prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: When did the London Paving and Lighting Act pass, which mandated the numbering of houses?

Read the following text and answer briefly.

Context 1: A Brief History of Benjamin Franklin's Residences on Craven Street, London: 1757 - 1775 - Journal of the American Revolution
Benjamin Franklin House, 36 Craven St, London  (Photo by Elliott Brown | Wikimedia Commons)
If one looked into Benjamin Franklin’s time on Craven Street, they might initially believe he lived at 36 Craven Street the entirety of his two stays in London based on the plethora of articles on the internet that say so  If they dug a little deeper they might read that he lived at No  27 Craven Street, previously numbered 7, but now numbered 36; or that he lived exclusively at No  7 Craven Street; or that he lived in multiple residences on Craven Street; or that he moved out of No  36 to another house on Craven Street and then moved back into No  36 the last year of his residence  What is one to believe with all of the conflicting accounts? What does the historical record have to say about Franklin’s time on Craven Street?
Figure 1  Spur Alley 1685  “A map of the parish of St Martins in the Fields, taken from ye last survey, with additions (1685)”  (© The British Library Board, Shelfmark: Maps Crace Port  13 2, Item number: 2)
Before Craven Street existed there was Spur Alley, a narrow passageway sandwiched between the Hungerford Market to the north (now Charing Cross Station) and Scotland Yard and the Northumberland House and Garden to the south  It was flanked on both ends by major thoroughfares, the Strand on the west, connecting Westminster to London by road, and the River Thames on the east, not only connecting the two cities to each other and to Southwark on the south side of the Thames, but connecting the entire metropolis to the rest of the world  Being located in the City of Westminster, Spur Alley had escaped the devastation of the Great Fire of London in 1666 leaving its wooden structures, built in the early part of seventeenth century, intact, but also in dire need of restoration or demolition  “The ratebooks show that during the last thirty years or so of their existence the houses in Spur Alley were in a very bad condition  Few of them were rated at more than a few shillings and many of them were unoccupied ”[1] The landowner, William, 5th Baron Craven, desiring to increase the profitability of his assets, tore down the derelict structures on Spur Alley around 1730 and leased the newly established lots to builders  By 1735, twenty brick houses in the Georgian style had been built on the west side and sixteen on the east side of the way now called Craven Street [2]
Figure 2  Craven Street 1746  (John Rocque London, Westminster and Southwark, First Edition 1746, Motco Enterprises Limited, motco com)
Letters to Franklin during his residence with Mrs  Margaret Stevenson, his landlady on Craven Street, were addressed rather vaguely; “Craven Street/Strand”, “Mrs  Stevensons in Craven Street”, or “Benjamin Franklin Esqr ” are but a few examples  Letters from Franklin referenced “London,” or sometimes “Cravenstreet,” but never included a number  Despite the absence of numbered addresses in Franklin’s correspondence, there was a sense of one’s place in the neighborhood based on entries in the Westminster Rate Books (tax assessments)  The Rate Books did not list house numbers during Franklin’s time there, but they did list the residents of Craven Street in a particular order that became the default numbering system for the street  Number one was associated with the first resident listed under “Craven Street” in the Rate Books and was the northernmost house on the west side of the street  The numbers increased counter-clockwise down the west side and up the east side in accordance with the list of residents  In 1748, the first year of Margaret Stevenson’s (Stevens in the Rate Books for that year) residence on Craven Street, she is listed as the twenty-seventh resident, the second house north of Court Street (later Craven Court, now Craven Passage) on the east side of the street [3]
In 1766, Parliament passed the London Paving and Lighting Act (6 Geo  3 c  26), “An act for the better paving, cleansing, and enlightening, the city of London, and the liberties thereof; and for preventing obstructions and annoyances within the same; and for other purposes therein mentioned ”[4] One of the other purposes therein mentioned was the numbering of houses  With an aim to bring order to the chaotic numbering systems or lack thereof on London streets the Act provided that “… the said commissioners … may also cause every house, shop, or warehouse, in each of the said streets, lanes, squares, yards, courts, alleys, passages, and places, to be marked or numbered, in such manner as they shall judge most proper for distinguishing the same ”[5] This was quite an undertaking that took years to accomplish  It was a decade later before numbered addresses on Craven Street in the City of Westminster appeared in The London Directory (1776)  The London Directory and its competitors were published primarily by booksellers or printers to supplement their income and were highly profitable  To say they were competitive is an understatement  “Some of the most hotly disputed struggles over copyright in the century concerned guidebooks  Many were optimistically emblazoned with a royal license and a notice that the work had been entered at Stationers’ Hall  Various struggles between rival guides intensified as the potential for profits became clear ”[6] The London Directory boldly proclaimed to contain “An ALPHABETICAL LIST OF THE NAMES and PLACES of ABODE of the MERCHANTS and PRINCIPAL TRADERS of the Cities of LONDON and WESTMINSTER, the Borough of SOUTHWARK, and their Environs, with the Number affixed to each House ”[7] Kent’s Directory made a similar proclamation: “An Alphabetical LIST OF THE Names and Places of Abode OF THE DIRECTORS of COMPANIES, Persons in Public Business, MERCHANTS, and other eminent TRADERS in the Cities of London and Westminster, and Borough of Southwark WITH THE NUMBERS as they are affixed to their Houses agreeable to the late Acts of Parliament ”[8] Mrs  Stevenson wasn’t included in the directories because she didn’t meet the criteria of being a merchant or trader, not because she was a woman  Although it is rare to see women listed in the directories, some examples do exist [9] If Mrs  Stevenson had appeared in the directories in 1776 it would not have been on Craven Street as she had moved to Northumberland Court, a stone’s throw away, the previous year [10] A comparison of Craven Street residents whose names and addresses do appear in the directories with the same residents as they appear in the Westminster Rate Books determines if the numbering systems were congruent  For the most part they were  For example, Joseph Bond at No  30, William Rowles at No  31, Samuel Sneyd at No  32, and Jonathan Michie at No  35 in The London Directory coincide with their places of residence in the Westminster Rate Books; however, errors did occur  The 1776 edition of The London Directory lists Brown & Whiteford, wine merchants, at No  9 Craven Street while the Westminster Rate Books list them as the twenty-ninth residents  Obviously, it makes no sense to have Brown & Whiteford at No  9 in The London Directory and their next-door neighbor, Joseph Bond, at No  30  The same error appears in Baldwin’s The New Complete Guide for 1783  The New Complete Guide may have “borrowed” the error from The London Directory  It was not uncommon for the owner of one directory to copy entries from another to save both time and money  Beginning in 1778 and contrary to The London Directory, Kent’s Directory faithfully followed the numbering system of the Westminster Rate Books in all of its editions and listed Brown & Whiteford at No  29 as did Bailey’s Northern Directory in 1781  Perhaps realizing their error, The London Directory changed their listing of Brown & Whiteford from No  9 to No  29 in their 1783 edition and maintained that listing thereafter 
Sometime prior to 1792, the embankment on the Thames at the south end of Craven Street had been sufficiently extended allowing for the construction of ten new houses below the original houses: “ … four houses, Nos  21–24, were built on the west side, and six houses, Nos  25–30, on the east side of the way ”[11] In a note in the same report, the new numbering system is explained  “The houses in the street, which had previously been numbered consecutively down the west side and up the east side, were then renumbered on the same system to include the additional houses ”[12] Because the new houses (21-24) on the west side were built below the existing houses (1-20), houses 1-20 retained their original numbering 
Figure 4  Craven Street 1799  (Richard Horwood’s Map of London, Westminster and the Borough of Southwark 1799, Motco Enterprises Limited, motco com)
One would think that the numbers of the sixteen original houses on the east side, Nos  21 – 36, would simply increase by ten with the addition of the ten new houses, but such was not the case; they increased by nine  How could that be? The only possible explanation is that No  21 of the original houses was demolished to make way for the construction of the northernmost of the six new houses on the east side (No  30)  Evidence of No  21’s demolition appears in the lease granted to Charles Owen by William, 7th Baron Craven, in 1792, which describes No  22 as: “All that messuage in Craven Street late in the occupation of Francis Deschamps undertaker … being the Southernmost house in the Old Buildings on the East Side of the said Street numbered with the No  22 ”[13] The lease describes No  22 as being the southernmost house in the old buildings on the east side of Craven Street  Clearly the house previously at No  21 did not exist when the lease granted to Charles Owen was written in 1792 as it used to be the southernmost house  It is also worth noting that in 1790, The London Directory listed Jacob Life at No  21 (original numbering)  In 1791-2, it listed him at No  6  With No  21 vacated, it would allow for its demolition and the construction of the tenth new house  By utilizing lot No  21 for the new construction, only nine additional lots were needed to build the ten houses, hence, Margaret Stevenson’s former residence at 27 became 36 (27 + 9) in the renumbering and not 37 
For nearly a century and a half after Franklin departed London for America in March of 1775 the scales were tipped heavily in favor of his residence having been No  7 Craven Street  As early as 1807 in London; Being An Accurate History And Description Of The British Metropolis And Its Neighborhood, Volume 4, one would have read: “In Craven Street is a house, No  7, remarkable for having been the residence of Dr  Benjamin Franklin [14] In 1815, the identical phrase appeared in The Beauties of England and Wales [15] After 23 editions of not mentioning Franklin, his name finally appeared in the 24th edition of The Picture of London in 1826: “The house, No  7, Craven Street, in the Strand, was once the residence of Dr  Benjamin Franklin ”[16] In 1840, Jared Sparks referred to Franklin’s Craven Street residence appearing in London guide books in his voluminous The Works of Benjamin Franklin: “In the London Guide Books, ‘No  7, Craven Street,’ is still indicated as the house in which Dr  Franklin resided ”[17] In 1846, George Gulliver F R S , in his book, The Works of William Hewson, wrote: “She [Polly] had been upon terms of the warmest friendship with Dr  Franklin
Figure 5  No  7 Craven Street with Memorial Tablet  (Photo courtesy of British History Online, and the Survey of London)
since she was eighteen years of age  That eminent philosopher resided with her mother, Mrs  Margaret Stevenson, at No  7, Craven Street, Strand, during the fifteen years of his abode in London ”[18] Guide books mentioning Franklin at No  7 continued to proliferate throughout the century: Handbook for London; Past and Present, Volume I (1849);”[19] Handbook for Modern London (1851);”[20] The Town; Its Memorable Characters and Events (1859);”[21] London and Its Environs (1879) [22] There was an anomaly when London In 1880 Illustrated With Bird’s-Eye Views of the Principal Streets, Sixth Edition (1880) placed Franklin at 27 Craven street [23] The anomaly lasted for six years until his place of residence was changed to No  7 in the revised edition, London  Illustrated by Eighteen Bird’s-Eye Views of the Principal Streets (1886) [24] London Past and Present; Its History, Associations, and Traditions, Volume 1 (1891), copied the 1849 Handbook for London almost word-for-word and included, “The house is on the right from the Strand ”[25] In October of 1867, The Society of Arts in London declared that: “In order to show how rich the metropolis is in the memory of important personages and events, which it would be desirable to mark by means of tablets on houses, the Council have caused an alphabetical list to be prepared, … ”[26] Franklin had been elected a corresponding member to the Society in 1756 and was a popular choice among Council members deciding who they were to memorialize [27] By January of 1870, a tablet honoring him was affixed to the house they believed to have been his residence while in London, No  7 Craven Street in the Strand on the west side of the street [28] A majority of historians writing about Franklin in the nineteenth and early twentieth century placed him at No  7: O  L  Holley, The Life of Benjamin Franklin (1848); E  M  Tomkinson, Benjamin Franklin (1885); John Torrey Morse, Benjamin Franklin (1891); Paul Elmer More, Benjamin Franklin (1900); John S  C  Abbot, Benjamin Franklin (1903); Sydney George Fisher, The True Benjamin Franklin (1903)  A notable exception is D  H  Montgomery’s His Life Written by Himself published in 1896  He has Franklin at No  27 Craven Street  It seems then that depending upon the source, Franklin was thought to have lived at either No  7 or No  27, but not both, the overwhelming majority favoring No  7  As late as 2011, Franklin is still mentioned as living at No  7 [29]
In 1913, No  7 was scheduled to be torn down  An article in the March 1914 edition of The Book News Monthly, describes the situation:
As is well known to informed American pilgrims, it has been possible for all admirers of the famous philosopher and statesman to pay their respects to his memory before that house, No  7 Craven Street, just off the Strand, which was his chief home during his two sojourns in the British capital, but even as these lines are being written the London newspapers are recording that that interesting shrine is soon to be pulled down to make room for a restaurant  It is some mitigation of this misfortune to remember that at the most the Craven Street house was nothing more than a reproduction of the one in which Franklin had his suite of four rooms, for the structure has been rebuilt since Franklin’s time  When, then, some one makes a piteous plea that at least the philosopher’s bedroom shall be preserved, the soothing answer is that the apartment in question is only a replica of that in which the illustrious American enjoyed his well-earned slumbers in 1757-62 and 1764-75  The restaurant-builder, however, with an eye doubtless to possible American patronage, has assured the world that every effort will be made to preserve as much as possible of the entire structure [30]
Concerned with the possible demolition of Franklin’s residence, the Royal Society of Arts (formerly the Society of Arts[31]) initiated an inquiry into the matter [32] The London County Council, having taken over the responsibility of placing memorial tablets on notable houses from the Royal Society, was charged with the investigation  It ultimately fell to Sir George Laurence Gomme, a clerk to the Council, to come up with a response  A few years earlier Sir George had discovered Margaret Stevenson residing at No  27 Craven Street in the Westminster Rate Books  He must have wondered why No  7 on the west side of Craven Street was being celebrated as Franklin’s residence when the evidence clearly showed otherwise 
Sir George and his staff examined the various London directories discussed earlier and came up with a novel explanation for the discrepancy  They concluded that there had been two numbering systems on Craven Street  An anonymous author echoes Sir George’s conclusion about the two numbering systems in an article in The Journal of the Royal Society of Arts:
…an inspection of the directories of that time proves that there were at least two systems of numbering in Craven Street before the erection of the additional houses  According to one of these the numbers started from the top (Strand end) on the west side of the street, and ran down to the bottom to No  20, then crossed over and went back to the Strand along the east side – 21 to 36  According to the other system, the east side of the street was numbered from the bottom upwards, starting at No 1  This was not apparently in general use, but there is evidence that this numbering was at all events occasionally used 
The evidence of these two systems of numbering, and for believing that Mrs  Stevenson’s house was first No  7 under the oldest system, next No  27 under the second system, and finally No  36 under the latest and existing system, is to be found in the various directories and the Westminster rate-books [33]
The “evidence” mentioned above consisted of The London Directory’s listing of Brown & Whiteford at No  9: “The rate-books for 1781 and 1786 show the house next but one to the north of Mrs  Stevenson’s house as in the occupation of Brown and ‘Whiteford,’ while the old directories mention the business of the firm as wine merchants, and give their address as 9, Craven Street – then a little later, down to 1791, as 29, Craven Street  Curiously enough, in the years 1778 to 1780, or 1781, Lowndes gives it as No  9, and Kent as 29 ”[34] Ignoring Kent’s Directory having Brown and Whiteford as 29 and The London Directory (Lowndes) having Brown and Whiteford “a little later” as 29, and knowing that Mrs  Stevenson lived two doors south of them, Sir George concluded that her house must have been numbered 7, even though there is no listing in any of the directories of her residence ever being No  7  He surmised that the No  7 on the west side of Craven Street with the memorial tablet thought to have been Franklin’s residence had simply been confused with number 7 (27) on the east side  Again from The Journal of the Royal Society of Arts:
Taking all the evidence together, there cannot be any doubt whatever that Mrs  Stevenson’s house, in which Franklin lodged, was the house two doors north from Craven Court, first numbered 7, afterwards 27, and finally 36, and consequently that the house in which Franklin lived was that now numbered 36, not the one now numbered 7, on which the tablet is placed [35]
A response to The Royal Society of Arts was issued: “… the London County Council … informed the Society that it had made a mistake and that No  36 Craven street was the building that deserved commemoration ”[36] The Society accepted the Council’s conclusion, and despite assurances of preservation by the restaurant builder, No  7 was torn down the following year 
Sir George’s assertion “that Mrs  Stevenson’s house, in which Franklin lodged, was the house two doors north from Craven Court” was correct, however, his assertion that it was “first numbered 7, afterwards 27”, was not  It was only by association with the errant entry of Brown & Whiteford at No  9 from 1776-1782 in The London Directory that Mrs  Stevenson’s address was conjured to be No  7  The problem with associating her address exclusively with that of Brown & Whiteford at No  9 during those years is that, as previously demonstrated, The London Directory also listed four other Craven Street residents, Bond, Rowles, Sneyd, and Michie, who’s addresses did conform to the numbering system in The Westminster Rate Books  If Brown & Whiteford at No  9 was indicative of a numbering system different from The Westminster Rate Books, Bond, Rowles, Sneyd, and Michie would have been listed as Nos  10, 11, 12, and 15, respectively  So on one hand Sir George was relying on the Westminster Rate Books to establish Mrs  Stevenson at No  27 and on the other hand he was dismissing the Westminster Rate Books to establish her at No  7  Instead of using the anomalous listing of Brown & Whiteford at No  9, he could have just as easily, and more logically, used the Bond et al  listings, or the post-1782 Brown & Whiteford listing in the London Directory at No  29 to establish Mrs  Stevenson at No  27  Even if there had been two numbering systems, his assertion that No  27 was first numbered 7 would still be false  The earliest numbering system was the Westminster Rate Books dating from the early 1730s when the houses were constructed  Brown & Whiteford at No  9 didn’t appear until 46 years later and then only for a brief period 
There is ample evidence in Franklin’s correspondence and in a memoir by Polly Hewson (Mrs  Stevenson’s daughter) that Benjamin and Mrs  Stevenson lived in not one, but two houses on Craven Street  On July 6, 1772, Polly wrote to Benjamin from her house at Broad Street North in London: “My Mother I must tell you went off last friday week, took our little Boy with her and left Mr  Hewson [Polly’s husband, William] the care of her House [27 Craven Street]  The first thing he did was pulling down a part of it in order to turn it to his own purpose, and advantage we hope  This Demolition cannot affect you, who at present are not even a Lodger [Benjamin was traveling at the time], your litterary apartment remains untouch’d, the Door is lock’d …”[37] In a memoir about her husband written after his death Polly writes: “He [William Hewson] began his Lectures Sept  30, 1772, in Craven-street, where he had built a Theatre adjoining a house which he intended for the future residence of his family ”[38] On October 7, 1772, Benjamin wrote to his son William: “I am very well  But we [Mrs  Stevenson and I] are moving to another House in the same street; and I go down tomorrow to Lord LeDespencer’s to [stay a] Week till things are settled ”[39] To his son-in-law, Richard Bache, on the same day he wrote: “We are moving to another House in the [street] leaving this to Mr  Hewson ”[40] Writing to a friend on October 30, 1772 he explained: “I should sooner have answered your Questions but that in the Confusion of my Papers, occasioned by removing to another House, I could not readily find the Memorandums …”[41] On November 4, 1772 Benjamin informed his wife Deborah of the move  “We are removed to a more convenient House in the same street, Mrs  Stevenson having accommodated her Son-in-Law with that we lived in  The Removing has been a troublesome Affair, but is now over ”[42]
An agreement had been struck between the parties  Margaret and Benjamin would move to another house on Craven Street and allow Polly and William to move into No  27, the large yard behind the house being spacious enough to accommodate the anatomy school William wished to build [43] Perhaps the idea was inspired by Margaret’s next-door neighbor at No  26, Dr  John Leake, a man-midwife and founder of the Westminster Lying-in Hospital, who had built a theater adjoining his residence in which he practiced anatomy and taught midwifery [44]
After Margaret and Benjamin vacated No  27, Polly, William, their son William Jr , and William’s younger sister, Dorothy Hewson, took up residence there [45] In the 1773 Westminster Rate Books for Craven Street, Mrs  Stevenson’s (Stephenson in the Rate Books) name has been crossed out and replaced with “William Hewson ”[46] Further proof that the Hewsons had indeed moved into 27 Craven Street has been confirmed by the discovery of human and animal remains buried in the basement of No  36 (formerly No  27 and now the Benjamin Franklin House), a by-product of the dissections that took place at William’s anatomy school [47]
So what house on Craven Street did Mrs  Stevenson and Benjamin move into after vacating No  27? An examination of the Westminster Rate Books for the years 1774 and 1775 reveal them living not at No  7 on the west side of Craven Street as one might expect from the overwhelming consensus of nineteenth century guidebooks and biographies, but surprisingly at No  1 [48] The controversy of No  7 being torn down was all for naught as it had never been Franklin’s residence  Sir George was correct on that point  Unfortunately, No  1 was torn down as well in the early part of the twentieth century  The first time No  1 is mentioned as Franklin’s second residence is in the Survey of London: Volume 18, St Martin-in-The-Fields II: the Strand published by the London County Council in 1937, ironically the same County Council that had declared No  36 as Franklin’s only residence twenty-four years earlier 
From 1748 until 1772 Margaret ‘Stephenson’ occupied this house [No  27 (36)], and it was there that Benjamin Franklin settled after his arrival in London in 1757 as Agent to the General Assembly of Pennsylvania … In October, 1772, Mrs  Stevenson and Franklin removed to No  1, Craven Street (now demolished), and No  36 was for the next two years occupied by William Hewson, surgeon, who had married Mary Stevenson [49]
In the spring of 1774, William Hewson died unexpectedly of septicemia two weeks after cutting himself while dissecting a cadaver  Polly was left to care for their two young sons and was pregnant with a daughter she would give birth to in August of the same year  Is it possible that Margaret and Benjamin moved back into No  27 to assist Polly after the death of her husband as suggested in The Americanization of Benjamin Franklin?[50]
If the Westminster Rate Books are to be believed, the answer is no  For the year 1774, the Rate Books list Margaret Stevenson at No  1 and William Hewson at No  27  For the year 1775, they list Margaret Stevenson at No  1 and Magnus Falkner (Falconer/Falconar) at No  27  Magnus was William’s assistant at the anatomy school and fiancé to William’s sister, Dorothy  On his death bed, William instructed Polly, “let Mr  Falconar be my successor ”[51] Magnus would immediately take over the running of the anatomy school and continue William’s unfinished research  Four months later, he and Dorothy would marry [52] Essentially only two things changed at 27 Craven Street after William’s death: Polly gave birth to her daughter, and Magnus replaced William as the lease holder, so even if Margaret and Benjamin had wished to move back into No  27, there would have been no room for them  It is also interesting to note that considering the multiple times Benjamin wrote of his move out of No  27 (and complained of it), he never once mentioned moving back into No  27 in any of his correspondence after Mr  Hewson’s death 
Figure 6  No  36 Craven Street  (Photo courtesy of David Ross, britainexpress com)
In sum, based on the Westminster Rate Books[53] and Franklin’s correspondence, Mrs  Stevenson is known to have resided at No  27 (36) Craven Street from 1748 to 1772  It follows that, aside from the two years Franklin spent in Philadelphia from 1762 to 1764, he resided there from 1757 to 1772  Franklin’s correspondence also reveals that in the autumn of 1772, he and Mrs  Stevenson moved to another house on Craven Street  The 1773 Westminster Rate Books show her name crossed off at No  27 and William Hewson’s inserted  The following year the Rate Books list her at No  1 Craven Street  Evidence for Mrs  Stevenson and Benjamin remaining at No  1 after William’s death appears in the Westminster Rate Books for 1775 which have Mrs  Stevenson still residing at No  1 and Magnus Falkner residing at No  27  Further evidence can be construed from the lack of any mention of a move back into No  27 in Franklin’s correspondence  Despite the many theories one could devise as to why Franklin was thought to have lived at No  7 Craven Street by so many guide books and Franklin biographers of the nineteenth century, one thing is certain; at some point after Franklin’s departure to America in March of 1775, and no later than 1807, someone mistakenly associated him with No  7 on the west side of Craven Street, and it soon became his de facto residence  Credit must go to D  H  Montgomery in 1896 and Sir George in 1913 for setting the record partially straight by placing Franklin at No  27(36)  In 1937, the London County Council gave us the first accurate account of Franklin’s residences on Craven Street in the Survey of London at No  27(36) and No  1  It has been shown conclusively that No  27 was never previously numbered 7  It was, however, renumbered 36 in 1792 after ten additional houses were built at the southern end of the street and remains No  36 to this day 
[1] “Craven Street and Hungerford Lane”, in Survey of London: Volume 18, St Martin-in-the-Fields II: the Strand, ed  G H Gater and E P Wheeler (London, 1937), 27-39, Early History of the Site 
http://www british-history ac uk/survey-london/vol18/pt2/pp27-39
[2] “England, Westminster Rate Books, 1634-1900,” from database with images, Craven Street – 1735, FamilySearch from database by FindMyPast and images digitized by FamilySearch; citing Westminster City Archives, London 
[3] Ibid , Craven Street – 1748 
[4] The Statutes at Large, From Magna Charta to the End of the Eleventh Parliament of Great Britain  Anno 1761 Continued, Vol  XXVII, ed  Danby Pickering, (Cambridge, John Archdeacon, 1767), 96 
[6] James Raven, Publishing Business in Eighteenth-Century England, (Woodbridge: The Boydell Press, 2014), 201 
[7] The London Directory For the Year 1776, Ninth Edition, (London: T  Lowndes, 1776), title page 
[8] Kent’s Directory For the Year 1778, Forty-Sixth Edition, (London: Richard and Henry Causton, 1778), title page 
[9] A listing in Kent’s Directory for the Year 1882 on p  28 reveals, “Brown Sarah, Leather-seller, 1, Westmoreland-buildings, Aldersgate-street”, and in Kent’s Directory for the Year 1883 on p  175, “Whiteland Mary, Wine & Brandy Mercht  Jermyn-str  St  James ”
[10] “The Papers of Benjamin Franklin,” Sponsored by The American Philosophical Society and Yale University, Digital Edition by The Packard Humanities Institute, 22:263a 
http://franklinpapers org/franklin
Mrs  Stevenson wrote to Benjamin Franklin a letter from her new home at 75 Northumberland Court on November 16, 1775: “In this Court I have a kind friend, Mr  Lechmoen he comes and seats with me and talks of you with a hiy regard and friendship ”
[11] Survey of London, Early History of the Site 
[12] Survey of London, Footnotes/n 10 
[13] Survey of London, Historical Notes/No  31 
[14] David Hughson, LL D , London; Being An Accurate History And Description Of The British Metropolis And Its Neighbourhood, To Thirty Miles Extent, From An Actual Perambulation, Vol  IV, (London: W  Stratford, 1807), 227 
[15] The Reverend Joseph Nightingale, The Beauties of England and Wales: Or, Original Delineations, Topographical, Historical, and Descriptive, of Each County, Vol  X, Part III, Vol  II (London: J  Harris; Longman and Co ; J  Walker; R  Baldwin; Sherwood and Co ; J  and J  Cundee; B  and R  Crosby and Co ; J Cuthell; J  and J  Richardson; Cadell and Davies; C  and J  Rivington; and G  Cowie and Co , 1815), 245 
[16] John Britton, F S A  & Co , ed , The Original Picture of London, Enlarged and Improved: Being A Correct Guide For The Stranger, As Well As For the Inhabitant, To The Metropolis Of The British Empire Together With A Description Of The Environs, The Twenty-Fourth Edition (London: Longman, Rees, Orme, Brown, and Green, 1826), 479 
[17] Jared Sparks, The Works of Benjamin Franklin, Vol  VII, (Philadelphia: Childs & Peterson, 1840), 151 
[18] George Gulliver, F R S , The Works of William Hewson, F  R  S , (London: Printed for the Sydenham Society, MDCCCXLVI), xx 
[19] Peter Cunningham, Handbook for London; Past and Present, Vol  I, (London: John Murray, 1849), 245 
[20] F  Saunders, Memories of the Great Metropolis: or, London, from the Tower to the Crystal Palace, (New York: G P  Putnam, MDCCCLII), 138 
[21] Leigh Hunt, The Town; Its Memorable Characters and Events, (London: Smith, Elder and Co , 1859), 185 
[22] K  Baedeker, London and Its Environs, Including Excursions To Brighton, The Isle of Wight, Etc : Handbook For Travelers, Second Edition, (London: Dulau and Co , 1879), 133 
[23] Herbert Fry, London In 1880 Illustrated With Bird’s-Eye Views of the Principal Streets, Sixth Edition, (New York: Scribner, Welford, & Co , 1880), 50 
[24] Herbert Fry, London  Illustrated By Eighteen Bird’s-Eye Views of the Principal Streets, (London: W  H  Allen and Co , 1886), 40 
[25] Henry B  Wheatley, F S A , London Past and Present; Its History, Associations, and Traditions, Vol  1, (London: John Murray, New York: Scribner & Welford, 1891), 473 
[26] The Journal of the Society of Arts, Vol  XV, No  778, (October 18, 1867): 717 
[27] D  G  C  Allen, “Dear and Serviceable to Each Other: Benjamin Franklin and the Royal Society of Arts,” American Philosophical Society, Vol  144, No  3, (September 2000): 248-249 
Franklin was a corresponding member in 1756 because he was still residing in Philadelphia  He became an active member the following year when he moved to London 
[28] The Journal of the Society of Arts, Vol  XVIII, No  894, (Jan  7, 1870): 137 
“Since the last announcement, the following tablets have been affixed on houses formerly occupied by – Benjamin Franklin, 7 Craven-street, Strand, W C ”
[29] Franklin in His Own Time, eds  Kevin J  Haytes and Isabelle Bour, (Iowa City, University of Iowa Press, 2011), xxxvii 
“Takes lodgings with Margaret Stevenson at No  7 Craven Street ” It is unknown if the editors are referring to No  7 on the west side of Craven Street or No  36 on the east side using Sir George’s explanation of No  36 being previously numbered 7 
[30] Henry C  Shelly, “American Shrines on English Soil, III  In the Footprints of Benjamin Franklin,” in The Book News Monthly, September, 1913 to August, 1914, (Philadelphia: John Wanamaker, 1914), 325 
[31] The Journal of the Royal Society of Arts, Vol  LVI, No  2,880, (Jan  31, 1908): 245 
http://babel hathitrust org/cgi/pt?id=mdp 39015058423073;view=1up;seq=251
“His Majesty the King, who is Patron of the Society, has granted permission to the Society to prefix to its title the term ‘Royal,’ and the Society will consequently be known in future as the ‘Royal Society of Arts ’”
[32] Nineteenth Annual Report, 1914, of the American Scenic and Historic Preservation Society, (Albany: J  B  Lyon Company, 1914), 293 
http://babel hathitrust org/cgi/pt?id=wu 89072985302;view=1up;seq=4;size=150
[33] The Journal of the Society of Arts, Vol  LXII, No  3,183, (Nov  21, 1913): 18 
http://babel hathitrust org/cgi/pt?id=mdp 39015058422968;view=1up;seq=26
[36] Allen, “Dear and Serviceable,” 263-264 
[37] Papers of Benjamin Franklin, 19:20 
[38] Thomas Joseph Pettigrew, F  L  S , Memoirs of the Life and Writings of the Late John Coakley Lettsom With a Selection From His Correspondence, Vol  I, (London: Nichols, Son, and Bentley, 1817), 144 of Correspondence 
[39] Papers of Benjamin Franklin, 19:321b 
[40] Ibid , 19:314 
[41] Ibid , 19:353a 
[43] Simon David John Chaplin, John Hunter and the ‘museum oeconomy’, 1750-1800, Department of History, King’s College London  Thesis submitted for the degree of Doctor of Philosophy of the University of London , 202 
“Following Falconar’s death [1778] the lease [27 Craven Street] was advertised, and the buildings were described as:
A genteel and commodious house, in good Repair, with Coach-house and Stabling for two Horses…consisting of two rooms and light closets on each floor, with outbuildings in the Yard, a Museum, a Compleat Theatre, and other conveniences  (Daily Advertiser, 27 August 1778)”
[44] Simon Chaplin, “Dissection and Display in Eighteenth-Century London,” in Anatomical Dissection in Enlightenment England and Beyond: Autopsy, Pathology and Display, ed  Dr  Piers Mitchell, (Burlington: Ashgate Publishing Company, 2012), 108 
“Given that a nearby building at 35 [ No  26 in Franklin’s time] was occupied by the man-midwife John Leake, who advertised lectures – including lessons in the art of making preparations – at his ‘theatre’ between 1764 and 1788, it is possible that some facilities were shared  In both cases, however, the buildings [Leake’s residence at No  26 and Hewson’s residence next door at 27] served a dual function as domestic accommodation and as sites for lecturing and dissection ”
[45] George Gulliver, F R S , The Works of William Hewson, F  R  S , (London: Printed for the Sydenham Society, MDCCCXLVI), xviii 
[46] Westminster Rate Books, Craven Street – 1773, courtesy of the City of Westminster Archives 
[47] S W  Hillson et al , “Benjamin Franklin, William Hewson, and the Craven Street Bones,” Archaeology International, Vol  2, (Nov  22, 1998): 14-16 
http://dx doi org/10 5334/ai 0206
[48] Westminster Rate Books, Craven Street – 1774, 1775, courtesy of the City of Westminster Archives 
[49] Survey of London, Historical Notes/No  36, Craven Street (not sourced) 
[50] Gordon S  Wood, The Americanization of Benjamin Franklin, (New York: The Penguin Press, 2004), 261 
[51] Pettigrew, Memoirs, 146 of Correspondence 
[52] http://founders archives gov/documents/Franklin/01-22-02-0178, note 7  “Falconar married Hewson’s sister five months after the Doctor’s death; most of the Craven Street circle attended the wedding, and BF gave away the bride: Polly to Barbara Hewson, Oct  4, 1774, APS” (American Philosophical Society); “England Marriages, 1538–1973 ,” database, FamilySearch (https://familysearch org/ark:/61903/1:1:V52W-TGS : accessed September 15, 2015), Magnus Falconar and Dorothy Hewson, September 12, 1774; citing Saint Martin In The Fields, Westminster, London, England, reference ; FHL microfilm 561156, 561157, 561158, 942 B4HA V  25, 942 B4HA V  66 
[53] I chose to rely on the Westminster Rate Books for the numbering system on Craven Street  The books were consistent throughout the eighteenth century in the ordering of residents on the street and were used as the basis for the 1792 re-numbering  For the most part, commercial directories aligned with them as well  If by chance a directory didn’t initially align, it would inevitably produce future editions that did 
Benjamin Franklin, Benjamin Franklin House, London
More from David Turnquist
If one looked into Benjamin Franklin’s time on Craven Street, they might   
I think it’s very ironic that on the street maps included in your excellent article, Craven Street is so close to Scotland Yard  Because following the back and forth juxtapositions of numbers 7, 27 and 36 Craven Street (throw in 75 Northumberland Court and 1 Craven Street, too) was a case that could confound Sherlock Holmes 
Excellent job of deciphering street renumbering material spanning sixty years, including that of a wrong house number (# 7) being erroneously identified and then perpetuated in subsequent street map printings  It’s gratifying at least to know that the present day #36 Craven Street is the correct house for Ben Franklin tourists to visit  Except for #1 Craven Street for the last three years Franklin was in London, but we won’t get into that 
Again, excellent article, David!
Context 2: \section{Introduction}
\label{sec:introduction}

Probabilistic models have proven to be very useful in a lot of applications in signal processing where signal estimation is needed \cite{rabiner1989tutorial,arulampalam2002tutorial,ji2008bayesian}  Some of their advantages are that 1) they force the designer to specify all the assumptions of the model, 2) they provide a clear separation between the model and the algorithm used to solve it, and 3) they usually provide some measure of uncertainty about the estimation 

On the other hand, adaptive filtering is a standard approach in estimation problems when the input is received as a stream of data that is potentially non-stationary  This approach is widely understood and applied to several problems such as echo cancellation \cite{gilloire1992adaptive}, noise cancellation \cite{nelson1991active}, and channel equalization \cite{falconer2002frequency} 

Although these two approaches share some underlying relations, there are very few connections in the literature  The first important attempt in the signal processing community to relate these two fields was the connection between a linear Gaussian state-space model (i e  Kalman filter) and the RLS filter, by Sayed and Kailath \cite{sayed1994state} and then by Haykin \emph{et al } \cite{haykin1997adaptive}  The RLS adaptive filtering algorithm emerges naturally when one defines a particular state-space model (SSM) and then performs exact inference in that model  This approach was later exploited in \cite{van2012kernel} to design a kernel RLS algorithm based on Gaussian processes 

A first attempt to approximate the LMS filter from a probabilistic perspective was presented in \cite{park2014probabilistic}, focusing on a kernel-based implementation  The algorithm of \cite{park2014probabilistic} makes use of a Maximum a Posteriori (MAP) estimate as an approximation for the predictive step  However, this approximation does not preserve the estimate of the uncertainty in each step, therefore degrading the performance of the algorithm 

In this work, we provide a similar connection between state-space models and least-mean-squares (LMS)  Our approach is based on approximating the posterior distribution with an isotropic Gaussian distribution  We show how the computation of this approximated posterior leads to a linear-complexity algorithm, comparable to the standard LMS  Similar approaches have already been developed for a variety of problems such as channel equalization using recurrent RBF neural networks \cite{cid1994recurrent}, or Bayesian forecasting \cite{harrison1999bayesian}  Here, we show the usefulness of this probabilistic approach for adaptive filtering 

The probabilistic perspective we adopt throughout this work presents two main advantages  Firstly, a novel LMS algorithm with adaptable step size emerges naturally with this approach, making it suitable for both stationary and non-stationary environments  The proposed algorithm has less free parameters than previous LMS algorithms with variable step size \cite{kwong1992variable,aboulnasr1997robust,shin2004variable}, and its parameters are easier to be tuned w r t  these algorithms and standard LMS  Secondly, the use of a probabilistic model provides us with an estimate of the error variance, which is useful in many applications 

Experiments with simulated and real data show the advantages of the presented approach with respect to previous works  However, we remark that the main contribution of this paper is that it opens the door to introduce more Bayesian machine learning techniques, such as variational inference and Monte Carlo sampling methods \cite{barber2012bayesian}, to adaptive filtering \\


\section{Probabilistic Model}

Throughout this work, we assume the observation model to be linear-Gaussian with the following distribution,

\begin{equation}
p(y_k|{\bf w}_k) = \mathcal{N}(y_k;{\bf x}_k^T {\bf w}_k , \sigma_n^2),
\label{eq:mess_eq}
\end{equation}
where  $\sigma_n^2$ is the variance of the observation noise, ${\bf x}_k$ is the regression vector and ${\bf w}_k$ is the parameter vector to be sequentially estimated, both $M$-dimensional column vectors 


In a non-stationary scenario, ${\bf w}_k$ follows a dynamic process  In particular, we consider a diffusion process (random-walk model) with variance $\sigma_d^2$ for this parameter vector:


\begin{equation}
p({\bf w}_k|{\bf w}_{k-1})= \mathcal{N}({\bf w}_k;{\bf w}_{k-1}, \sigma_d^2 {\bf I}),
\label{eq:trans_eq}
\end{equation}
where $\bf I$ denotes the identity matrix  In order to initiate the recursion, we assume the following prior distribution on ${\bf w}_k$

\begin{equation}
p({\bf w}_0)= \mathcal{N}({\bf w}_0;0, \sigma_d^2{\bf I}) \nonumber
\end{equation}

\section{Exact inference in this model: Revisiting the RLS filter}

Given the described probabilistic SSM, we would like to infer the posterior probability distribution $p({\bf w}_k|y_{1:k})$ 
Since all involved distributions are Gaussian, one can perform exact inference, leveraging the probability rules in a straightforward manner  The resulting probability distribution is
\begin{equation}
p({\bf w}_k|y_{1:k}) =  \mathcal{N}({\bf w}_k;{\bf\boldsymbol\mu}_{k}, \boldsymbol\Sigma_{k}), \nonumber
\end{equation}
in which the mean vector ${\bf\boldsymbol\mu}_{k}$ is given by
\begin{equation}
{\bf\boldsymbol\mu}_k = {\bf\boldsymbol\mu}_{k-1} + {\bf K}_k (y_k - {\bf x}_k^T {\bf\boldsymbol\mu}_{k-1}){\bf x}_k, \nonumber
\end{equation}
where we have introduced the auxiliary variable
\begin{equation}
{\bf K}_k = \frac{ \left(\boldsymbol\Sigma_{k-1} + \sigma_d^2 {\bf I}\right)}{{\bf x}_k^T  \left(\boldsymbol\Sigma_{k-1} + \sigma_d^2 {\bf I}\right)  {\bf x}_k + \sigma_n^2}, \nonumber
\end{equation}
and the covariance matrix $\boldsymbol\Sigma_k$ is obtained as
\begin{equation}
\boldsymbol\Sigma_k = \left( {\bf I} -  {\bf K}_k{\bf x}_k {\bf x}_k^T \right) ( \boldsymbol\Sigma_{k-1} +\sigma_d^2), \nonumber
\end{equation}
Note that the mode of $p({\bf w}_k|y_{1:k})$, i e  the maximum-a-posteriori estimate (MAP), coincides with the RLS adaptive rule
\begin{equation}
{{\bf w}}_k^{(RLS)} = {{\bf w}}_{k-1}^{(RLS)} + {\bf K}_k (y_k - {\bf x}_k^T {{\bf w}}_{k-1}^{(RLS)}){\bf x}_k  
\label{eq:prob_rls}
\end{equation}
This rule is similar to the one introduced in \cite{haykin1997adaptive} 

Finally, note that the covariance matrix $\boldsymbol\Sigma_k$ is a measure of the uncertainty of the estimate ${\bf w}_k$ conditioned on the observed data $y_{1:k}$  Nevertheless, for many applications a single scalar summarizing the variance of the estimate could prove to be sufficiently useful  In the next section, we show how such a scalar is obtained naturally when $p({\bf w}_k|y_{1:k})$ is approximated with an isotropic Gaussian distribution  We also show that this approximation leads to an LMS-like estimation 
 


\section{Approximating the posterior distribution: LMS filter }

The proposed approach consists in approximating the posterior distribution $p({\bf w}_k|y_{1:k})$, in general a multivariate Gaussian distribution with a full covariance matrix, by an isotropic spherical Gaussian distribution 

\begin{equation}
\label{eq:aprox_post}
\hat{p}({\bf w}_{k}|y_{1:k})=\mathcal{N}({\bf w}_{k};{\bf \hat{\boldsymbol\mu}}_{k}, \hat{\sigma}_{k}^2 {\bf I} ) 
\end{equation}

In order to estimate the mean and covariance of the approximate distribution $\hat{p}({\bf w}_{k}|y_{1:k})$, we propose to select those that minimize the Kullback-Leibler divergence with respect to the original distribution, i e , 

\begin{equation}
\{\hat{\boldsymbol\mu}_k,\hat{\sigma}_k\}=\arg \displaystyle{  \min_{\hat{\boldsymbol\mu}_k,\hat{\sigma}_k}} \{ D_{KL}\left(p({\bf w}_{k}|y_{1:k}))\| \hat{p}({\bf w}_{k}|y_{1:k})\right) \}  \nonumber
\end{equation}

The derivation of the corresponding minimization problem can be found in Appendix A  In particular, the optimal mean and the covariance are found as
\begin{equation}
{\hat{\boldsymbol\mu}}_{k} = {\boldsymbol\mu}_{k};~~~~~~ \hat{\sigma}_{k}^2 = \frac{{\sf Tr}\{ \boldsymbol\Sigma_k\} }{M} 
\label{eq:sigma_hat}
\end{equation}


We now show that by using \eqref{eq:aprox_post} in the recursive predictive and filtering expressions we obtain an LMS-like adaptive rule  First, let us assume that we have an approximate posterior distribution at $k-1$, $\hat{p}({\bf w}_{k-1}|y_{1:k-1}) =  \mathcal{N}({\bf w}_{k-1};\hat{\bf\boldsymbol\mu}_{k-1}, \hat{\sigma}_{k-1}^2 {\bf I} )$  Since all involved distributions are Gaussian, the predictive distribution
is obtained as %
\begin{eqnarray}
\hat{p}({\bf w}_k|y_{1:k-1}) &=& \int p({\bf w}_k|{\bf w}_{k-1}) \hat{p}({\bf w}_{k-1}|y_{1:k-1}) d{\bf w}_{k-1} \nonumber\\
&=& \mathcal{N}({\bf w}_k;{\bf\boldsymbol\mu}_{k|k-1}, \boldsymbol\Sigma_{k|k-1}), 
\label{eq:approx_pred}
\end{eqnarray}
where the mean vector and covariance matrix are given by
\begin{eqnarray}
\hat{\bf\boldsymbol\mu}_{k|k-1} &=& \hat{\bf\boldsymbol\mu}_{k-1} \nonumber \\
\hat{\boldsymbol\Sigma}_{k|k-1} &=& (\hat{\sigma}_{k-1}^2 + \sigma_d^2 ){\bf I}\nonumber 
\end{eqnarray}

From \eqref{eq:approx_pred}, the posterior distribution at time $k$ can be computed using Bayes' Theorem and standard Gaussian manipulations (see for instance \cite[Ch  4]{murphy2012machine})  Then, we approximate the posterior $p({\bf w}_k|y_{1:k})$ with an isotropic Gaussian,
\begin{equation}
\hat{p}({\bf w}_k|y_{1:k}) =  \mathcal{N}({\bf w}_k ; {\hat{\boldsymbol\mu}}_{k}, \hat{\sigma}_k^2 {\bf I} ),\nonumber
\end{equation}
where 
\begin{eqnarray}
{\hat{\boldsymbol\mu}}_{k} &= & {\hat{\boldsymbol\mu}}_{k-1}+ \frac{ (\hat{\sigma}_{k-1}^2+ \sigma_d^2)  }{(\hat{\sigma}_{k-1}^2+ \sigma_d^2)  \|{\bf x}_k\|^2 + \sigma_n^2} (y_k - {\bf x}_k^T {\hat{\boldsymbol\mu}}_{k-1}){\bf x}_k  \nonumber  \\
&=& {\hat{\boldsymbol\mu}}_{k-1}+ \eta_k (y_k - {\bf x}_k^T {\hat{\boldsymbol\mu}}_{k-1}){\bf x}_k   
\label{eq:prob_lms}
\end{eqnarray}
Note that, instead of a gain matrix ${\bf K}_k$ as in Eq ~\eqref{eq:prob_rls}, we now have a scalar gain $\eta_k$ that operates as a variable step size 


Finally, to obtain the posterior variance, which is our measure of uncertainty, we apply \eqref{eq:sigma_hat} and the trick ${\sf Tr}\{{\bf x}_k{\bf x}_k^T\}= {\bf x}_k^T{\bf x}_k= \|{\bf x}_k \|^2$,

\begin{eqnarray}
\hat{\sigma}_k^2 &=& \frac{{\sf Tr}(\boldsymbol\Sigma_k)}{M} \\
&=& \frac{1}{M}{\sf Tr}\left\{ \left( {\bf I} -  \eta_k {\bf x}_k {\bf x}_k^T \right) (\hat{\sigma}_{k-1}^2 +\sigma_d^2)\right\} \\
&=& \left(1 - \frac{\eta_k \|{\bf x}_k\|^2}{M}\right)(\hat{\sigma}_{k-1}^2 +\sigma_d^2) 
\label{eq:sig_k}
\end{eqnarray}
If MAP estimation is performed, we obtain  an adaptable step-size LMS estimation

\begin{equation}
{\bf w}_{k}^{(LMS)} = {\bf w}_{k-1}^{(LMS)} + \eta_k (y_k - {\bf x}_k^T {\bf w}_{k-1}^{(LMS)}){\bf x}_k, 	
\label{eq:lms}
\end{equation}
with
\begin{equation}
\eta_k = \frac{ (\hat{\sigma}_{k-1}^2+ \sigma_d^2)  }{(\hat{\sigma}_{k-1}^2+ \sigma_d^2)  \|{\bf x}_k\|^2 + \sigma_n^2} \nonumber
\end{equation}
At this point, several interesting remarks can be made:

\begin{itemize}

\item The adaptive rule \eqref{eq:lms} has linear complexity since it does not require us to compute the full matrix $\boldsymbol\Sigma_k$ 

\item For a stationary model, we have $\sigma_d^2=0$ in \eqref{eq:prob_lms} and \eqref{eq:sig_k}  In this case, the algorithm remains valid and both the step size and the error variance, $\hat{\sigma}_{k}$, vanish over time $k$  

\item Finally, the proposed adaptable step-size LMS has only two parameters, $\sigma_d^2$ and $\sigma_n^2$, (and only one, $\sigma_n^2$, in stationary scenarios) in contrast to other variable step-size algorithms \cite{kwong1992variable,aboulnasr1997robust,shin2004variable}  More interestingly, both $\sigma_d^2$ and $\sigma_n^2$ have a clear underlying physical meaning, and they can be estimated in many cases  We will comment more about this in the next section  
\end{itemize}



\section{Experiments}
\label{sec:experiments}

We evaluate the performance of the proposed algorithm in both stationary and tracking experiments  In the first experiment, we estimate a fixed vector ${\bf w}^{o}$ of dimension $M=50$  The entries of the vector are independently and uniformly chosen in the range $[-1,1]$  Then, the vector is normalized so that $\|{\bf w}^o\|=1$  Regressors $\boldsymbol{x}_{k}$ are zero-mean Gaussian vectors with identity covariance matrix  The additive noise variance is such that the SNR is $20$ dB
Context 3: \section{introduction}
In the past few years, the synthesis of ferromagnetic semiconductors has become a major challenge for spintronics  Actually, growing a magnetic and semiconducting material could lead to promising advances like spin injection into non magnetic semiconductors, or electrical manipulation of carrier induced magnetism in magnetic semiconductors \cite{ohno00,Bouk02}  Up to now, major efforts have focused on diluted magnetic semiconductors (DMS) in which the host semiconducting matrix is randomly substituted by transition metal (TM) ions such as Mn, Cr, Ni, Fe or Co \cite{Diet02}  However Curie temperatures ($T_{C}$) in DMS remain rather low and TM concentrations must be drastically raised in order to increase $T_{C}$ up to room temperature  That usually leads to phase separation and the formation of secondary phases  It was recently shown that phase separation induced by spinodal decomposition could lead to a significant increase of $T_{C}$ \cite{Diet06,Fuku06}  For semiconductors showing $T_{C}$ higher than room temperature one can foresee the fabrication of nanodevices such as memory nanodots, or nanochannels for spin injection  Therefore, the precise control of inhomogeneities appears as a new challenge which may open a way to industrial applications of ferromagnetism in semiconductors 

The increasing interest in group-IV magnetic semiconductors can also be explained by their potential compatibility with the existing silicon technology  In 2002, carrier mediated ferromagnetism was reported in  MBE grown Ge$_{0 94}$Mn$_{0 06}$ films by Park \textit{et al } \cite{Park02}  The maximum critical temperature was 116 K  Recently many publications indicate a significant increase of $T_{C}$ in Ge$_{1-x}$Mn$_{x}$ material depending on growth conditions \cite{Pint05,Li05,tsui03}  Cho \textit{et al } reported a Curie temperature as high as 285 K \cite{Cho02}  
Taking into account the strong tendency of Mn ions to form intermetallic compounds in germanium, a detailed investigation of the nanoscale structure is required  Up to now, only a few studies have focused on the nanoscale composition in Ge$_{1-x}$Mn$_{x}$ films  Local chemical inhomogeneities have been recently reported by Kang \textit{et al } \cite{Kang05} who evidenced a micrometer scale segregation of manganese in large Mn rich stripes  Ge$_3$Mn$_5$ as well as Ge$_8$Mn$_{11}$ clusters embedded in a germanium matrix have been reported by many authors  However, Curie temperatures never exceed 300 K \cite{Bihl06,Morr06,Pass06,Ahle06}  Ge$_3$Mn$_5$ clusters exhibit a Curie temperature of 296 K \cite{Mass90}  This phase frequently observed in Ge$_{1-x}$Mn$_{x}$ films is the most stable (Ge,Mn) alloy  The other stable compound Ge$_8$Mn$_{11}$ has also been observed in nanocrystallites surrounded with pure germanium \cite{Park01}  Ge$_8$Mn$_{11}$ and Ge$_3$Mn$_5$ phases are ferromagnetic but their metallic character considerably complicates their potential use as spin injectors 
Recently, some new Mn-rich nanostructures have been evidenced in Ge$_{1-x}$Mn$_{x}$ layers  Sugahara \textit{et al } \cite{Sugh05} reported the formation of high Mn content (between 10 \% and 20 \% of Mn) amorphous Ge$_{1-x}$Mn$_x$ precipitates  in a Mn-free germanium matrix  Mn-rich coherent cubic clusters were observed by Ahlers \textit{et al } \cite{Ahle06} which exhibit a Curie temperatures below 200 K  Finally, high-$T_{C}$ ($>$ 400 K) Mn-rich nanocolumns have been evidenced \cite{Jame06} which could lead to silicon compatible room temperature operational devices \\
In the present paper, we investigate the structural and magnetic properties of Ge$_{1-x}$Mn$_x$ thin films for low growth temperatures ($<$ 200$^{\circ}$C) and low Mn concentrations (between 1 \% and 11 \%)  By combining TEM, x-Ray diffraction and SQUID magnetometry, we could identify different magnetic phases  We show that depending on growth conditions, we obtain either Mn-rich nanocolumns or Ge$_{3}$Mn$_{5}$ clusters embedded in a germanium matrix  We discuss the structural and magnetic properties of these nanostructures as a function of manganese concentration and growth temperature  We also discuss the magnetic anisotropy of nanocolumns and  
Ge$_3$Mn$_5$ clusters  

\section{Sample growth}

Growth was performed using solid sources molecular beam epitaxy (MBE) by co-depositing Ge and Mn evaporated from standard Knudsen effusion cells   Deposition rate was low ($\approx$ 0 2 \AA s$^{-1}$)  Germanium substrates were epi-ready Ge(001) wafers with a residual n-type doping and resistivity of 10$^{15}$ cm$^{-3}$ and 5 $\Omega cm$ respectively  After thermal desorption of the surface oxide, a 40 nm thick Ge buffer layer was grown at 250$^{\circ}$C, resulting in a 2 $\times$ 1 surface reconstruction as observed by reflection high energy electron diffraction (RHEED) (see Fig  1a)  Next, 80 nm thick Ge$_{1-x}$Mn$_{x}$ films were subsequently grown  at low substrate temperature (from 80$^{\circ}$C to 200$^{\circ}$C)  Mn content has been determined by x-ray fluorescence measurements performed on thick samples ($\approx$ 1 $\mu m$ thick) and complementary Rutherford Back Scattering (RBS) on thin Ge$_{1-x}$Mn$_{x}$ films grown on silicon  Mn concentrations range from 1 \% to 11\% Mn 

For Ge$_{1-x}$Mn$_{x}$ films grown at substrate temperatures below 180$^{\circ}$C, after the first monolayer (ML) deposition, the  2 $\times$ 1 surface reconstruction almost totally disappears  After depositing few MLs, a slightly diffuse 1 $\times$ 1 streaky RHEED pattern and a very weak 2 $\times$ 1 reconstruction (Fig  1b) indicate a predominantly two-dimensional growth  For growth temperatures above 180$^{\circ}$C additional spots appear in the RHEED pattern during the Ge$_{1-x}$Mn$_{x}$ growth (Fig  1c)  These spots may correspond to the formation of very small secondary phase crystallites  The nature of these crystallites will be discussed below 

Transmission electron microscopy (TEM) observations were performed using a JEOL 4000EX microscope with an acceleration voltage of 400 kV  Energy filtered transmission electron microscopy (EFTEM) was done using a JEOL 3010 microscope equipped with a Gatan Image Filter   Sample preparation was carried out by standard mechanical polishing and argon ion milling for cross-section investigations and plane views were prepared by wet etching with H$_3$PO$_4$-H$_2$O$_2$ solution \cite{Kaga82} 

\begin{figure}[htb]
    \center
    \includegraphics[width= 29\linewidth]{ /fig1a eps}
    \includegraphics[width= 29\linewidth]{ /fig1b eps}
    \includegraphics[width= 29\linewidth]{ /fig1c eps}
    \caption{RHEED patterns recorded during the growth of Ge$_{1-x}$Mn$_{x}$ films: (a) 2 $\times$ 1 surface reconstruction of the germanium buffer layer  (b) 1 $\times$ 1 streaky RHEED pattern obtained at low growth temperatures ($T_g<$180$^{\circ}$C)  (c) RHEED pattern of a sample grown at $T_g=$180$^{\circ}$C  The additional spots reveal the presence of Ge$_3$Mn$_5$ clusters at the surface of the film }
\label{fig1}
\end{figure}

\section{Structural properties \label{structural}}

\begin{figure}[htb]
    \center
	\includegraphics[width= 49\linewidth]{ /fig2a eps}
	\includegraphics[width= 49\linewidth]{ /fig2b eps}
	 \includegraphics[width= 49\linewidth]{ /fig2c eps}
	 \includegraphics[width= 49\linewidth]{ /fig2d eps}
    \caption{Transmission electron micrographs of a Ge$_{1-x}$Mn$_{x}$ film grown at 130$^{\circ}$C and containing 6 \% of manganese  (a) cross-section along the [110] axis : we clearly see the presence of nanocolumns elongated along the growth axis  (b) High resolution image of the interface between the Ge$_{1-x}$Mn$_{x}$ film and the Ge buffer layer  The Ge$_{1-x}$Mn$_{x}$ film exhibits the same diamond structure as pure germanium  No defect can be seen which could be caused by the presence of nanocolumns  (c) Plane view micrograph performed on the same sample confirms the columnar structure and gives the density and size distribution of nanocolumns  (d) Mn chemical map obtained by energy filtered transmission electron microcopy (EFTEM)  The background was carefully substracted from pre-edge images  Bright areas correspond to Mn-rich regions }
\label{fig2}
\end{figure}

In samples grown at 130$^{\circ}$C and containing 6 \% Mn, we can observe vertical elongated nanostructures \textit{i e } nanocolumns as shown in Fig  2a  Nanocolumns extend through the whole Ge$_{1-x}$Mn$_{x}$ film thickness  From the high resolution TEM image shown in Fig  2b, we deduce their average diameter around 3 nm  Moreover in Fig  2b, the interface between the Ge buffer layer and the Ge$_{1-x}$Mn$_{x}$ film is flat and no defect propagates from the interface into the film  The Ge$_{1-x}$Mn$_{x}$ film is a perfect single crystal in epitaxial relationship with the substrate  In Fig  2c is shown a plane view micrograph of the same sample confirming the presence of nanocolumns in the film  From this image, we can deduce the size and density of nanocolumns  The nanocolumns density is 13000 $\rm{\mu m}^{-2}$ with a mean diameter of 3 nm which is coherent with cross-section measurements  In order to estimate the chemical composition of these nanocolumns, we further performed chemical mapping using EFTEM  In Fig  2d we show a cross sectional Mn chemical map of the Ge$_{1-x}$Mn$_{x}$ film  This map shows that the formation of nanocolumns is a consequence of Mn segregation  Nanocolumns are Mn rich and the surrounding matrix is Mn poor  However, it is impossible to deduce the Mn concentration in Ge$_{1-x}$Mn$_{x}$ nanocolumns from this cross section  Indeed, in cross section observations, the columns diameter is much smaller than the probed film thickness and the signal comes from the superposititon of the Ge matrix and Mn-rich nanocolumns  In order to quantify Mn concentration inside the nanocolumns and inside the Ge matrix, EELS measurements (not shown here) have been performed in a plane view geometry \cite{Jame06}  These observations revealed that the matrix Mn content is below 1 \% (detection limit of our instrument)  Measuring the surface occupied by the matrix and the nanocolumns in plane view TEM images, and considering the average Mn concentration in the sample (6 \%), we can estimate the Mn concentration in the nanocolumns  The Mn concentration measured by EELS being between 0\% and 1\%, we can conclude that the Mn content in the nanocolumns is between 30 \% and 38 \% \\
For samples grown between 80$^\circ$C and 150$^\circ$C cross section and plane view TEM observations reveal the presence of Mn rich nanocolumns surrounded with a Mn poor Ge matrix  In order to investigate the influence of Mn concentration on the structural properties of Ge$_{1-x}$Mn$_{x}$ films, ten samples have been grown at 100$^\circ$C and at 150$^\circ$C with Mn concentrations of 1 3 \%, 2 3 \%, 4 \%, 7 \% and 11 3 \%  Their structural properties have been investigated by plane view TEM observations  

\begin{figure}[htb]
    \center
    \includegraphics[width= 98\linewidth]{ /fig3a eps}
	\includegraphics[width= 45\linewidth]{ /fig3b eps}
		\includegraphics[width= 45\linewidth]{ /fig3c eps}
    \caption{Nanocolumns size and density as a function of growth conditions  Samples considered have been grown at 100$^{\circ}$C and 150$^{\circ}$C respectively  (a) Mn concentration dependence of the size distribution  (b) columns density as a function of Mn concentration  (c) Volume fraction of the nanocolumns as a function of Mn concentration }
 \label{fig3}
\end{figure}

For samples grown at 100$^\circ$C with Mn concentrations below 5 \% the nanocolumns mean diameter is 1 8$\pm$0 2 nm  The evolution of columns density as a fonction of Mn concentration is reported in figure 3b  By increasing the Mn concentration from 1 3 \% to 4 \% we observe a significant increase of the columns density from 13000 to 30000 $\rm{\mu m}^{-2}$  For Mn concentrations higher than 5 \% the density seems to reach a plateau corresponding to 35000 $\rm{\mu m}^{-2}$ and their diameter slightly increases from 1 8 nm at 4 \% to 2 8 nm at 11 3 \%  By plotting the volume fraction occupied by the columns in the film as a function of Mn concentration, we observe a linear dependence for Mn contents below 5 \%  The non-linear behavior above 5 \% may indicate that the mechanism of Mn incorporation is different in this concentration range, leading to an increase of Mn concentration in the columns or in the matrix  For samples grown at 100$^\circ$C, nanocolumns are always fully coherent with the surrounding matrix (Fig  4a)  

Increasing the Mn content in the samples grown at 150$^\circ$C from 1 3 \% to 11 3 \% leads to a decrease of the columns density (fig 3b)  Moreover, their average diameter increases significantly and size distributions become very broad (see Fig  3a)  For the highest Mn concentration (11 3 \%) we observe the coexistence of very small columns with a diameter of 2 5 nm and very large columns with a diameter of 9 nm  In samples grown at 150$^\circ$C containing 11 3 \% of Mn, the crystalline structure of nanocolumns is also highly modified  In plane view TEM micrographs, one can see columns exhibiting several different crystalline structures  We still observe some columns which are fully coherent with the Ge matrix like in the samples grown at lower temperature  Nevertheless, observations performed on these samples grown at 150$^\circ$C and with 11 3\% Mn reveal some uniaxially \cite{Jame06} or fully relaxed columns exhibiting a misfit of 4 \% between the matrix and the columns and leading to misfit dislocations at the interface between the column and the matrix (see fig  4b)  Thus we can conclude that coherent columns are probably in strong compression and the surrounding matrix in tension



ground_truth: 1766.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What types of data did the authors use in their experiments?

Read the following text and answer briefly.

Context 2:  We compare our algorithm with standard RLS and three other LMS-based algorithms: LMS, NLMS \cite{sayed2008adaptive}, VSS-LMS \cite{shin2004variable} \footnote{The used parameters for each algorithm are: for RLS $\lambda=1$, $\epsilon^{-1}=0 01$; for LMS $\mu=0 01$; for NLMS $\mu=0 5$; and for VSS-LMS $\mu_{max}=1$, $\alpha=0 95$, $C=1e-4$ } The probabilistic LMS algorithm in \cite{park2014probabilistic} is not simulated because it is not suitable for stationary environments 

In stationary environments, the proposed algorithm has only one parameter, $\sigma^2_n$  We simulate both the scenario where we have perfectly knowledge of the amount of noise (probLMS1) and the case where the value $\sigma^2_n$ is $100$ times smaller than the actual value (probLMS2)  The Mean-Square Deviation (${\sf MSD} = {\mathbb E} \| {\bf w}_0 - {\bf w}_k \|^2$), averaged out over $50$ independent simulations, is presented in Fig  \ref{fig:msd_statationary} 



\begin{figure}[htb]
\centering
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[width=\textwidth]{results_stationary_MSD}}
\end{minipage}
\caption{Performance in terms of MSD of probabilistic LMS with both optimal (probLMS1) and suboptimal (probLMS2) compared to LMS, NLMS, VS-LMS, and RLS }
\label{fig:msd_statationary}
\end{figure}

The performance of probabilistic LMS is close to RLS (obviously at a much lower computational cost) and largely outperforms previous variable step-size LMS algorithms proposed in the literature  Note that, when the model is stationary, i e  $\sigma^2_d=0$ in \eqref{eq:trans_eq},  both the uncertainty $\hat{\sigma}^2_k$, and the adaptive step size $\eta_k$, vanish over time  This implies that the error tends to zero when $k$ goes to infinity  Fig  \ref{fig:msd_statationary} also shows that the proposed approach is not very sensitive to a bad choice of its only parameter, as demonstrated by the good results of probLMS2, which uses a $\sigma^2_n$ that is $100$ times smaller than the optimal value  


\begin{figure}[htb]
\centering
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[width=\textwidth]{fig2_final}}
\end{minipage}
\caption{Real part of one coefficient of the measured and estimated channel in experiment two  The shaded area represents two standard deviations from the prediction {(the mean of the posterior distribution)} }
\label{fig_2}
\end{figure}


\begin{table}[ht]
\begin{footnotesize}
\setlength{\tabcolsep}{2pt}
\def1 5mm{1 5mm}
\begin{center}
\begin{tabular}{|l@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|}
\hline
Method &  LMS &  NLMS & LMS-2013 & VSSNLMS & probLMS & RLS \\
\hline
\hline
MSD (dB) &-28 45 &-21 07 &-14 36 &-26 90 &-28 36 &-25 97\\
\hline                                                                     
\end{tabular}
\end{center}
\caption{Steady-state MSD of the different algorithms for the tracking of a real MISO channel }
\label{tab:table_MSD}
\end{footnotesize}

\end{table}
\newpage
In a second experiment, we test the tracking capabilities of the proposed algorithm with {real} data of a wireless MISO channel acquired in a realistic indoor scenario  More details on the setup can be found in \cite{gutierrez2011frequency}  Fig  \ref{fig_2} shows the real part of one of the channels, and the estimate of the proposed algorithm  The shaded area represents the estimated uncertainty for each prediction, i e  $\hat{\mu}_k\pm2\hat{\sigma}_k$  Since the experimental setup does not allow us to obtain the optimal values for the parameters, we fix these parameters to their values that optimize the steady-state mean square deviation (MSD)  \hbox{Table \ref{tab:table_MSD}} shows this steady-state MSD of the estimate of the MISO channel with different methods  As can be seen, the best tracking performance is obtained by standard LMS and the proposed method  





\section{Conclusions and Opened Extensions}
\label{sec:conclusions}

{We have presented a probabilistic interpretation of the least-mean-square filter  The resulting algorithm is an adaptable step-size LMS that performs well both in stationary and tracking scenarios  Moreover, it has fewer free parameters than previous approaches and these parameters have a clear physical meaning  Finally, as stated in the introduction, one of the advantages of having a probabilistic model is that it is easily extensible:}

\begin{itemize}
\item If, instead of using an isotropic Gaussian distribution in the approximation, we used a Gaussian with diagonal covariance matrix, we would obtain a similar algorithm with different step sizes and measures of uncertainty, for each component of ${\bf w}_k$  Although this model can be more descriptive, it needs more parameters to be tuned, and the parallelism with LMS vanishes 
\item Similarly, if we substitute the transition model of \eqref{eq:trans_eq} by an Ornstein-Uhlenbeck process, 

\begin{equation}
p({\bf w}_k|{\bf w}_{k-1})= \mathcal{N}({\bf w}_k;\lambda {\bf w}_{k-1}, \sigma_d^2), \nonumber
\label{eq:trans_eq_lambda}
\end{equation}
a similar algorithm is obtained but with a forgetting factor $\lambda$ multiplying ${\bf w}_{k-1}^{(LMS)}$ in \eqref{eq:lms}  This algorithm may have improved performance under such a kind of autoregresive dynamics of ${\bf w}_{k}$, though, again, the connection with standard LMS becomes dimmer 

\item As in \cite{park2014probabilistic}, the measurement model \eqref{eq:mess_eq} can be changed to obtain similar adaptive algorithms for classification, ordinal regression, and Dirichlet regression for compositional data  

\item A similar approximation technique could be applied to more complex dynamical models, i e  switching dynamical models \cite{barber2010graphical}  The derivation of efficient adaptive algorithms that explicitly take into account a switch in the dynamics of the parameters of interest is a non-trivial and open problem, though the proposed approach could be useful 

\item Finally, like standard LMS, this algorithm can be kernelized for its application in estimation under non-linear scenarios 

\end{itemize}


\begin{appendices}

\section{KL divergence between a general gaussian distribution and an isotropic gaussian}
\label{sec:kl}

 We want to approximate  $p_{{\bf x}_1}(x) = \mathcal{N}({\bf x}; \boldsymbol\mu_1,\boldsymbol\Sigma_1)$ by $p_{{\bf x}_2}({\bf x}) = \mathcal{N}({\bf x}; \boldsymbol\mu_2,\sigma_2^2 {\bf I})$  In order to do so, we have to compute the parameters of $p_{{\bf x}_2}({\bf x})$, $\boldsymbol\mu_2$ and $\sigma_2^2$, that minimize the following Kullback-Leibler divergence,

\begin{eqnarray}
D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) &=&\int_{-\infty}^{\infty} p_{{\bf x}_1}({\bf x}) \ln{\frac{p_{{\bf x}_1}({\bf x})}{p_{{\bf x}_2}({\bf x})}}d{\bf x} \nonumber  \\
&= &  \frac{1}{2} \{ -M + {\sf Tr}(\sigma_2^{-2} {\bf I}\cdot \boldsymbol\Sigma_1^{-1})  \nonumber \\
  & &  + (\boldsymbol\mu_2 - \boldsymbol\mu_1 )^T \sigma^{-2}_2{\bf I} (\boldsymbol\mu_2 - \boldsymbol\mu_1 )  \nonumber \\
 & &   +  \ln \frac{{\sigma_2^2}^M}{\det\boldsymbol\Sigma_1} \}   
\label{eq:divergence}
\end{eqnarray}
Using symmetry arguments, we obtain 
\begin{equation}
\boldsymbol\mu_2^{*} =\arg \displaystyle{  \min_{\boldsymbol\mu_2}} \{ D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) \} = \boldsymbol\mu_1 
\end{equation}
Then, \eqref{eq:divergence} gets simplified into 

\begin{eqnarray}
D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) = \frac{1}{2}\lbrace { -M + {\sf Tr}(\frac{\boldsymbol\Sigma_1}{\sigma_2^{2}}) + \ln \frac{\sigma_2^{2M}}{\det\boldsymbol\Sigma_1}}\rbrace 
\end{eqnarray}
The variance $\sigma_2^2$ is computed in order to minimize this Kullback-Leibler divergence as

\begin{eqnarray}
\sigma_2^{2*} &=& \arg\min_{\sigma_2^2} D_{KL}(P_{x_1}\| P_{x_2}) \nonumber \\
 &=& \arg\min_{\sigma_2^2}\{ \sigma_2^{-2}{\sf Tr}\{\boldsymbol\Sigma_1\} + M\ln \sigma_2^{2} \}  
\end{eqnarray}
Deriving and making it equal zero leads to

\begin{equation}
\frac{\partial}{\partial \sigma_2^2} \left[ \frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{\sigma_2^{2}} + M \ln \sigma_2^{2} \right] = \left  {\frac{M}{\sigma_2^{2}}-\frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{(\sigma_2^{2})^2}}\right|_{\sigma_2^{2}=\sigma_2^{2*}}\left  =0 \right   
\nonumber
\end{equation}
Finally, since the divergence has a single extremum in $R_+$,
\begin{equation}
\sigma_2^{2*} = \frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{M} 
\end{equation}




\end{appendices}

\vfill
\clearpage

\bibliographystyle{IEEEbib}

Context 3:  On the same samples (T$_g$=150$^{\circ}$C, 11 3\% Mn), we also observe a large number of highly disordered nanocolumns leading to an amorphous like TEM contrast(fig  4c) 

\begin{figure}[htb]
    \center
   \includegraphics[width= 31\linewidth]{ /fig4a eps}
	\includegraphics[width= 31\linewidth]{ /fig4b eps}
	\includegraphics[width= 31\linewidth]{ /fig4c eps}
    \caption{Plane view high resolution transmission electron micrographs of different types of nanocolumns : (a) typical structure of a column grown at 100$^{\circ}$C  The crystal structure is exactly the same as germanium   (b) Partially relaxed nanocolumn  One can see dislocations at the interface between the columns and the matrix leading to stress relaxation  (c) Amorphous nanocolumn  These columns are typical in samples grown at 150$^{\circ}$C with high Mn contents }
 \label{fig4}
\end{figure}

In conclusion, we have evidenced a complex mechanism of Mn incorporation in Mn doped Ge films grown at low temperature  In particular Mn incorporation is highly inhomogeneous  For very low growth temperatures (below 120$^\circ$C) the diffusion of Mn atoms leads to the formation of Mn rich, vertical nanocolumns  Their density mostly depends on Mn concentration and their mean diameter is about 2 nm  These results can be compared with the theoretical predictions of Fukushima \textit{et al } \cite{Fuku06}: they proposed a model of spinodal decomposition in (Ga,Mn)N and (Zn,Cr)Te based on layer by layer growth conditions and a strong pair attraction between Mn atoms which leads to the formation of nanocolumns  This model may also properly describe the formation of Mn rich nanocolumns in our samples  Layer by layer growth conditions can be deduced from RHEED pattern evolution during growth  For all the samples grown at low temperature, RHEED observations clearly indicate two-dimensional growth  Moreover, Ge/Ge$_{1-x}$Mn$_{x}$/Ge heterostructures have been grown and observed by TEM (see Fig  5)  Ge$_{1-x}$Mn$_{x}$/Ge (as well as Ge/Ge$_{1-x}$Mn$_{x}$) interfaces are very flat and sharp thus confirming a two-dimensional, layer by layer growth mode  Therefore we can assume that the formation of Mn rich nanocolumns is a consequence of 2D-spinodal decomposition 

\begin{figure}[htb]
    \center
	\includegraphics[width= 7\linewidth]{ /fig5 eps}
    \caption{Cross section high resolution micrograph of a Ge/Ge$_{1-x}$Mn$_{x}$/Ge/Ge$_{1-x}$Mn$_{x}$/Ge heterostructure  This sample has been grown at 130 $^{\circ}$C with 6\% Mn  Ge$_{1-x}$Mn$_{x}$ layers are 15 nm thick and Ge spacers 5 nm thick  We clearly see the sharpness of both Ge$_{1-x}$Mn$_{x}$/Ge and Ge/Ge$_{1-x}$Mn$_{x}$ interfaces  Mn segregation leading to the columns formation already takes place in very thin Ge$_{1-x}$Mn$_{x}$ films }
\label{fig5}
\end{figure}

For growth temperatures higher than 160$^\circ$C, cross section TEM and EFTEM observations (not shown here) reveal the coexistence of two Mn-rich phases: nanocolumns and Ge$_{3}$Mn$_{5}$ nanoclusters embedded in the germanium matrix  A typical high resolution TEM image is shown in figure 6  
Ge$_{3}$Mn$_{5}$ clusters are not visible in RHEED patterns for temperatures below 180$^\circ$C  To investigate the nature of these clusters, we performed x-ray diffraction in $\theta-2\theta$ mode  Diffraction scans were acquired on a high resolution diffractometer using the copper K$_\alpha$ radiation and on the GMT station of the BM32 beamline at the European Synchrotron Radiation Facility (ESRF)  Three samples grown at different temperatures and/or annealed at high temperature were investigated  The two first samples are Ge$_{1-x}$Mn$_{x}$ films grown at 130$^\circ$C and 170$^\circ$C respectively  The third one has been grown at 130$^\circ$C and post-growth annealed at 650$^\circ$C  By analysing x-ray diffraction spectra, we can evidence two different crystalline structures  For the sample grown at 130$^\circ$C, the $\theta-2\theta$ scan only reveals the (004) Bragg peak of the germanium crystal, confirming the good epitaxial relationship between the layer and the substrate, and the absence of secondary phases in the film in spite of a high dynamics of the order of 10$^7$  For both samples grown at 170$^\circ$C and annealed at 650$^\circ$C, $\theta-2\theta$ spectra are identical  In addition to the (004) peak of germanium, we observe three additional weak peaks  The first one corresponds to the (002) germanium forbidden peak which probably comes from a small distortion of the germanium crystal, and the two other peaks are respectively attributed to the (002) and (004) Bragg peaks of a secondary phase  The $c$ lattice parameter of Ge$_3$Mn$_5$ hexagonal crystal is 5 053 \AA \ \cite{Fort90} which is in very good agreement with the values obtained from diffraction data for both (002) and (004) lines assuming that the $c$ axis of Ge$_3$Mn$_5$ is along the [001] direction of the Ge substrate 

\begin{figure}[htb]
    \center
	\includegraphics[width= 7\linewidth]{ /fig6 eps}
	\caption{Cross section high resolution transmission electron micrograph of a sample grown at 170$^{\circ}$C  We observe the coexistence of two different Mn-rich phases: Ge$_{1-x}$Mn$_{x}$ nanocolumns and Ge$_3$Mn$_5$ clusters }
\label{fig6}
\end{figure}

In summary, in a wide range of growth temperatures and Mn concentrations, we have evidenced a two-dimensional spinodal decomposition leading to the formation of Mn-rich nanocolumns in Ge$_{1-x}$Mn$_{x}$ films  This decomposition is probably the consequence of: $(i)$ a strong pair attraction between Mn atoms, $(ii)$ a strong surface diffusion of Mn atoms in germanium even at low growth temperatures and $(iii)$ layer by layer growth conditions  We have also investigated the influence of growth parameters on the spinodal decomposition: at low growth temperatures (100$^{\circ}$C), increasing the Mn content leads to higher columns densities while at higher growth temperatures (150$^{\circ}$C), the columns density remains nearly constant whereas their size increases drastically  By plotting the nanocolumns density as a function of Mn content, we have shown that the mechanism of Mn incorporation in Ge changes above 5 \% of Mn  Finally, using TEM observations and x-ray diffraction, we have shown that Ge$_3$Mn$_5$ nanoclusters start to form at growth temperatures higher than 160$^\circ$C 

\section{Magnetic properties \label{magnetic}}

We have thoroughly investigated the magnetic properties of thin Ge$_{1-x}$Mn$_{x}$ films for different growth temperatures and Mn concentrations  In this section, we focus on Mn concentrations between 2 \% and 11 \%  We could clearly identify four different magnetic phases in Ge$_{1-x}$Mn$_{x}$ films : diluted Mn atoms in the germanium matrix, low $T_{C}$ nanocolumns ($T_{C}$ $\leq$ 170 K), high $T_{C}$ nanocolumns ($T_{C}$ $\geq$ 400 K) and Ge$_{3}$Mn$_{5}$ clusters ($T_{C}$ $\thickapprox$ 300 K)  The relative weight of each phase clearly depends on the growth temperature and to a lesser extend on Mn concentration  For low growth temperature ($<$ 120$^{\circ}$C), we show that nanocolumns are actually made of four uncorrelated superparamagnetic nanostructures  Increasing T$_{g}$ above 120$^{\circ}$C, we first obtain continuous columns exhibiting low $T_{C}$ ($<$ 170 K) and high $T_{C}$ ($>$ 400 K) for $T_{g}\approx$130$^{\circ}$C  The larger columns become ferromagnetic \textit{i e } $T_{B}>T_{C}$  Meanwhile Ge$_{3}$Mn$_{5}$ clusters start to form  Finally for higher $T_{g}$, the magnetic contribution from Ge$_{3}$Mn$_{5}$ clusters keeps increasing while the nanocolumns signal progressively disappears 

\begin{figure}[htb]
\center
   \includegraphics[width= 6\linewidth]{ /fig7a eps}
   \includegraphics[width= 3\linewidth]{ /fig7b eps}
\caption{(a) Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{0 93}$Mn$_{0 07}$ samples for different growth temperatures  The magnetic field is applied in the film plane  The inset shows the temperature dependence of a sample grown at 130$^{\circ}$C and annealed at 650$^{\circ}$C for 15 minutes  After annealing, the magnetic signal mostly arises from Ge$_{3}$Mn$_{5}$ clusters  (b) ZFC-FC measurements performed on Ge$_{0 93}$Mn$_{0 07}$ samples for different growth temperatures  The in-plane applied field is 0 015 T  The ZFC peak at low temperature ($\leq$150 K) can be attributed to the superparamagnetic nanocolumns  This peak widens and shifts towards high blocking temperatures when increasing growth temperature  The second peak above 150 K in the ZFC curve which increases with increasing growth temperature is attributed to superparamagnetic Ge$_{3}$Mn$_{5}$ clusters  The increasing ZFC-FC irreversibility at $\approx$ 300 K is due to the increasing contribution from large ferromagnetic Ge$_{3}$Mn$_{5}$ clusters  The nanocolumns signal completely vanishes after annealing at 650$^{\circ}$C for 15 minutes }
\label{fig7}
\end{figure}

In Fig  7a, the saturation magnetization at 2 Tesla in $\mu_{B}$/Mn of Ge$_{1-x}$Mn$_{x}$ films with 7 \% of Mn is plotted as a function of temperature for different growth temperatures ranging from $T_{g}$=90$^{\circ}$C up to 160$^{\circ}$C  The inset shows the temperature dependence of the magnetization at 2 Tesla after annealing at 650$^{\circ}$C during 15 minutes  Figure 7b displays the corresponding Zero Field Cooled - Field Cooled (ZFC-FC) curves recorded at 0 015 Tesla  In the ZFC-FC procedure, the sample is first cooled down to 5 K in zero magnetic field and the susceptibility is subsequently recorded at 0 015 Tesla while increasing the temperature up to 400 K (ZFC curve)  Then, the susceptibility is recorded under the same magnetic field while decreasing the temperature down to 5 K (FC curve)  Three different regimes can be clearly distinguished  \\
For $T_{g}\leq$120$^{\circ}$C, the temperature dependence of the saturation magnetization remains nearly the same while increasing growth temperature  The overall magnetic signal vanishing above 200 K is attributed to the nanocolumns whereas the increasing signal below 50 K originates from diluted Mn atoms in the surrounding matrix  The Mn concentration dependence of the saturation magnetization is displayed in figure 8  For the lowest Mn concentration (4 \%), the contribution from diluted Mn atoms is very high and drops sharply for higher Mn concentrations (7 \%, 9 \% and 11 3 \%)  Therefore the fraction of Mn atoms in the diluted matrix decreases with Mn concentration probably because Mn atoms are more and more incorporated in the nanocolumns  In parallel, the Curie temperature of nanocolumns increases with the Mn concentration reaching 170 K for 11 3 \% of Mn  This behavior may be related to different Mn compositions and to the increasing diameter of nanocolumns (from 1 8 nm to 2 8 nm) as discussed in section \ref{structural} 

\begin{figure}[htb]
\center
   \includegraphics[width= 7\linewidth]{ /fig8 eps}
    \caption{Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{1-x}$Mn$_{x}$ films grown at 100$^{\circ}$C plotted for different Mn concentrations: 4 1 \%; 7 \%; 8 9 \% and 11 3 \% }
\label{fig8}
\end{figure}

ZFC-FC measurements show that the nanocolumns are superparamagnetic  The magnetic signal from the diluted Mn atoms in the matrix is too weak to be detected in susceptibility measurements at low temperature  In samples containing 4 \% of Mn, ZFC and FC curves superimpose down to low temperatures  As we do not observe hysteresis loops at low temperature, we believe that at this Mn concentration nanocolumns are  superparamagnetic in the whole temperature range and the blocking temperature cannot be measured  For higher Mn contents, the ZFC curve exhibits a very narrow peak with a maximum at the blocking temperature of 15 K whatever the Mn concentration and growth temperature (see Fig  7b)  Therefore the anisotropy barrier distribution is narrow and assuming that nanocolumns have the same magnetic anisotropy, this is a consequence of the very narrow size distribution of the nanocolumns as observed by TEM  To probe the anisotropy barrier distribution, we have performed ZFC-FC measurements but instead of warming the sample up to 400 K, we stopped at a lower temperature $T_{0}$  

\begin{figure}[htb]
\center
   \includegraphics[width= 6\linewidth]{ /fig9 eps}
\caption{Schematic drawing of the anisotropy barrier distribution n($E_{B}$) of superparamagnetic nanostructures  If magnetic anisotropy does not depend on the particle size, this distribution exactly reflects their magnetic size distribution  In this drawing the blocking temperature ($T_{B}$) corresponds to the distribution maximum  At a given temperature $T_{0}$ such that 25$k_{B}T_{0}$ falls into the anisotropy barrier distribution, the largest nanostructures with an anisotropy energy larger than 25$k_{B}T_{0}$ are blocked whereas the others are superparamagnetic }
\label{fig9}
\end{figure}

If this temperature falls into the anisotropy barrier distribution as depicted in Fig  9, the FC curve deviates from the ZFC curve  Indeed the smallest nanostructures have become superparamagnetic at $T_{0}$ and when decreasing again the temperature, their magnetization freezes along a direction close to the magnetic field and the FC susceptibility is higher than the ZFC susceptibility  Therefore any irreversibility in this procedure points at the presence of superparamagnetic nanostructures  The results are given in Fig  10a  ZFC and FC curves clearly superimpose up to $T_{0}$=250 K thus the nanocolumns are superparamagnetic up to their Curie temperature and no Ge$_{3}$Mn$_{5}$ clusters could be detected  Moreover for low $T_{0}$ values, a peak appears at low temperature in FC curves which evidences strong antiferromagnetic interactions between the nanocolumns \cite{Chan00} 

\begin{figure}[htb]
\center
    \includegraphics[width= 35\linewidth]{ /fig10a eps}
    \includegraphics[width= 63\linewidth]{ /fig10b eps}
\caption{(a) ZFC-FC measurements performed on a Ge$_{0 887}$Mn$_{0 113}$ sample grown at 115$^{\circ}$C  The in-plane applied field is 0 015 T  Magnetization was recorded up to different T$_{0}$ temperatures: 30 K, 50 K, 100 K, 150 K, 200 K and 250 K  Curves are shifted up for more clarity  (b) ZFC-FC curves for in-plane and out-of-plane applied fields (0 015 T) }
\label{fig10}
\end{figure}

In order to derive the magnetic size and anisotropy of the Mn-rich nanocolumns embedded in the Ge matrix, we have fitted the inverse normalized in-plane (resp  out-of-plane) susceptibility: $\chi_{\parallel}^{-1}$ (resp  $\chi_{\perp}^{-1}$)  The corresponding experimental ZFC-FC curves are reported in Fig  10b  Since susceptibility measurements are performed at low field (0



ground_truth: The authors used simulated data and real data from a wireless MISO channel.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What are some potential applications of ferromagnetic semiconductors?

Read the following text and answer briefly.

Context 3: 015 T), the matrix magnetic signal remains negligible  In order to normalize susceptibility data, we need to divide the magnetic moment by the saturated magnetic moment recorded at 5 T  However the matrix magnetic signal becomes very strong at 5 T and low temperature so that we need to subtract it from the saturated magnetic moment using a simple Curie function  From Fig  10b, we can conclude that nanocolumns are isotropic  Therefore to fit experimental data we use the following expression well suited for isotropic systems or cubic anisotropy: $\chi_{\parallel}^{-1}= \chi_{\perp}^{-1}\approx 3k_{B}T/M(T)+\mu_{0}H_{eff}(T)$  $k_{B}$ is the Boltzmann constant, $M=M_{s}v$ is the magnetic moment of a single-domain nanostructure (macrospin approximation) where $M_{s}$ is its magnetization and $v$ its volume  The in-plane magnetic field is applied along $[110]$ or $[-110]$ crystal axes  Since the nanostructures Curie temperature does not exceed 170 K, the temperature dependence of the saturation magnetization is also accounted for by writting $M(T)$  Antiferromagnetic interactions between nanostructures are also considered by adding an effective field estimated in the mean field approximation \cite{Fruc02}: $\mu_{0}H_{eff}(T)$ 
The only fitting parameters are the maximum magnetic moment (\textit{i e } at low temperature) per nanostructure: $M$ (in Bohr magnetons $\mu_{B}$) and the maximum interaction field (\textit{i e } at low temperature): $\mu_{0}H_{eff}$ 

\begin{figure}[htb]
\center
   \includegraphics[width= 7\linewidth]{ /fig11 eps}
\caption{Temperature dependence of the inverse in-plane (open circles) and out-of-plane (open squares) normalized susceptibilities of a Ge$_{0 887}$Mn$_{0 113}$ sample grown at 115$^{\circ}$C  Fits were performed assuming isotropic nanostructures or cubic anisotropy  Dashed line is for in-plane susceptibility and solid line for out-of-plane susceptibility }
\label{fig11}
\end{figure}

In Fig  11, the best fits lead to $M\approx$1250 $\mu_{B}$ and $\mu_{0}H_{eff}\approx$102 mT for in-plane susceptibility and $M\approx$1600 $\mu_{B}$ and $\mu_{0}H_{eff}\approx$98 mT for out-of-plane susceptibility  It gives an average magnetic moment of 1425 $\mu_{B}$ per column and an effective interaction field of 100 mT  Using this magnetic moment and its temperature dependence, magnetization curves could be fitted using a Langevin function and $M(H/T)$ curves superimpose for $T<$100 K  However, from the saturated magnetic moment of the columns and their density (35000 $\rm{\mu m}^{-2}$), we find almost 6000 $\mu_{B}$ per column  Therefore, for low growth temperatures, we need to assume that nanocolumns are actually made of almost four independent elongated magnetic nanostructures  The effective field for antiferromagnetic interactions between nanostructures estimated from the susceptibility fits is at least one order of magnitude larger than what is expected from pure magnetostatic coupling  This difference may be due to either an additional antiferromagnetic coupling through the matrix which origin remains unexplained or to the mean field approximation which is no more valid in this strong coupling regime  As for magnetic anisotropy, the nanostructures behave as isotropic magnetic systems or exhibit a cubic magnetic anisotropy  First we can confirm that nanostructures are not amorphous otherwise shape anisotropy would dominate leading to out-of-plane anisotropy  We can also rule out a random distribution of magnetic easy axes since the nanostructures are clearly crystallized in the diamond structure and would exhibit at least a cubic anisotropy (except if the random distribution of Mn atoms within the nanostructures can yield random easy axes)  Since the nanostructures are in strong in-plane compression (their lattice parameter is larger than the matrix one), the cubic symmetry of the diamond structure is broken and magnetic cubic anisotropy is thus unlikely  We rather believe that out-of-plane shape anisotropy is nearly compensated by in-plane magnetoelastic anisotropy due to compression leading to a \textit{pseudo} cubic anisotropy  From the blocking temperature (15 K) and the magnetic volume of the nanostructures , we can derive their magnetic anisotropy constant using $Kv=25k_{B}T_{B}$: K$\approx$10 kJ m$^{-3}$ which is of the same order of magnitude as shape anisotropy 

\begin{figure}[htb]
\center
    \includegraphics[width= 35\linewidth]{ /fig12a eps}
    \includegraphics[width= 63\linewidth]{ /fig12b eps} 
\caption{(a) ZFC-FC measurements performed on a Ge$_{0 93}$Mn$_{0 07}$ sample grown at 122$^{\circ}$C  The in-plane applied field is 0 015 T  Magnetization was recorded up to different T$_{0}$ temperatures: 50 K, 100 K, 150 K, 200 K and 250 K  Curves are shifted up for more clarity  (b) ZFC-FC curves for in-plane and out-of-plane applied fields  (0 015 T) }
\label{fig12}
\end{figure}

For growth temperatures $T_{g}\geq$120$^{\circ}$C and Mn concentrations $\geq$ 7 \%, samples exhibit a magnetic signal above 200 K corresponding to Ge$_{3}$Mn$_{5}$ clusters (see Fig  7a)  As we can see, SQUID measurements are much more sensitive to the presence of Ge$_{3}$Mn$_{5}$ clusters, even at low concentration, than TEM and x-ray diffraction used in section \ref{structural}  We also observe a sharp transition in the ZFC curve (see Fig  7b, Fig  12a and 12b): the peak becomes very large and is shifted towards high blocking temperatures (the signal is maximum at $T=$23 K)  This can be easily understood as a magnetic percolation of the four independent nanostructures obtained at low growth temperatures into a single magnetic nanocolumn  Therefore the magnetic volume increases sharply as well as blocking temperatures  At the same time, the size distribution widens as observed in TEM  In Fig  12a, we have performed ZFC-FC measurements at different $T_{0}$ temperatures  The ZFC-FC irreversibility is observed up to the Curie temperature of $\approx$120 K meaning that a fraction of nanocolumns is ferromagnetic (\textit{i e } $T_{B}\geq T_{C}$) 
In Fig  12b, in-plane and out-of-plane ZFC curves nearly superimpose for $T\leq$150 K due to the isotropic magnetic behavior of the nanocolumns: in-plane magnetoelastic anisotropy is still compensating out-of-plane shape anisotropy  Moreover the magnetic signal above 150 K corresponding to Ge$_{3}$Mn$_{5}$ clusters that start to form in this growth temperature range is strongly anisotropic  This perpendicular anisotropy confirms the epitaxial relation: (0002) Ge$_{3}$Mn$_{5}$ $\parallel$ (002) Ge discussed in Ref \cite{Bihl06}  The magnetic easy axis of the clusters lies along the hexagonal $c$-axis which is perpendicular to the film plane 

\begin{figure}[ht]
\center
   \includegraphics[width= 35\linewidth]{ /fig13a eps}
   \includegraphics[width= 63\linewidth]{ /fig13b eps} 
\caption{(a) ZFC-FC measurements performed on a Ge$_{0 887}$Mn$_{0 113}$ sample grown at 145$^{\circ}$C  The in-plane applied field is 0 015 T  Magnetization was recorded up to different T$_{0}$ temperatures: 50 K, 100 K, 150 K, 200 K, 250 K and 300 K  Curves are shifted up for more clarity  (b) ZFC-FC curves for in-plane and out-of-plane applied fields (0 015 T) }
\label{fig13}
\end{figure}

For growth temperatures $T_{g}\geq$145$^{\circ}$C the cluster magnetic signal dominates (Fig  13b)  Superparamagnetic nanostructures are investigated performing ZFC-FC measurements at different $T_{0}$ temperatures (Fig  13a)  The first ZFC peak at low temperature \textit{i e } $\leq$ 150 K is attributed to  low-$T_{C}$ nanocolumns ($T_{C}\approx$130 K)  This peak is wider than for lower growth temperatures and its maximum is further shifted up to 30 K  These results are in agreement with TEM observations: increasing $T_{g}$ leads to larger nanocolumns (\textit{i e } higher blocking temperatures) and wider size distributions  ZFC-FC irreversibility is observed up to the Curie temperature due to the presence of ferromagnetic columns  The second peak above 180 K in the ZFC curve is attributed to Ge$_{3}$Mn$_{5}$ clusters and the corresponding ZFC-FC irreversibility persisting up to 300 K means that some clusters are ferromagnetic  We clearly evidence the out-of-plane anisotropy of Ge$_{3}$Mn$_{5}$ clusters and the isotropic magnetic behavior of nanocolumns (Fig  13b)  In this growth temperature range, we have also investigated the Mn concentration dependence of magnetic properties  

\begin{figure}[ht]
\center
    \includegraphics[width= 49\linewidth]{ /fig14a eps}
    \includegraphics[width= 49\linewidth]{ /fig14b eps} 
\caption{Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{1-x}$Mn$_{x}$ films grown at 150$^{\circ}$C plotted for different Mn concentrations: 2 3 \%; 4 \%; 7 \%; 9 \%; 11 3 \%  (b) ZFC-FC measurements performed on Ge$_{1-x}$Mn$_{x}$ films grown at 150$^{\circ}$C  The in-plane applied field is 0 025 T for 2 3 \% and 4 \% and 0 015 T for 8 \% and 11 3 \%  }
\label{fig14}
\end{figure}

In Fig  14a, for low Mn concentrations (2 3 \% and 4 \%) the contribution from diluted Mn atoms in the germanium matrix to the saturation magnetization is very high and nearly vanishes for higher Mn concentrations (7 \%, 9 \% and 13 \%) as observed for low growth temperatures  Above 7 \%, the magnetic signal mainly comes from nanocolumns and Ge$_{3}$Mn$_{5}$ clusters  We can derive more information from ZFC-FC measurements (Fig  14b)  Indeed, for 2 3 \% of Mn, ZFC and FC curves nearly superimpose down to low temperature meaning that nanocolumns are superparamagnetic in the whole temperature range  Moreover the weak irreversibility arising at 300 K  means that some Ge$_{3}$Mn$_{5}$ clusters have already formed in the samples even at very low Mn concentrations  For 4 \% of Mn, we can observe a peak with a maximum at the blocking temperature (12 K) in the ZFC curve  We can also derive the Curie temperature of nanocolumns: $\approx$45 K  The irresversibility arising at 300 K still comes from Ge$_{3}$Mn$_{5}$ clusters  Increasing the Mn concentration above 7 \% leads to: higher blocking temperatures (20 K and 30 K) due to larger nanocolumns and wider ZFC peaks due to wider size distributions in agreement with TEM observations (see Fig  3a)  Curie temperatures also increase (110 K and 130 K) as well as the contribution from Ge$_{3}$Mn$_{5}$ clusters \\
Finally when increasing $T_{g}$ above 160$^{\circ}$C, the nanocolumns magnetic signal vanishes and only Ge$_{3}$Mn$_{5}$ clusters and diluted Mn atoms coexist  The overall magnetic signal becomes comparable to the one measured on annealed samples in which only Ge$_{3}$Mn$_{5}$ clusters are observed by TEM (see Fig  7a) \\
The magnetic properties of high-$T_{C}$ nanocolumns obtained for $T_{g}$ close to 130$^{\circ}$C are discussed in detail in Ref \cite{Jame06} \\
In conclusion, at low growth temperatures ($T_{g}\leq$120$^{\circ}$C), nanocolumns are made of almost 4 independent elongated magnetic nanostructures  For $T_{g}\geq$120$^{\circ}$C, these independent nanostructures percolate into a single nanocolumn sharply leading to higher blocking temperatures  Increasing $T_{g}$ leads to larger columns with a wider size distribution as evidenced by ZFC-FC measurements and given by TEM observations  In parallel, some Ge$_{3}$Mn$_{5}$ clusters start to form and their contribution increases when increasing $T_{g}$  Results on magnetic anisotropy seems counter-intuitive  Indeed Ge$_{3}$Mn$_{5}$ clusters exhibit strong out-of-plane anisotropy whereas nanocolumns which are highly elongated magnetic structures are almost isotropic  This effect is probably due to compensating in-plane magnetoelastic coupling (due to the columns compression) and out-of-plane shape anisotropy  

\section{Conclusion}

In this paper, we have investigated the structural and magnetic properties of thin Ge$_{1-x}$Mn$_{x}$ films grown by low temperature molecular beam epitaxy  A wide range of growth temperatures and Mn concentrations have been explored  All the samples contain Mn-rich nanocolumns as a consequence of 2D-spinodal decomposition  However their size, crystalline structure and magnetic properties depend on growth temperature and Mn concentration  For low growth temperatures, nanocolumns are very small (their diameter ranges between 1 8 nm for 1 3 \% of Mn and 2 8 nm for 11 3 \% of Mn), their Curie temperature is rather low ($<$ 170 K) and they behave as almost four uncorrelated superparamagnetic nanostructures  Increasing Mn concentration leads to higher columns densities while diameters remain nearly unchanged  For higher growth temperatures, the nanocolumns mean diameter increases and their size distribution widens  Moreover the 4 independent magnetic nanostructures percolate into a single magnetic nanocolumn  Some columns are ferromagnetic even if Curie temperatures remain quite low  In this regime, increasing Mn concentration leads to larger columns while their density remains nearly the same  In parallel, Ge$_{3}$Mn$_{5}$ nanoclusters start to form in the film with their $c$-axis perpendicular to the film plane  In both temperature regimes, the Mn incorporation mechanism in the nanocolumns and/or in the matrix changes above 5 \% of Mn and nanocolumns exhibit an isotropic magnetic behaviour due to the competing effects of out-of-plane shape anisotropy and in-plane magnetoelastic coupling  Finally for a narrow range of growth temperatures around 130$^{\circ}$C, nanocolumns exhibit Curie temperatures higher than 400 K  Our goal is now to investigate the crystalline structure inside the nanocolumns, in particular the position of Mn atoms in the distorted diamond structure, which is essential to understand magnetic and future transport properties in Ge$_{1-x}$Mn$_{x}$ films 

\section{Aknowledgements}
The authors would like to thank Dr  F  Rieutord for grazing incidence x-ray diffraction measurements performed on the GMT station of BM32 beamline at the European Synchrotron Radiation Facility 





ground_truth: Spin injection into non magnetic semiconductors, or electrical manipulation of carrier induced magnetism in magnetic semiconductors.