A J, Lonzarich G G and Grigera S A
2002 Phys. Rev. Lett. 88 217204[7] Green et al A G 2005 Phys. Rev. Lett. 95 086402[8] Kee H Y and Kim Y B 2005 Phys. Rev. B 71 184402[9] Yamase H and Katanin A 2007 J. Phys. Soc. Jpn. 76
073706[10] Yamase H 2007 Phys. Rev. B 76 155117[11] Puetter C, Doh H and Kee H Y 2007 Phys. Rev. B 76
235112[12] Berridge A M, Green A G, Grigera S A and Simons B D
2009 Phys. Rev. Lett. 102 136404[13] Berridge A M, Grigera S A, Simons B D and Green A G
2010 Phys. Rev. B 81 054429[14] Lee W C and Wu C 2009 Phys. Rev. B 80 104438[15] Raghu S et al 2009 Phys. Rev. B 79 214402[16] Puetter C M, Rau J G and Kee H Y 2010 Phys. Rev. B 81
081105[17] Fischer M H and Sigrist M 2010 Phys. Rev. B 81 064435[18] Lee W C, Arovas D P and Wu C 2010 Phys. Rev. B 81
184403[19] Wu C, Bergman D, Balents L and Das Sarma S 2007 Phys.
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65 220502[29] Hammel P C et al 1998 Phys. Rev. B 57 R712[30] Teitel’baum G B, Büchner B and de Gronckel H 2000 Phys.
Rev. Lett. 84 2949[31] Singer P M, Hunt A W and Imai T 2002 Phys. Rev. Lett.
88 047602[32] Lang G et al 2010 Phys. Rev. Lett. 104 097001[33] Ishida K et al 1997 Phys. Rev. B 56 R505
037201-5
Supplemental Material for
“Microscopic Theory of the Thermodynamic Properties of Sr3Ru2O7”
(See CHIN.PHYS. LETT. Vol. 33, No. 3 (2016) 037201)
Wei-Cheng Lee1, 2, ∗ and Congjun Wu1, †
1Department of Physics, University of California, San Diego, California, 92093, USA2Department of Physics, Applied Physics, and Astronomy,
Binghamton University-State University of New York, Binghamton, USA
I. THE TIGHT-BINDING MODEL
In this section, we explain the detailed band structureSr3Ru2O7, which is complicated by the t2g-orbital struc-ture (e.g. dxz, dyz, dxy), the bilayer splitting, the stag-gered distortion of the RuO octahedra, and spin-orbitcoupling. We have constructed a detailed tight-bindingHamiltonian which gives rise to band structures in agree-ment with the ARPES data in a previous work1. Wefound that a difference of the on-site potential betweenthe two adjacent RuO layers, Vbias, should be added1 inorder to fit the shape of the Fermi surfaces observed inthe ARPES experiments2. This term appears becauseARPES is a surface probe and this bilayer symmetrybreaking effect is important near the surface. Since wefocus on the thermodynamic properties which are all bulkproperties, Vbias is set to be zero in this paper. Below wewill start from this model and refer readers to Ref. [1]for more detailed information.The tight-binding band Hamiltonian H0 can be re-
duced to block forms classified by kz = 0, π correspond-ing to bonding and anti-bonding bands with respect tolayers as:
H0 = h0(kz = 0) + h0(kz = π), (1)
with h0(kz) defined as
h0(kz)=∑k
′Φ†
k,kz,s
(h0s(k, kz) g†(k, kz)
g(k, kz) h0s(k + Q, kz)
)Φk,kz,s
,
(2)
where the spinor Φ†k,kz,s
operator is defined as
Φ†k,kz,s
=(dyz †k,s,kz
, dxz †k,s,kz
, dxy †k,−s,kz
,
dyz †k+Q,s,kz
, dxz †k+Q,s,kz
, dxy †k+Q,−s,kz
); (3)
dαs,kz(k) annihilates an electron with orbital α and spin
polarization s at momentum (k, kz); Q = (π, π) is thenesting wavevector corresponding to unit cell doublinginduced by the rotations of RuO octahedra;
∑k′means
that only half of the Brillouin zone is summed. Pleasenote the opposite spin configurations s and -s for thedxz, dyz and dxy-orbitals in Eq. 3, which is convenientfor adding spin-orbit coupling later.
The diagonal matrix kernels h0s in Eq. 2 are definedas
h0s(k, kz) = As(k) + B1 cos kz − µI, (4)
where
As(k) =
ϵyzk
ϵoffk
+ isλ − sλ
ϵoffk
− isλ ϵxzk
iλ
−sλ −iλ ϵxyk
, (5)
and
B1 =
−t⊥ 0 00 −t⊥ 00 0 0
; (6)
where t⊥ is the longitudinal inter-layer hopping for thedxz and dyz orbitals. λ is the spin-orbit coupling strengthwhich comes from the on-site spin-orbit coupling term
as Hso = λ∑
i Li · Si; µ is the chemical potential; thedispersions for the dyz, dxz, and dxy bands in Eq. 5 aredefined as
ϵyzk
= −2t2 cos kx − 2t1 cos ky,
ϵxzk
= −2t1 cos kx − 2t2 cos ky,
ϵxyk
= −2t3(cos kx + cos ky
)− 4t4 cos kx cos ky
−2t5(cos 2kx + cos 2ky
)− Vxy
ϵoffk
= −4t6 sin kx sin ky, (7)
which includes longitudinal (t1) and transverse (t2) hop-ping for the the dxz and dyz orbitals, respectively, as wellas are nearest neighbor (t3), next-nearest neighbor (t4),and next-next-nearest neighbor (t5) hopping for the dxyorbital. Following the previous LDA calculations3, Vxy isintroduced to account for the splitting of the dyz and dxzstates relative to the dxy states. While symmetry forbidsnearest-neighbor hopping between different t2g orbitalsin a perfect square lattice without the rotation of Ru oc-tahedra, a term describing hopping between dxz and dyzorbitals on next-nearest neighbor sites (t6) is allowed andput into the tight-binding model.
The off-diagonal matrix kernel g(k, kz) in Eq. 2 reads
g(k, kz) = G(k)− 2B2 cos kz, (8)
where
2
B2 =
0 t⊥INT 0−t⊥INT 0 00 0 0
, (9)
and
G(k) =
0 −2tINT γ(k) 0
2tINT γ(k) 0 00 0 0
, (10)
with γ(k) = cos kx + cos ky. tINT and t⊥INT describe theintra- and inter-layer hopping between dxz and dyz in-duced by the rotations of RuO octahedra, providing the
coupling between k and k + Q.When describing the Zeeman energy, we can choose
the magnetic field B to lie on the xz plane and define
θ as the angle between B and the c-axis of the samplewithout loss of the generality. Consequently, the Zeemanterm becomes:
HZeeman = HorbitalZeeman +Hspin
Zeeman,
HorbitalZeeman = −µB B
∑i,a
(Lz,ia cos θ + Lx,ia sin θ
),
HspinZeeman = −2µB B
∑i,a,α
(Sαz ia cos θ + Sα
x ia sin θ),
(11)
where a is the layer index, B = |B|, and the matricesLx,z can be found in Ref. [1].For θ = 0, the extra Zeeman terms from x-component
of L and S couple Φ†k,kz,↑
and Φ†k,kz,↓
. Defining ϕ†k,kz
≡(Φ†
k,kz,↑,Φ†
k,kz,↓
), the Zeeman term can be written in the
matrix form as:
HZeeman = µBB∑k
′ ∑kz
ϕ†k,kz
HZ(θ)ϕk,kz
,
(12)
where
HZ(θ)=
HD
Z (θ,+) 0 HO †Z (θ) 0
0 HDZ (θ,+) 0 HO †
Z (θ)
HOZ (θ) 0 HD
Z (θ,−) 0
0 HOZ (θ) 0 HD
Z (θ,−)
,
(13)
HDZ (θ, s) = cos θ ×
−s −i 0i −s 00 0 s
, (14)
with s = ±1 and
HOZ (θ) = sin θ ×
−1 0 00 −1 −i0 i −1
. (15)
In realistic band structures measured by ARPES2,there exists an additional δ-band arising from the dx2−y2-orbital which is not covered by the current model. Theparticle filling in the t2g-orbitals is not fixed. For theconvenience of calculation, we fix the chemical potentialµ = 0.94t1 instead of fixing particle filling in the t2g-orbitals while changing magnetic fields and orientations.The corresponding fillings inside the t2g-orbitals per Ruatom varies within the range between 4.05 and 4.06 inFigs. 2, 4, 5 in the main text. This treatment does notchange any essential qualitative physics.
II. THE MEAN-FIELD THEORY
In this part, we present the process of mean-field the-ory solution.
The standard mean-field decomposition of Hint leadsto
HMFint =
∑i,a,α
∑s
Wαs d
α †s,a(i)d
αs,a(i)− USα
x dα †s,a(i)d
αs,a(i) ,
(16)where
Wαs = U
(12n
α − s Sαz
)+ V
∑β =α
nβ , (17)
with the assumptions of nαa = nα, Sα
x,z a = Sαx,z.
Since the order parameters {nαa} are non-zero even
without magnetic field, we require that the renormalizedFermi surface at zero field to be the one given in Fig. 1 inthe main text. As a result, in addition to the optimizedparameters obtained in our previous work1, we need tosubtract the following term from Eq. 16:
Hshift =∑i,a,α
∑s
Wαs (0)d
α †s,a(i)d
αs,a(i), (18)
where
Wαs (0) = 1
2Unα(0) + V∑β =α
nβ(0), (19)
and nα(0) is the occupation number in orbital α corre-sponding to the Fermi surfaces shown in Fig. 1 in themain text. This is an effect of the renormalization of thechemical potential µ and Vxy due to interactions. Afterputting all the pieces together, we finally arrive at themean-field Hamiltonian as:
HMF = H0 +HMFint −Hshift +HZeeman
≡∑k
′ ∑kz
ϕ†k,kz
HMF (k)ϕk,kz
. (20)
The order parameters are computed self-consistently.
3
FIG. 1: Fermi surface evolution for the case of magnetic field parallel to c axis (a) before (µBB = 0.0048t1) (b) inside(µBB = 0.00544t1), and (c)after (µBB = 0.006t1) the nematic phase. Significant changes in the Fermi surface topology underthe magnetic field can be seen (see Fig. 1 in the main text for the Fermi surfaces at zero field). The nematic distortion is mostobvious in the Fermi surfaces near (±π, 0) and (0,±π) as indicated by the yellow areas in (b). These parts of Fermi surfacesare composed mostly of quasi-1D bands, supporting the intimacy of nematic phase to the orbital ordering.
III. THE CASE OF THE PERPENDICULARMAGNETIC FIELD (θ = 0)
In this section we present supplemental informationfor thermodynamic properties at low temperatures arecalculated within the mean-field theory for the case of
B ∥ c, i.e., θ = 0. They can be reasonably reproducedand understood by the singular behavior of D(ϵF ) underthe magnetic field.
A. Evolution of Fermi surface
We plot the part of Fermi surfaces mostly composedof quasi-1D bands as shown in the yellow areas in Fig. 1(b), which is close to the van Hove singularities at (π, 0)and (0, π). The evolution of the Fermi surface structuresas increasing the B-field across the nematic phase bound-aries is presented in Fig. 1 a (before), b (inside), and c(after) at µBB/t1 = 0.0048, 0.00544, 0.006, respectively.Before and after the nematic phases, the Fermi surfaceshave the 4-fold rotational symmetry as exhibited in Fig.1 (a) and (c), while the 4-fold symmetry is broken into2-fold in the nematic phase.It is worthy of mentioning that the onsite spin-orbit
coupling Hso = λ∑
i Li · Si has important effects onthe Fermi surface evolutions. Hso hybridizes the oppo-site spins between quasi-1D bands dyz,xz and the 2-Dband dxy. As the spin Zeeman energy is present, thespin majority (minority) bands of dyz,xz couples to thespin minority (majority) band of dxy. Moreover, theorbital Zeeman energy provides more hybridizations be-tween quasi-1D dxz,yz bands. Combined with the abovetwo effects, the addition of the spin and orbit Zeeman en-ergies causes reconstruction of the Fermi surfaces ratherthan just chemical potential shifts. These results show
that the complexity and the sensitivity of the Sr3Ru2O7
band structure can be captured very well by our tight-binding model with a reasonable quantitative accuracy.In the following, the same model will be used to fur-ther investigate some novel physical properties observedin experiments.
B. Total density of states
Iwaya et al. has measured the STM tunneling differen-tial conductance dI
dV in the B-field for Sr3Ru2O7, whichcorresponds to measurement of the DOS. It has been ob-served that while DOS at higher and lower energy doesnot change, the DOS at the Fermi energy (D(ϵF )) in-crease significantly under the application of the magneticfield, demonstrating the violation of the rigid band pic-ture.
In our model the total DOS can be evaluated using:
ρtot(ω) = 1πN
∑k
′TrIm
[GMF (k, ω)
]GMF (k, ω) ≡
(ω + iη −HMF (k)
)−1(21)
where HMF (k) is given in Eq. 20 with the self-consistentorder parameters and N is the total number of sites inthe bilayer square lattices.
The profiles of the total DOS at several differentmagnetic field strength are plotted in Fig. 2(a), andclearly a rigid band picture does not apply here. D(ϵF )(ρtot(ω = 0) in the plot) increases significantly as thenematic phase is approached. This feature can also bedirectly understood by the picture of Fermi surface recon-struction, since the changes in the Fermi surface topol-ogy inevitably lead to the non-monotonic behavior in
4
FIG. 2: (a) The total DOS ρtot(ω) as a function of µBB.The broadening factor η is set to be η = 0.002t1. ρtot(ω)does not follow the rigid band picture and ρtot(ω = 0) hasa sudden increase near the nematic region. Inset: the totalDOS at zero field for a wider range of |ω|/t1 ≤ 0.4. The peakscorresponding to van Hove singularities are near ω/t1 ≈ −0.2and ω/t1 ≈ 0.35. (b) The DOS of the quasi-1D bands. Inset:the DOS of quasi-1D bands at zero field for a wider rangeof |ω|/t1 ≤ 0.4. Only the peak around ω/t1 ≈ 0.35 remains,meaning that this peak is due to the van Hove singularitiesin quasi-1D bands.
ρtot(ω = 0). In particular, comparing Fig. 1 in themain text and Fig. 1 in this Supplemental Material, it isstraightforward to see that more Fermi surfaces appearnear the (±π, 0) and (0,±π) as the magnetic field in-creases. Since there are van Hove singularities near these
four k points, ρtot(ω = 0) is expected to increase. Asthe magnetic field is increased further so that the vanHove singularities are all covered below the Fermi sur-faces, ρtot(ω = 0) starts to drop (not shown here).
At the first glance, the entropy measurement and ourresults of the total DOS seem to contradict with the STMmeasurement. While both the entropy measurement andour results develop a maximum around the nematic re-gion in D(ϵF ), the STM measurement showed insteadthat D(ϵF ) keeps increasing even after the nematic re-gion is passed. To resolve this discrepancy, several realis-
tic features need to be considered before comparing ourcalculations with the STM results. Since the STM is asurface probe and the surface of the material is usuallycleaved to have the oxygen atoms in the outermost layer,there is an oxygen atom lying above each uppermost Ruatom. Consequently, the tunneling matrix element willbe mostly determined by the wavefunction overlaps be-tween the p-orbitals of the oxygen atom and the d-orbitalsof the Ru atom, resulting in a much smaller matrix ele-ment for dxy orbital compared to dxz,yz orbitals1. Theminimal model to take this effect into account is to ex-tract the DOS only from the quasi-1D orbitals, which isplotted in Fig. 2(b). Although the overall profile in Fig.2(b) is not exactly the same as that in Ref. 4, which is at-tributed to more complicated momentum dependence oftunneling matrix elements5–7 not considered here, it cap-tures the increasing DOS with the magnetic field whichis more consistent with Ref. 4.
The insets in Fig. 2 plot the total and quasi-1D bandDOSs at zero field within a wider range of |ω|/t1 ≤ 0.4.The peaks corresponding to the van Hove singularitiesreside at ω/t1 ≈ −0.2 and ω/t1 ≈ 0.35, far away from theFermi energy. The reason why the small energy scale likeZeeman energy (∼ 0.003t1) can push the system to getcloser to the van Hove singularities at energies far awayfrom the Fermi energy is the help of the metamagnetism.In the mean-field theory, the magnetization produces aneffective chemical potential shift as sUSα
z for electrons atorbital α and spin s. As a result, under the magnetic fieldthe jump in the magnetization gives Sα
z ∼ 0.05 withinthe range of experimental interests. This leads to theeffective chemical potential shift about ±0.18t1 for U =3.6t1, which is large enough to push the system closer tothe van Hove singularities. This renormalization of thechemical potential by the interaction is also part of thecause for the violation of the rigid band picture.
IV. THE CASE OF THE TILTED MAGNETICFIELDS (θ = 0)
We present the Fermi surface configuration in the pres-ence of tilted magnetic field. The Fermi surfaces without
any interaction for B ∥ x with strength µB |B| = 0.1t1is plotted in Fig. 3, and an anisotropy can be seen.Although such anisotropy in the band structure is notimportant at low field, it can be amplified by the effectof the interactions, driving the system more susceptibleto the nematic phase as the critical points is approached.As a result, the portion of the nematic phase in the phasediagram is enlarged as θ increases from 0◦ to 90◦ as seenin our calculations.
V. SUMMARY AND MORE DISCUSSIONS
The failure of a rigid band picture is essentially a con-sequence of the interplay between spin-orbit coupling and
5
FIG. 3: The non-interacting Fermi surfaces for B ∥ x andµBB = 0.1t1. The anisotropy in Fermi surfaces is alreadyvisible, especially for the areas near (±π, 0) and (0,±π).
the Zeeman energy, despite the strong correlation effectcould also result in the violation of the rigid band shiftupon doping8. Because the spin-orbit coupling hybridizesthe quasi-1D bands and 2-D bands with opposite spins,the Zeeman energy naturally induces the reconstructionof the Fermi surfaces instead of just rigid chemical poten-tial shifts. This singular behavior in D(ϵF ) also resultsin the divergences in the entropy and specific heat land-scapes, since at very low temperature both quantities areapproximately proportional to D(ϵF ). Because the diver-gence observed by Rost el. al.9 start approximately at6-7 Tesla which is not very close to the quantum criti-cal point residing about 8 Tesla, a direct application ofthe quantum critical scaling seems to be inappropriate.The explanation of the critical exponent associated withthis divergence could not be complete without taking theband structure singularity into account in this particularmaterial10.As the magnetic field is tilted away from the c axis
(θ = 0), we find that nematic region expands instead ofshrinking as the resistivity measurement has indicated.From the theoretical viewpoints, the tilt of the magneticfield induces an extra in-plane component of the orbitalZeeman energy which explicitly breaks the C4 symmetrydown to the C2 symmetry. As argued above that thissystem is very sensitive to small energy scale, the effectof this extra Zeeman energy is not important at low fieldbut could amplify the effect of interaction to drive thesystem toward nematicity as the quantum critical pointis approached. As a result, the nematic phase is more fa-vored and stable in the presence of the in-plane magneticfield and it requires another Fermi surface reconstruc-
tion at even higher magnetic field in order to weaken thenematic phase by reduced D(ϵF ).
To explain this discrepancy between our theory and theresistivity measurement, we adopt the domain scatteringargument proposed by Raghu et. al.11. Furthermore,we have given an explanation for another experimentalpuzzle that the easy axis for the current flow is alwaysperpendicular to the in-plane magnetic field. Measure-ments like quasiparticle interference in the spectroscopicimaging STM and NQR which could detect the orbitalordering directly have been proposed to be more reliableprobes for the nematicity in this material than the resis-tivity measurement.
Finally we would like to comment on limitation of thepresent theory. Although we have found the ’re-entry’ be-havior of the nematic phase, i.e., the appearance of thenematic phase only at the finite temperature but not atthe zero temperature, near the upper-field boundary, theexperiments showed this behavior near both upper- andlower- field boundary. In our calculations, the re-entrybehavior is due to the increase of the density of stateswithin the narrow energy window around the Fermi en-ergy opened by thermal energy, but we do not reject otherschemes for the re-entry behavior. One possible schemeis an analogue of ferromagnetism without exchange split-ting proposed by Hirsch12. He showed that the nearestneighbor interactions could result in a spin-dependentrenormalization on the bandwidth (equivalently, the ef-fective mass). As a result, the filling for different spinbands can be different because of the unequal effectivemasses, leading to the ferromagnetism even without theexchange splitting as in the Stoner model.
In Hirsch’s original paper, the re-entry of the ferromag-netism at higher temperature was found. Since we onlyconsidered the on-site interactions in our model, suchan effect is beyond the scope of the current theory. Itis possible that after including the nearest neighbor in-teraction, the renormalizations of the bandwidths havenovel temperature-dependences, leading to a phase dia-gram better consistent with the experiments. If this isthe correct scheme, the re-entry of the nematic statesshould be accompanied by a change in the kinetic energydue to the effective mass renormalization, which could beexamined by the sum rules for the optical properties12–16.
Another possible scheme for the re-entry behavior isthe quantum critical fluctuations. It is well-known thatthe influences of the critical fluctuations extend from thequantum critical point to finite temperature in a V -shaperegion in the phase diagram. Moreover, the critical fluc-tuations in this material contain not only the ferromag-netic but also the nematic ones. As a result, it is notsurprising that the competition between these two typesof critical fluctuations leads to a intriguing phase dia-gram at the finite temperature, and the study towardthis direction is currently in progress.
6
∗ Electronic address: [email protected]† Electronic address: [email protected] W.-C. Lee, D. P. Arovas, and C. Wu, Phys. Rev. B 81,184403 (2010).
2 A. Tamai et al., Phys. Rev. Lett. 101, 026407 (2008).3 D. Singh and I. Mazin, Phys. Rev. B 63, 165101 (2001).4 K. Iwaya et al., Phys. Rev. Lett. 99, 057208 (2007).5 J. Tersoff and D. Hamann, Phys. Rev. Lett. 50, 1998(1983).
6 Y. Zhang et al., Nat. Phys. 4, 627 (2008).7 W.-C. Lee and C. Wu, Phys. Rev. Lett. 103, 176101(2009).
8 J. Farrell et al., Phys. Rev. B 78, 180409 (2008).9 A. W. Rost et al., Science 325, 1360 (2009).
10 J. Lee et al., Nat. Phys. 5, 800 (2009).11 S. Raghu et al., Phys. Rev. B 79, 214402 (2009).12 J. E. Hirsch, Phys. Rev. B 59, 6256 (1999).13 Y. Okimoto et al., Phys. Rev. Lett. 75, 109 (1995).14 Y. Okimoto et al., Phys. Rev. B 55, 4206 (1997).15 D. N. Basov and T. Timusk, Rev. Mod. Phys. 77, 721
(2005).16 A. D. LaForge et al., Phys. Rev. Lett. 101, 097008 (2008).
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%----------------------- 2014--7--11------------------------------ \documentclass[a4paper,twoside]{article} \usepackage{AMS2LA,fancyhdr,CJK,multicol,graphics,greekup,bm,ccmap} %\usepackage{amsfonts}\usepackage{mathrsfs} \usepackage{amssymb}\usepackage{gensymb,mathcomp} %-------------------hyperline------------------------------- \usepackage[dvipdfm, a4paper, pdfstartview=FitH, bookmarks=false, colorlinks=false, linkcolor=blue, urlcolor=blue, pdfborder=001, citecolor=blue, pdftitle={Changepdftitle}, pdfauthor={Changepdfauthor}, pdfcreator=CHIN. PHYS. LETT., pdfproducer=CPL, pdfkeywords={changePACS}]{hyperref} %------------Page layout and margin and Headrule------------------- \headsep=6pt \headheight=4mm \topmargin=0pt \voffset= -5mm \oddsidemargin=-0.5cm \evensidemargin=-0.5cm \marginparwidth=0pt \marginparsep= 0pt \marginparpush=0pt \textheight=24.4cm \textwidth=17cm \footskip=20pt \columnsep=6mm % put Page layout before \pagestyle{fancy} \pagestyle{fancy} \renewcommand{\headrulewidth}{0.42pt} %-------------------user-defined------------------------------- \def\headrule{\kern 1mm \hrule width 17cm \kern -1mm}% \def\footnoterule{\kern 1mm \hrule width 7cm \kern 2.2mm}% \def\REF#1{\par\hangindent\parindent\indent\llap{#1\enspace}\ignorespaces}% %----------Row spacing of Text and Table and *Footnote------------- \renewcommand{\thefootnote}{\fnsymbol{footnote}} \parindent=15pt \nofiles % \setlength{\parskip}{0pt} \renewcommand{\baselinestretch}{1.03} % text-distance \renewcommand{\arraystretch}{1.1} % table-distance \abovedisplayskip=9.0pt plus 2.0pt minus 2.0pt \belowdisplayskip=9.0pt plus 2.0pt minus 2.0pt %---------------No.page and Odd and Even------------------- \newcommand{\cplyear}{2016} \newcommand{\cplvol}{33} \newcommand{\cplno}{10} \newcommand{\cplpagenumber}{10{????}} \setcounter{page}{1} \newcommand{\cplpage}{\cplpagenumber-\thepage} \cfoot{\cplpage} \chead{\small{\href{http://cpl.iphy.ac.cn} {CHIN.\,PHYS.\,LETT.}~~Vol.\,\cplvol, No.\,\cplno\,({\cplyear})\,\cplpagenumber}}% %---------------------------------------------------------- \begin{document} \begin{CJK}{GBK}{song}\vspace* {-6mm} \begin{center} %-------------------------Title---------------------------- \large\bf{\boldmath{Relativistic Brueckner-Hartree-Fock Theory for Finite Nuclei}} %------------------------Footnote-------------------------- \footnote{Correspondence author: [email protected] \\ \bf \% Tel.: (O) 010-62765620;(Mobile) 13601366422 \\ \% Email of all authors: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected] %Email: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected] \hspace*{1.8mm}$^{**}$Corresponding author. Email: \hspace*{1.8mm}\copyright\,{\cplyear} \href{http://www.cps-net.org.cn}{Chinese Physical Society} and \href{http://www.iop.org}{IOP Publishing Ltd}} \\[4mm]
%------------------------Authors---------------------------- \normalsize \rm{}Shihang Shen(申时行), Jinniu Hu(胡金牛), Haozhao Liang(梁豪兆), Jie Meng(孟杰) , Peter Ring, Shuangquan Zhang (张双全) %----------------------COM. or University------------------- \\[1mm]\small\sl $^{1}$State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University, Beijing 100871 $^{2}$RIKEN Nishina Center, Wako 351-0198, Japan $^{3}$Department of Physics, Nankai University, Tianjin 300071 $^{4}$Department of Physics, Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan $^{5}$School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191 $^{6}$Department of Physics, University of Stellenbosch, Stellenbosch, South Africa $^{7}$Physik-Department der Technischen Universit\"at M\"unchen, D-85748 Garching, Germany %------------------------Received date---------------------- \\[4mm]\normalsize\rm{}(Received ????) \end{center} \end{CJK} %----------------------Abstract and PACS-------------------- \vskip -1mm \noindent{\narrower\small\sl{}Starting with a bare nucleon-nucleon interaction, for the first time the full relativistic Brueckner-Hartree-Fock equations are solved for finite nuclei in a Dirac-Woods-Saxon basis. No free parameters are introduced to calculate the ground-state properties of finite nuclei. The nucleus $^{16$O is investigated as an example. The resulting ground-state properties, such as binding energy and charge radius, are considerably improved as compared with the non-relativistic Brueckner-Hartree-Fock results and much closer to the experimental data. This opens the door for \emph{ab initio} covariant investigations of heavy nuclei. \par}\vskip 3mm \normalsize\noindent{\narrower\sl{PACS: 21.60.De, %:.Ab.initio.methods 21.10.Dr, %:.Binding.energies.and.masses} {\rm\hspace*{13mm}DOI: 10.1088/0256-307X/\cplvol/\cplno/\cplpagenumber} \par}\vskip 3mm %-------------------TEXT TEXT TEXT TEXT--------------------- \begin{multicols}{2} \textit{Ab initio} calculations, i.e., the proper description of finite nuclei with a bare nucleon-nucleon ($NN$) interaction adjusted only to the scattering data of free nucleons, form a central problem of theoretical nuclear physics since the middle of last century. These realistic $NN$ interactions are characterized by a repulsive core at short distance,$^{[1]}$ a strong attraction at intermediate range, and are dominated by one-pion exchange at large distance.$^{[2]}$ Many methods have been proposed in the past to treat their singular behavior, such as Brueckner theory$^{[3,4]}$ and variational methods.$^{[5,6]}$ Recently with the great progress of the high-precision $NN$ interactions, such as Reid93,$^{[7]}$ AV18,$^{[8]}$ CD Bonn,$^{[9]}$ and chiral potentials,$^{[10,11]}$ and with increasing computer technology, more and more \textit{ab initio} methods have been developed to study the nuclear many-body system, e.g., the Green's function Monte Carlo method,$^{[12]}$ the self-consistent Green's function method,$^{[13]}$ the coupled-cluster method,$^{[14]}$ the lattice chiral effective field theory,$^{[15]}$ and the no-core shell model.$^{[16]}$
ground-state properties in RBHF theory are improved considerably as compared with the non-relativistic results. The deviation from the experimental values have been decreased from 18\% to 6\% in the case of the energy and from 16 \% to 7 \% in the case of the charge radius, which is consistent with the conclusions in the infinite nuclear matter.$^{[50]}$ This energy of $^{16}$O is also very close to the value of $E=-119.7$ MeV obtained within the No Core Shell Model (NCSM) using the chiral $NN$ interaction N$^{3}$LO$^{[51]}$ and to the value of $E=-121.0$ MeV obtained within the Coupled Cluster (CC) method.$^{[52]}$ The spin-orbit splittings in RBHF theory is only slightly smaller than the experimental data. For the $1p$ proton shell we have a deviation of about $5\%$. Of course the results of the calculations with PKO1 which has been fitted to these data shows only a very small deviation of 0.5 \% for the energy, of 1.6 \% for the radius and the spin-orbit splitting. \vskip 2mm \noindent{\footnotesize {\bf Table 1.} (Color online) Total energy $E$ and charge radius $r_c$ of $^{16}$O as a function of energy cut-off $\varepsilon_{\rm cut}$ calculated within RBHF theory with the realistic $NN$ interaction Bonn A.$^{[2]}$ \label{Fig.\,1 \vskip 2mm \tabcolsep 8pt \centerline{\footnotesize \begin{tabular}{lcccc} &\multicolumn{1}{c}{$E$ (MeV)} & \multicolumn{1}{c}{$r_c$ (fm)} & \multicolumn{1}{c}{$r_m$ (fm)} & \multicolumn{1}{c}{$\Delta E_{\pi1p}^{ls}$ (MeV)} \\ \hline Exp.$^{[47-49,53]}$ & $-127.6$ & $2.70$ & $2.54$ & $6.3$ \\ RBHF & $-120$.$7$ & $2.52$ & $2.38$ & $6$.$0$ \\ BHF$^{[28]}$ & $-105.0$ & $2.29$ & $-$ & $7.5$ \\ DDRH$^{[54]}$& $-106.4$ & $2.72$ & $-$ & $-$ \\ DDRHF$^{[54]}$ & $-142.6$ & $2.62$ & $-$ & $4.5$ \\ NCSM$^{[51]}$ & $-119.7$ & $-$ & $-$ & $-$ \\ CC$^{[52]}$ & $-121.0$ & $-$ & 2.30 & $-$ \\ PKO1$^{[38]}$ & $-128.3$ & $2.68$ & 2.54 & $6.4$ \\ \end{tabular}}} \medskip Next we compare in Fig.\,2 our self-consistent results in finite nuclei with those obtained in Ref.\,[54] by two "\textit{ab initio}" calculations based on the LDA. There the full RBHF equations are solved for nuclear matter at various densities and the corresponding scalar and vector self-energies are derived. Then density-dependent coupling strengths for the exchange of various mesons in a relativistic Hartree (DDRH) or a Hartree-Fock (DDRHF) model have been adjusted to these results. In this case it is possible to investigate finite nuclei in an \textit{ab inito} approach without any phenomenological parameters. \vskip 4mm \centerline{\includegraphics{0941fig2}} \vskip 2mm \centerline{\footnotesize \begin{tabular}{p{7.5 cm}}\bf Fig.\,2. \rm (Color online) Energy per particle and charge radius of $^{16}$O by (relativistic) BHF theories compared with experimental data and other calculations. See text for details. \label{Fig.\,2} \end{tabular}} \medskip The results of these calculations based on the LDA also listed in the 4$^{th}$ and the 5$^{th}$ rows of Table 1. The charge radii are rather well reproduced in these local density
deviates by 6 \% from the experimental value and the charge radius agrees with the experimental value up to 7 \%. Also the spin-orbit splitting is well reproduced. Despite the good agreement of these results, there is room for improvements. The RBHF-theory presented here is no exact solution of the nuclear many-body problem. So far, rearrangement terms are not taken into account and higher order diagrams in the hole-line expansion are not included. Those effects have been taken into account in some approximation in non-relativistic calculations,$^{[59]}$ but for relativistic theories they are left for future investigations. On the other side, our method has the potential to investigate heavier nuclei, where exact solutions are impossible, in particular systems without spin saturation and with large neutron excess. In this case we hope to be able to gain a parameter-free, microscopic understanding of open questions in modern phenomenological density functional theories, such as their isospin dependence or the importance of the tensor terms.$^{[37]}$ \begin{acknowledgments} We thank Wenhui Long for the discussions and for providing the RHF code. This work was partly supported by the Major State 973 Program of China No.~2013CB834400, Natural Science Foundation of China under Grants No.~11175002, No.~11335002, No.~11405090, No.~11375015, and No.~11621131001, the Research Fund for the Doctoral Program of Higher Education under Grant No.~20110001110087, the DFG cluster of excellence \textquotedblleft Origin and Structure of the Universe\textquotedblright\ (www.universe-cluster.de), the CPSC Grant No. 2012M520100, and the RIKEN IPA and iTHES projects. \end{acknowledgments} \section*{\Large\bf References} \vspace*{-0.8\baselineskip}\frenchspacing \hskip 7pt {\footnotesize \REF{[1]} Jastrow R 1951 {\it Phys. Rev.} {\bf 81} 165 %DOI:10.1103/PhysRev.81.165 \REF{[2]} Machleidt R 1989 {\it Adv. Nucl. Phys.} {\bf 19} 189 %DOI:10.1007/978-1-4613-9907-0_2 \REF{[3]} Brueckner K A, Levinson C A and Mahmoud H M 1954 {\it Phys. Rev.} {\bf 95} 217 %DOI:10.1103/PhysRev.95.217 \REF{[4]} Day B D 1967 {\it Rev. Mod. Phys.} {\bf 39} 719 %DOI:10.1103/RevModPhys.39.719 \REF{[5]} Jastrow 1955 {\it Phys. Rev.} {\bf 98} 1479 %DOI:10.1103/PhysRev.98.1479 \REF{[6]} Day B D 1978 {\it Rev. Mod. Phys.} {\bf 50} 495 %DOI:10.1103/RevModPhys.50.495 \REF{[7]} Stoks V G J, Klomp R A M, Terheggen C P F and de Swart J J 1994 {\it Phys. Rev.} C {\bf 49} 2950 %DOI:10.1103/PhysRevC.49.2950 \REF{[8]} Wiringa R B, Stoks V G and Schiavilla R 1995 {\it Phys. Rev.} C {\bf 51} 38 %DOI:10.1103/PhysRevC.51.38 %%Changed Year, the old is 1995 \REF{[9]} Machleidt R 2001 {\it Phys. Rev.} C {\bf 63} 024001 %DOI:10.1103/PhysRevC.63.024001 \REF{[10]} Epelbaum E, Hammer H-W and Meissner U-G 2009 {\it Rev. Mod. Phys.} {\bf 81} 1773 %DOI:10.1103/RevModPhys.81.1773 \REF{[11]} Machleidt R and Entem D R 2011 {\it Phys. Rep.} {\bf 503}
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CHIN.PHYS. LETT. Vol. 33, No. 12 (2016) 124207
A Single Mode Hybrid III–V/Silicon On-Chip Laser Based on Flip-Chip Bonding1
Technology for Optical Interconnection *2
Hai-Ling Wang(�° )1, Wan-Hua Zheng(x�u)1,2∗∗3
1Laboratory of Solid State Optoelectronic Information Technology, Institute of Semiconductors,4
Chinese Academy of Sciences, Beijing 10008352State Key Laboratory on Integrated Optoelectronics, Institute of Semiconductors,6
Chinese Academy of Sciences, Beijing 1000837
(Received 14 October 2016)8
A single mode hybrid III–V/silicon on-chip laser based on the flip-chip bonding technology for on-chip optical9
interconnection is demonstrated. A single mode Fabry–Perot laser structure with micro-structures on an InP10
ridge waveguide is designed and fabricated on an InP/AlGaInAs multiple quantum well epitaxial layer structure11
wafer by using i-line lithography. Then, a silicon waveguide platform including a laser mounting stage is designed12
and fabricated on a silicon-on-insulator substrate. The single mode laser is flip-chip bonded on the laser mounting13
stage. The lasing light is butt-coupling to the silicon waveguide. The laser power output from a silicon waveguide14
is 1.3mW, and the threshold is 37mA at room temperature and continuous wave operation.15
PACS: 42.55.Px DOI: 10.1088/0256-307X/33/12/12420716
Silicon photonics is now widely accepted as a key17
technology in next-generation communication systems18
and data interconnects.[1−3] For silicon photonics, one19
of major obstacles is the absence of a compact and20
power-efficient silicon-based light source, due to the21
indirect band gap of silicon, especially single mode hy-22
brid silicon lasers with very simple fabrication. As the23
key component, the light source has been attracting24
numerous attention, because it is difficult to generate25
light for silicon itself.26
Recently, various strategies for light source in sil-27
icon photonic circuits have been demonstrated.[4−13]28
For example, integrating direct-band III–V semicon-29
ductor materials on the silicon-on-insulator (SOI) sub-30
strate through a direct bonding technology,[8−13] and31
integrating the III–V laser diode (LD) on the SOI sub-32
strate by using the flip-chip bonding technology.[14−19]33
The flip-chip bonding and butt-coupling are amongst34
the best technologies to achieve the record device35
performance when the footprint is not a concern.36
The efficient flip-chip bonded hybrid lasers are the37
devices demonstrated by researchers from Fujitsu[14]38
and Arakawa.[15] The high precision flip-chip bond-39
ing technology with exceptionally low alignment error40
(∼0.1µm) was a crucial factor in misalignment loss41
reduction. The highest reported wall plug efficiency42
(WPE) to date (9.5%) has been achieved in a hybrid43
laser by Kotura et al.[18] Kotura’s design utilizes an ex-44
ternal cavity reflective semiconductor optical amplifier45
(SOA), butt-coupled to a silicon waveguide Bragg mir-46
ror on SOI chip. However, most of these devices have47
the complicated spot-size converter (SSC) to minimize48
coupling losses and to improve the coupling power and49
need expensive high resolution processing.50
Generally, the stable single mode lasers are ob-51
tained by using distributed feedback (DFB),[20,21] or52
external cavity structures (ECLs),[22,23] while manu-53
facturing of these structures require additional pro-54
cessing steps of epitaxial regrowth and electron beam55
lithography. To avoid the complex regrowth steps and56
expensive electron-beam lithography, a single-mode57
laser has been presented in theory and experiment,58
which depends on reflective defects (micro-structures59
or slot structures).[24,25] Using slots in a standard mul-60
timode Fabry–Perot (FP) laser to sufficiently select a61
single mode. Fabricating the slot structures does not62
need high resolution processing, and only need i-line63
lithography, dry etching into the ridge waveguide.64
Here we propose a single mode hybrid III–V/silicon65
on-chip laser based on the flip-chip bonding tech-66
nology. Firstly, a single mode laser with slots on67
InP ridge waveguide is designed and fabricated on an68
InP/AlGaInAs multiple quantum well epitaxial layer69
structure. The slot structures are designed as the high70
order grating and the whole fabrication processing are71
achieved by the standard 1:1 photolithography to re-72
duce the cost. Then, a waveguide platform including73
a simple tapered silicon waveguide and a laser mount-74
ing stage is designed and fabricated on an SOI sub-75
strate. The single mode laser is flip-chip bonded on76
the laser mounting stage. Lastly, the lasing light is77
butt-coupling to the tapered silicon waveguide. The78
laser power output from silicon waveguide is 1.3mW,79
the threshold is 37mA at room temperature and con-80
tinuous wave operation, and the side-mode suppres-81
sion ratio (SMSR) is 35.7 dB.82
The schematic diagram of the proposed hybrid83
integrated light source is shown in Fig. 1(a), which84
is a hybrid integration structure consisting a single85
mode LD on a silicon waveguide platform. The sili-86
*Supported by the National Basic Research Program of China under Grant No 2012CB933501, and the National Natural ScienceFoundation of China under Grant Nos 61307033, 61274070, 61137003 and 61321063.
∗∗Corresponding author. Email: [email protected]© 2016 Chinese Physical Society and IOP Publishing Ltd
124207-1
CHIN.PHYS. LETT. Vol. 33, No. 12 (2016) 124207
con waveguide platform includes a silicon waveguide87
and a laser mounting stage. An edge emitting LD las-88
ing wavelength of about 1.55µm flip-chip bonded on89
the mounting stage. The laser has the same epitaxial90
layer structure as the 1550 nm commercial epitaxial91
wafer, which consists of an AlGaInAs/InP MQWs ac-92
tive layer, a p-InP cladding layer, an n-cladding layer,93
and the emission peak around 1550 nm. The lasing94
light is butt-coupling into silicon waveguide. We de-95
sign a simple tapered silicon waveguide on SOI to cou-96
ple the lasing light of the LD. All of the silicon waveg-97
uides designed in this study are based on silicon-on-98
insulator (SOI) wafer with a 0.34µm top silicon layer99
and a 2µm buried silicon oxide layer. Figure 1(b)100
show the schematic diagram of the proposed single101
mode LD. The single longitudinal mode is obtained102
by the slot structures on the 4.5µm-wide InP ridge103
waveguide. The slot structures are located the middle104
of the InP waveguide surface. The two cleaved facets105
as the feedback facets to form the laser cavity.106
x
y
z
InP ri
dge Slo
t P-cladding
N-cladding
InP substrate
AlgalnAs MQWs
Laser diode Silic
on sub
stra
te
Buried
oxide
(a) (b)
Fig. 1. (a) The schematic diagram of hybrid integratedlight source with a spot-size converter, in which the siliconwaveguide is labeled by purple circle, and the single modelaser diode is labeled by the red circle. (b) The schematicdiagram of the single mode laser with the slot structureson the InP ridge waveguide.
To obtain the single longitudinal mode laser diode,107
we should to design the optimum geometrical param-108
eters of slots such as the slot period, slot width and109
slot etch depth. In fact, the proposed slot structure is110
based on Bragg’s condition with high-order grating.[26]111
The slot period is defined as 2neffΛ = aλB, where q112
is a integer representing the order of the grating, Λ113
is the period of the slot structure, neff is the effective114
index of the slot structure waveguide region, and λB115
is the Bragg wavelength. To determine the slot pa-116
rameters, we use the FDTD method to simulate the117
effect of the slot period and the slot etch depth on the118
reflection of a group slots. In the simulation, under119
the effective index approximation, the p-InP cladding120
layer of 1.85µm and the AlGaInAs quantum well has121
an effective index of 3.18 and 3.52 in the 2D situation,122
respectively. Figures 2(a) and 2(b) show the contours123
of the simulated amplitude reflection of the slot struc-124
ture versus the slot etch depth and the slot period125
fixed at 500µm cavity length, 24 slot number, and126
1.6µm InP ridge waveguide height. We can see that a127
number of local maximum values of amplitude reflec-128
tion at different wavelengths vary with the depth and129
period of slots. As the reflection amplitudes vary with130
wavelength, single mode can be realized under special131
injection currents. From Fig. 2(a), we can see that132
amplitude reflection (labeled with the green circle) is133
about 35% at the wavelength 1.55µm and 8.9µm slot134
period, which is about the 37th order grating. This135
period has a about 37 nm free spectral range. From136
Fig. 2(b), we can see that the 1.1µm slots etch depth137
(labeled with the black circle) leads to about 30% re-138
flectivity amplitude in the 1545–1560 nm region. Ac-139
cording to the simulating result, we determine the slot140
structure period and etch depth as 8.9µm and 1.1µm,141
respectively. Moreover, the duty cycle (slot width)142
of the slot structure period is another parameter we143
should consider. In this study, we are interested in the144
slot width around 1.1µm that can be easily fabricated145
by standard photolithography.146
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6
1.56
1.555
1.55
1.545
1.54
1.56
1.555
1.55
1.545
1.54
7 7.5 8 8.5 9 9.5 10
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
-0.35
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
Etch depth of slots (mm)
Wavele
ngth
(mm
)W
avele
ngth
(mm
)Period of slots (mm)
(a)
(b)
Fig. 2. Contours of the simulated amplitude reflection.(a) Wavelength versus slot period with a slot etch depthof 1.1µm and a slot width of 1.1µm. (b) Wavelength ver-sus depth of slot structure with a period of 8.9µm, a slotwidth of 1.1µm and period number of 24.
1.0
1.0-10 0 10
2000
1000
00.5 0.0
0.0
X (mm)
Z (mm
)
Monitor value (arb. units)
Pathway
monitor
Pow
er
(a) (b)
Fig. 3. The transmission map and the lasing light cou-pling efficiency of the proposed tapered silicon waveguide.
To keep the single mode transmission in the sili-147
con waveguide (shown in Fig. 1(a)), we need to design148
the silicon waveguide parameters, including the silicon149
waveguide width, length and etch depth. The tapered150
silicon waveguide length was designed as 2000µm,151
then the silicon waveguide width was tapered from152
10µm to 450 nm. The BPM method was used to sim-153
124207-2
CHIN.PHYS. LETT. Vol. 33, No. 12 (2016) 124207
ulate the coupling efficiency of the silicon waveguide,154
as shown in Fig. 3. The butt coupling efficiency is155
more than 50%.156
Figures 4(a) and 4(b) show the schematic diagram157
of the proposed tapered silicon waveguide output ter-158
minal and the TE mode pattern at the end of the159
silicon waveguide with the 0.45µm tip width. Sim-160
ulation was performed at 1550 nm by using the rig-161
orous coupled-wave analysis method. From Fig. 4(b),162
a single guiding transverse electric (TE) mode in the163
silicon waveguide was obtained. The waveguide width164
at the output facet was therefore set to 450 nm in the165
fabrication process.166
-2 -1 0 1 2
0
1
2
3
40.45 mm
0.34 mm
2 mm
Si
Buried oxide
Silicon substrate
x
y
X direction (mm)
Y d
irection (mm
)
3.55552T1016
1.94618T10-8
(a)(b)
PECVD SiO2
Fig. 4. (a) The schematic diagram of the proposed ta-pered silicon waveguide output terminal, and (b) thetransverse electric mode pattern for the 0.45µm tip widthof the silicon waveguide.
(a) (b)
(c)
Fig. 5. The microscopy photographs and the scanningelectron microscope (SEM) image of the silicon platformand the hybrid laser. (a) The microscopy of the siliconplatform with metal electrode, (b) the microscopy of hy-brid silicon laser, and (c) the SEM image of the hybridsilicon laser.
Firstly, the single mode laser is fabricated by using167
standard i-line lithography and inductively coupled168
plasma (ICP) reactive ion etching. Then, a silicon169
waveguide and a laser diode mounting stage is fabri-170
cated on the SOI substrate by using the electron beam171
lithography and ICP etching. On the LD mounting172
stage, there are four silicon pedestals. A layer of SiO2173
insulator is deposited on the silicon waveguide to pas-174
sivation. Finally, the LD is flip-chip bonded on the LD175
mounting by using the flip-chip bonder. The vertical176
alignment accuracy of LD was precisely adjusted by177
the surface position of the etched silicon pedestals and178
the thickness of evaporation AuSn solder. The verti-179
cal alignment accuracy was made better than ±0.1µm180
by dry etching and evaporation controllability. Fig-181
ures 5(a) and 5(b) are the microscopy photography182
of the silicon waveguide platform and the hybrid sil-183
icon laser with an LD mounted on it. Figure 5(c) is184
the SEM image of the hybrid integrated light source.185
From Fig. 5(c), we can see that the gap between the186
LD and silicon waveguide input facet was about 1µm187
and was not filled with AuSn.188
1545 1550 1555 1560 1565-90
-80
-70
-60
-50
-40(b)
(a)
Outp
ut
pow
er
(dB
m)
Wavelength (nm)
SMSR=35.7 dB
0 20 40 60 80 100 120 1400.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Injection current (mA)O
utp
ut
pow
er
(mW
)
0
1
2
3
4
5
Volt
age (
V)
Fig. 6. (a) Measured P–I–V curve under the continuouscurrent. (b) The spectrum at the current of 90mA of thehybrid silicon laser.
The measured P–I–V curve and the optical spec-189
trum of the fabricated hybrid laser are shown in190
Figs. 6(a) and 6(b). At room temperature, the laser191
has a threshold of 30mA and a peak output power of192
1.3mW under 150mA continuous wave operation, as193
shown in Fig. 6(a). At the current of 90mA, the las-194
ing wavelength is 1556 nm. Over 35 dB SMSR is real-195
ized at room temperature, as shown in Fig. 6(b). The196
measured power coupling efficiency is lower than the197
simulating results as shown in Fig. 3. Theoretically,198
there may be several factors that affect the coupling199
efficiency. Firstly, there is a different mode size in200
laser diode and the silicon waveguide leading to mode201
scattering and reflecting, and thus reducing the cou-202
pling efficiency. The other reasons may be that the203
reflectivity and scattering of the silicon waveguide in-204
put facet due to Fresnel reflection and the imperfect205
waveguide facet. It is possible to reduce the reflectiv-206
ity by filling the gap between the LD facet and silicon207
waveguide input facet with a material matched to the208
refractivity of the silicon waveguide input facet. The209
other method is making SSC on the silicon platform210
and using CMOS processing in the future.211
In summary, we have proposed the design, fab-212
rication and measurement of a heterogeneously inte-213
grated III–V-on-Silicon laser based on a single mode214
124207-3
CHIN.PHYS. LETT. Vol. 33, No. 12 (2016) 124207
laser with the slot structure on InP ridge waveguide215
and the flip-chip bonding technology. A single mode216
hybrid III–V/silicon laser, lasing at 1556 nm and with217
1.3mW on-chip output power, 35.7 dB SMSR, 30mA218
threshold, is obtained.219
References220
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